REVISTA DE LA

UNION

MATEMATICA ARGENTINA

Director: Dario J. Picco.

Redactores: A. Diego, E. Gentile, R. Panzone. H. Porta,

E. Oklander; C. Trejo, O. Villamayor.

S6~retarios de Redacci6n: M. L. Gastaminza, A. G. de Pousa.

VOLUMEN 25, NUMEROS 1 y 2

DEDICADO AL PROF. ALBERTO GONZALEZ,DOMINGUEZ

1970

BAHIA BLANCA

1970

UNION MA TEMA TICA ARGENTINA

La U. M. A. reconoce cuatro categorias de miembros: honorarios, protec­tores, titulares y adherentes. EI miE;!mbro protector paga una cuota anual de .$ 4000, por 10 menos; el titular una cuota anual de $ 2000 y el adherente (e's­tudiante solamente) una cuota anual de .$ 1000. Los pagos deberan efectuarse por cheque, giro u otro medio de gastos, a la orden de UNION MATEMA'l'1CA ARGEN1'INA, Casilla de Correo 3588, Buer,os Aires.

Por ser la U. M. A. miembro del patronato de la MatheIlllCItical Reviews (sponsoring member), los socios de la U.M.A. tienen derecho a suscribirse a esa importante revista: de' bibliografia y critica con 50 % de rebaja sobre el precio de suscripci6n que es de 50 d61ares por ano. Los socios de la U. M· A. pagaran por tanto s610 25 d61ares por ano.

Los autores de trabajos reciben gratuitcnnente una tirada aporte de 50 ejemplares. Las correcciones extraordinarias de pruebas son por cuenta de los autores.

JUNTA DIRECI1V A

Presidente: Dr. Alberto Gonzillez Dominguez; Vieepresidentes: lng. Eduardo Gaspar e Ing. Orlando Villamayor; Seeretaria: Dra. Beatriz Margolis; Tesorera: Lie. Josefina A. Alonso; Protesorera: Lie. Norma Pietroeola; Director de Publi­eaciones: Dr. Dario Picco; Seeretarlos Locales: Bahia Blanca: Lie. Maria I. Plat· zeck; Buenos Aires: Lie. Angel Larotonda; Cordoba: Ing. Arcadio Niell; Mendoza: Dr. Eduardo Zarantonello; Nordeste: Ing. Marcos Marangunic; La Plata: Dra. Sara Salvioli; Rosario: Dr. Miguel Ferrero; Salta: Ing. Roberto Ovejero; Sun Luis: Dr. Osvaldo Borghi; Tucuman. Lie. Guillermo Hansen.

WEMBROS HONORARIOS

Tulio Levi-Civita (t); Beppo Levi (t n Alejandro Terracini (t); Ga'orge D. Birkhoff (t); Marshall H. Stone; Georges Valiron (t); Antoni Zygmund; Godofredo Garcia; Wilhelm Blaschke (t); Laurent Schwartz; Charles Ehres­mann; Jean Dieudonne; Alexandre Ostrowski; Jose Babini; Marcel Brelo!.

REPRESENT ANTES EN EL EXTRANJERO

Ing. Rafael Laguardia (Uruguay), Ing. Jose Luis Massera (Uruguay), Dr. Ces~r Carranza (Peru), Dr. Leopoldo Naehbin (Brasil), Dr. Roberto Frucht (Chile), Dr Mario Gonzalez (Cuba), Dr. Alfonso Napoles Gandara (Mexico).

Abonnement a l'etranJger (comprenant un volu~ complet) : 12 dollars (Etats-Unis).

Priere d'adresser toute la correspondence' scientifique. administrative et les echanges a l'adresse ci-dessous:

REVISTA DE LA UNION MATEMATICA ARGENTINA

Casilla de Correo 3588

Buenos Aires. (Argentina)

REVISTA SEMESTRAL

REVISTA DE LA

UNION .,

MATEMATICA ARGENTINA

VOLUMEN 25

DEDICADO AL PROF. ALBERTO GONZALEZ DOMINGUEZ

BAHIA BLANCA

1970

~!.

I I I I I I I I I I I I I I I I I I I I I I I I I I I I

I I I I

I I

I I I I

I I I I I I I

I I I I I I I

I I I

I I

I I I I

Alberto Gonzalez Dom(nguez

DEDICATORIA

En 1969 el Dr. ALBERTO GONZALEZ DOMINGUEZ cumpliO los se-

8enta y cinco .anos de edad, limite que el Estatuto de la Universidad de Buenos

Aires 8Bfiala para sus profesores titulares.

La UNION MATEMATICA ARGENTINA ha querido aprovechar la opor­

tflnidad para dedicar en su homena;e el presente Volumen 25 de su Revista.

La actuaci6n del Profesor Gonzcilez Dominguez en la UniOn Matematica

Argentina ha aido continuada y fundamental: miembro fundador, varias ve­

ces Presidente y siempre el alma de sus Comisiones Directivas. La creaci6n.

IJ subsistencia de la UniOn Matematica Argentina y de su Revista, si bien ha

contado siempre con el aliento de muchos, ha aido la obra tenaz y perseveran­

tede unos pocos, entre los cuales el Profesor Gonzalez Dominguez ocupa un

lugar de excepciOn.

Desde cualquier lingulo que se mire, la matematica argentina ha girado

pn los (dUmos cincuenta anos alrededor de la obra de Rey Pastor y de su prin­

cipal colaborador y continuador Gonzalez Dominguez. Nada nace esponta-_

neamente y detras de las realidades aleanzadas y las promisorias perspectivas

que se vislumbran para la Matematica en el pais, se percibe la mano soUcita y

cuidadosa, la sombra protectora del Profesor Gonzalez Dominguez, presente

siempre para limar asperezas, pregonando transigencia en la vida de relaciOn,

manteniendo con firmeza la seriedad y el nivel deltrabaio cientifico, poniendo

orden en los iuicios de valores y colocando a personas y cosa~ en su sitio. Tra­

bajo sutil y delicado, que solo puede realizarse cuando se posee suficiente au­

toridad cientifica y una inagotable capacidad afectiva para dar amor a manos

llenas y para absorver ingratitudes sin desmayo, aunque no sin dolor.

La UniOn Matematica Argentina quiere expresar su mayor agradecimien­

to a todos los autores de los trabajos de estevolumen enviados especial mente

como cordial adhesiOn a la ohra y persona de Gonzalez Dominguez. Gracias

a ellos, el volumen adauiere 1a jerarqufa que deseabamos y que el homena;ea-"

do merece.

El volumen ha sido dividido en dos partes, 10, primera constitufda por los

presentes fasciculos 1 - 2. La segunda parte, con los fascfculos 3 - 4, tambien

dedicados al Profesor Gonzalez Dominguez, aparecera a continuaci6n.

LA OBRA CIENTIFICA Y DOCENTE

DEL Dr. A. GONZALEZ DOMINGUEZ

I. CONSIDERACIONES GENERALES

Para hacer una valoraci6n de la obra del Dr. Gonzllez Dominguez se debe atender a los dos aspectos fundamentales que definen a una personalidad cientifica: los aportes originales realizados en el campo de su especialidad y la formaci6n de discipulos que hayan

continuado y ampliado esa obra~

Pero ambos aspectos presentan en este caso cualidades especificas

sin las cuales no podria considerarse, por 10 menDs en nuestro me dio, que aquellos hayan alcanzado especial significaci6n en el a~­

biente matemltico. Estas cualidades pueden enunciarse diciendo ,

_~r_imeL(),_Au~S_U ()br_a ha.sJdo .prec!l1"§ora_en~. selltid~gue _ilumil1~ nuevos rumbos a las ideas que hasta un cierto momenta venian pri­mando en la especialidad, expresamente, el Anllisis Matemltico y

sus aplicaciones, y despues, que su personalidad tuvo, sobre todos los que trabajaron en contacto con el, una influencia de intensi­

dad y amplitud tales que aseguraron el numero y la cali dad de los discipulos necesarios para impulsar la singular empresa de la acti

vidad creadora en el mundo de la ciencia.

Un poco mls adelante haremos ver la importancia de sus contribucio nes a la Matemltica,su destacada obra docente y otros notables me­

ritos que figuran enumerados en su extenso curriculum vitae, pero podemos afirmar desde ya que su presencia en la matemltica argenti na marca claramente el comienzo de una nueva era en el estudio del

Anllisis que su maestro Rey Pastor habia fundado entre los- -anos 1929 y 1940 Y que sufriria una radical modernizaci6n con las ideas

que entonces comenzaba a madurar y exponer en sus lecciones, inspi radas en las mls avanzadas corrientes de ese momento,como eran las

que surgian de los trabajos de Bochner, Zygmund , Titchmarsh y 0 -

tros autores de renombre universal, que iniciaban la introducci6n

del Anllisis Arm6nicu en toda la matemltica.

Pero no solamente en el dominic particular del Anllisis, sino tam­bien en los de la Fisica Matemltica, la Electr6nica Te6rica, y el Cilculo de Probabilidades, que recien empezaban a cultivarse en nuestro pais, se iba a sentir el influjo de las ideas renovadoras

contehidas en los resultados de su fecunda actividad.

Resulta asi sintomltico queel mismo Profesor Rey Pastor, en opor­

vii

tunidad de dar termino a su obra fundamental "Teoria de los algo -ri.tmos 1 ineales de convergencia y sumaci6n" (Trabaj os del Semina -rio Matematico, Facultad, de Ciencias Exactas, Fisicas y Naturales de Buenos Aires, Serie' B, 1932), considerando de extraordinario in teres las observaciones de su discipulo sobre la posibilidad de, comp1etar ciertas condiciones que Se exponian alIi, de manera que su demostraci6n se desprendl.ese de, una sola idea central, 1a de las Integrales Singu1ares de Lebesgue (desconocidas hasta ese mo -mento en el ambiente matematico de habla hispana), decidi6 hacer un agregado aclaratorio que u1teriormente incluy6 en esetrabajo.

Puede decirse que esa iba a serla t6riica que marc a ia permanente preocupaci6n del pensamiento del Dr. Gonzalez Dominguez, un hondo sentido de captaci6n y valoraci6n de las grarides ideas transform~ doras que insensiblemente venian gest,ando una renovaci6n total en los enfoques y las tecnicas de la investigaci6n contemporanea.

Asi, en sus 1ecciones y seminarios y en continuo dialogo en los que expuso sus resultados, transparent6 frente a sus alumnos y c~ 1aboradores su constante desvelo por la aclaraci6n de cuestiones b4sicas del An4lisis Funcional moderno y sus aplicaciones a. la Fi sica Cuantica, que estuvieron mucho tiempo sumidas en una gran os

curidad.

De esa manera explic6 por primera vez en sus exposl.cl.ones, a tra­yeS de la teoria de la Transformada de Hilbert y de los Nucleos Singu1ares, sus conjetutas sobre muchas ideas que mas tarde se -rian formuladas en 1a Teoria de las Distribuciones de Laurent Schwartz, cuya pub1icaci6n marc6 un acontecim:lento hist6rico en el desa,rroll0 posterior del AniHisis Funcional en el plano mun -dial.

II. OBRA CIENTIFICA

Pueden ahora hacerse desfi1ar, ya maS espec!ficamente, los apor -tes definitivos que ha hecho el Dr .. Gonzalez Dominguez a 1a 1ite­ratura matematica mundial.

Sus primeras notas: "Sur un theor~me de Glivenko" y "Sur les Int! grales de Laplace" aparecidas en los Comptes Rendues de l' Academie des Sciences de Parts en 1936 y 1937; "The representation of func-

tions by Fourier Integrals" en el Duke Mathematical Journal en 1940; "Some theorems on the Hermite Kernel" en el Bulletin of the the American Mathematical Society en 1940, indicaban ya el hecho remarcable de una producci6n netamente argentina publicada en las mas acreditadas revistas extranjeras de la especialidad, en una

temprana epoca del historial cientifico de nuestro pais.

A partir de alIi las contribuciones a publicaciones locales como son la Revista de la Uni6n Matematica Argentina , Mathematicae N~ tae (Rosario), Ciencia y Tecnica (Centro de Estudiantes de Inge­nieria), Publicaciones del Instituto de Matematica de la Universi dad del Litoral, la Serie de Contribuciones Cientificas de la Fa­cultad de Ciencias Exactas, Fisicas y Naturales de Buenos Aires, la Revista del Centro de Cooperaci6n para America Latina de UNES­CO, la Serie de Cursos y Seminarios de Matematica de la Facultad de Ciencias Exactas y Naturales de Buenos Aires, prestan un vital aliento a la producci6n matematica del pais por medio de casi ci~ cuenta memorias y notas en las que, aparte del merito intrinseco que significa la elaboraci6n local, debe destacarse el hecho que en algunas de elIaS empiezan a aparecer los nombres de varios de sus colaboradores que recibian asi decisivo impulso en la practi­ca de la investigaci6n ..

Entre tanto, y tambien mas recientemente, fueron apareciendo otros trabajos en caracterizadas publicaciones internacionales , entre elIas, los Proceedings of the International Congress of Ma­thematiciens en 1954 y 1962, el Seminaire de Theories Physiques de la Sorbonne (Seminaire de Broglie) en 1954, los Comptes Ren­dues de l'Academie des Sciences de Paris' lya mencionados) en 1955,

el Journal of Mathemat;cal Physics en 1964, los Proceedings of the First Conference of Systems Scienc~s en 1968, formando un nu­trido conjunto de importantes resultad~s varios de los cuales fi­guran definitivamente incorporados a obras y tratados fundamenta­les de la literatura especializada y muchos son citados en las re ferencias bibliograficas de memorias de matematicos de reconocida autoridad.

Basta citar, para probar 10 antedicho, los hechos siguientes: al­gunos teoremas de su trabajo "Sur les Integrales de Laplace" (Com.

Rend. 1937) son utilizados en el ya clasico libro de G. Doetsch, "Handbuch der Laplace Transformationen", 1950, en las pag. 305 y 306 del vol. I; un teorema contenido en el trabajo "Sobre ciertas

ix

formulas de inversion" (Publ. Inst. Mat. Univ. Lit. 1946, IV) iIIe­rece todo un paragrafo, el NO"7, en la pag. 250'del vol. II de la

misma obra; diversos resultados de la nota "Sobre los valores li­mites de funciones analiticas" (en colaboracion coh'A; P: Calde -

ron y A. Zygmund) son reproducidos en el libro funditmental de uno de los tratadist'as indiscutidos en la especialldad, ;'TrigotlOine­trical Serie,s" de A.Zygmund,1959,vol.I; otros'autoresde primer

rango mundial en lao Matemati,ca,la Fisica ,Teorica y otras aplic~ ciones 10 ci tan en ias re,ferencias de sus memorias"entre ~llos,

R.P. Boas y D.W. Widder (1940), H. Pollard (1943),J. R. Shoat y J. D. Tamarkin (1943), W. Rudin (1951),H. Berens y P. L. Butzen'

(1963), Olli Lehto (1964), Makuto Ohkusta (1955), K., No.sito (1960), M. Heins (1960), H. Zeemanian (1968), Eugene Speer (1~69).

I I I. FORMACION DE DISCIPULOS

Como ha sido dicho mas arriba no queda consagrado un investigador en el campo cientifico si no se ve aparecer una generacion de di~ cipulos que en el hayan encontrado la orientacion para realizar su propia obra.

En el caso del Dr. Gonzalez Dominguez debe ademas agregarse que ~ sa accion tiene un rasgo caracteristico que califica a una perso­nalidad con el titulo de verdadero maestro: la generosidad con

que ha prodigado la,s ideas que permi tieron hacer surgir una real escuela de jovenes que fue renovandose en el transcurso del tiem­

po a traves de una destacadafactuacion en el campo local e inter­nacional.

En prueba de ella puede citarse, en primer lugar, a A. P. Calde -

ron, actualmente John Blok Professor de la Universidad de Chicago y Academico de la National Academie of Sciences de Estados Unidos,

cuya brillante carrera 10 llevo a ser virtual candidato del pre • mio internacional de matematica que confiere la Mathematical In -ternational Union en oportunidad de los congresos, mundiales cad'a

cuatro afios a la obra, realizada en esos periodos, y cuyas origin~ les y profundas ideas, con resonancia mundial, sobre la aplica -cion de las Integrales Singulares a la Teoria de las Ecuaciones en Derivadas Parciales, tienen, si se llega a un ultimo analisis, su raiz y motivacion en las orientaciones que hal16 en el Semina-

rio que el Dr. Gonzalez DomInguez dirigi6 en la Universidad de Bu~ i nos y al que Calder6n asistia como docente auxiliar entre los arros 1940 y 1943.

Debe mencionarse despues a R. Scarfiello, colaborador en varios de sus trabajos, cuya memoria "Sur les transformees de Fourier

des courants", Nuovo Cimento, 1954, figura casi Integramente transcripta en el capitulo final de la Qltima edici6n de 1966 del ya citado "Theorie des Distributions" de L. Schwartz, debiendo d~

cirse que la genesis de su elaboraci6n fue deb ida a las ideas que el Dr. Gonzalez Dominguez venia explicando en esa epoca, sobre

los problemas matematicos de la Electrodinamica Cuantica y en las que aqu~l habia participado activamente.

Tambien a J. J. Giambiagi, destacado exponente de la joven Fisica

Te6rica argentina, cuyo tema de tesis "Sobre las ecuaciones del ~ lectr6n", propuesto por el Dr. Gonzalez Dominguez, revela la im -

pronta de las ideas de su tutor con qui en continu6 mas tarde cola borando en diversos trabajos tales como, para no citar sino uno

5610 de ellos, "Analytic regularisation and the divergences in Quantum Field Theories", Nuovo Cimento, vol. 11, 1964, en el cual colabor6 tambien C. G. Bollini, trabajo que es especialmente cita

do por E. R. Speer en el volumen sobre "Generalised Feynmann Am -plitudes" de los Ann.of Math. Studies, N° 62, 1969.

Otros investigadores que experimentaron definitivamente su influen

cia fueron: S. V~gi, ahoraprofesor en la De Paul University, Ill.

EE. UU. Y los ya mas j6venes, graduados en nuestra facultad: Calu

to Calder6n que redact6 su tesis nSobre la sumabilidad de las se­ries de Hermite y Laguerre en n variables", publicada despues en

Studia Mathematica (1968) con tema propuesto por el Dr. G6nzalez Dominguez; C. Merlo, cuyo trabajo "Sobre los nucleos singulares",

Serie de Cursos y Seminarios de la F.C.E.N. de Bs. As., naci6 de las sugerencias que Ie hiciera tambien el Dr. Gonzalez Dominguez;

y por fin otros como C. Segovia y R. Riviere, todos ellos de bri­llante actuaci6n en la actividad matematica del pais y del extran

jero.

Adem~s debe recordarse que. algunas obras publicadas en el dominic de otras especialidades que usan la matematica como herramienta

fundamental sintieron tambien su influencia. Valga como ejemplo el libro "Servomecanismos lineales. Teoria y disefio" de L. M. Fr~

diani, alumno suyo en los cursos de Analisis Matematico de la es-

Xl

cuela Superior Tecnica del Ejercito Argentino, que fue escrito con su provechoso consejo.

IV. ACTIVIDAD DOCENTE

Si se pasa ahoraa su labor docente puede descartarse una extensa dedic8ci6n que va desde su cargo de Ayudante del Seminario Matern! tico en 1933 hasta el de Profesor Titular Plenario, mostrando una fidelidad a sus tareas que 10 ha mantenido ligado a la Facultad de Ciencias Exactas,Fisicas y Naturales de la Universidad de Buenos

Aires pOI' mas de 35 afios casi ininterrumpidamente, excepci6n hecha de algunos perIodos de ausencia motivados pOI' invitaciones a dictar cursos y conferencias 0 participar en la actividad de universidades

e institutos extranjeros de gran prestigio como el Instituto H.Poi!!. care de la Universidad de Paris (Seminarios Schwartz y Dc Broglie en 1954 y 1958), la Universidad de Illinois, la Universidad de Cali

fornia (Berkeley) y la Universidad de Hawaii.

Tambien a su importante obra docente y educativa debe asignarse la amplia lista de treinta y cinco titulos entre conferencias y

articulos de caracter general, dictadas en renombradas institucio nes del pais, como la Sociedad Cientifica Argentina, la Academia Nacional de Ciencias Exactas y Naturales de Buenos Aires, y del extranjero, como la Real Academia de Ciencias Exactas, Fisicas y Naturales de Madrid, la Facultad de Ciencias de Sevilla y el Ins­tituto H. Poincare de la Univ~sidadde Paris, 0 publicados en di ferentes revistas menos especializadas como Ciencia e Investiga -ci6n,mostrando la dilatada gama de la cultura humanistica e hist~ rica que han distinguido siempre su relevante actividad.

V. OTRAS ACTUACIONES.

POI' \iltimocab.e ha..cer not.ar que el Dr. Gon .. zalez Dominguez perten~

ce 0 ha pertenecido a numeros8s instituciones nacionales y extra!!. jeras en cuyo sene ha cumplido una ponderada y valiosa actuaci6n,. entre elIas, la Academia Nacional de Ciencias Exactas y Naturales de Buenos Aires, la Real Academia de Ciencias Exactas, Fisicas y

Naturales de Madrid, la Academia Nacional de Ciencias Exactas, F! sicas y Naturales de Lima, la Uni6n Matemitica Argentina, la Com,!. si6n Nacional de la Energia At6mica, la Escuela S1,Iperlor Tecnica del Ejercito Argentino, el Centro Regional para America Latina de Ma'temitica (UNESCO), el Consejo Nacional de Investigaciones Cien­tificas y Tecnicas, la Asociaci6n Argentina para el Progreso de las Ciencias y lao Fun'daci6n Bariloche.

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TRABAJOS DE INVESTIGACION

1. Aptiaaaion de "La teor1.a de "Las inte(1ra"Les singu"Lares a "La de­mostraaion de una formu"La de Stie"LtJes. Bolettn del Semina­rio Matematico, vol. III (1932-1933), pag. 35.

2. Sur un th'or'me de M. GZivenko. Comptes rendus de L'Acad~mie des Sciences de Paris, vol. 204 (1936), pag. 577.

3. Sur "Les integra"Les de Lap"Laae. Comptes rendus de L'Academie des Sciences de Paris, vol. 205 (1937), pag. 1035.

4. Genera"Lizaai6n de un teorema de Cante"L"Li. Revista de la U­ni6n Matematica Argentina, vol. II (1937), pag. 63.

5. Sobre "Las series de funaiones de Hermite. Revista de la U­ni6n Matematica Argentina, vol. II (1938-1939), pag. 1-16.

6. Sobre"La inversion de integra"Les de Lap"Laae abso"Lutamente ao~ vergentes. Revista de la Uni6n Matematica Argentina, vol. II (1938-1939), pag. 23-26.

7. Una nUeva demostraaion del segundo teorema limite de"L aalculo de probabilidades. Revista de la Uni6n Matematica Argentina, Publicaci6n N° 4 (1938).

8. Condiaiones necesarias y sUficientes para que una funcion sea integra"L de Lap "Lace. Revista de la Union Matematica Argenti­na. vol II (1938).

9. Un nuevo teorema limite del Ca"Lcu"Lo de Probabilidades. Bue­nos Aires, 1939. Tesis para optar al grado de Doctor en Cien cias Fisicomatematicas (no fue impresa).

10. The representation of functions by Fourier Integrals. Duke Mathematical Journal. vol. VI (1940), pag. 580.

11. Some theorems on the Hermite KerneL Bulletin of the Ameri­can Mathemat'ical Society, 46 (1940), pag. 580.

12. Contpibuaion a la teoria de "Las funciones de Hille. y T€cnica", 476 (1941), pag. 487-537.

"Ciencia

13. Sobre una eauaaion integraL Revista de la Union Matematica Argentina, vol. VIII (1942), pag. 111.

14. Sobre"La eauacion integra"L de Hille. Revista de la Uni6n Ma­tematica Argentina. vol. VIII (1942), pag. 39.

15. Sobre aiertas formu"Las de inversion. Publicaciones del Insti tuto de Matematica de la Universidad Nacional deL Litoral voT. VI (1946) , pag. 207-214. .

16 Sobre series aonJugadas de Legendre. Revista de la Union Ma­tematica Argentina, vol. XII (1946), pag. 46.

17. Un m'todo genera"L para s£ntesis de impedanaias. Revista de la Uni6n Matematica Argentina, vol. XIII (1948), pag. 35.

18. Un m'todo para s£ntesis de impedancias. Mathematicae Notae, vol. VII (1947), pag. 146-161.

19. Teoria de Za funci6n deZta compZeja. Revista de la Union Ma­tematica Argentina, vol. XIII (1948), pag. 57.

20. Demostraci6n rigurosa deZ ZZamado Teorema fundamentaZ de Zas comunicaciones eZectricas. Revista de la Union Matematica Ar gentina, vol. XIII (1948), pag. 49-54.

21. Un teorema sobre Za teoria de Za estabiZidad. Revista de la Union Matematica Argentina, vol. XIII(1948), pag. 89.

22. Sobre un metodo de sintesis de circuitos. Revista de la U­nion Matematica Argentina, vol. XIII (1948), pag. 169.

23. Sobre eZ transitorio en fiZtros. Revista de la Union Matema­tica Argentina, vol. XIV (1949), pag. 80.

24. ReZaciones n6dulo-fase en un intervalo finito de frecuencias. Revista de la Union Matematica Argentina, vol. XIV (1949), pag. 80.

25. Las funciones singulares de la Fisica. Revista de la Union Matematica Argentina, vol. XIV (1949), pag. 89.

26. Nota sobre los valores limites de funciones anaZiticas. (en colaboracion can A.P.Calderon y A.Zygmund). Revista de la U­nion Matematica Argentina, vol. XIV. (1949), pag. 16-19.

27. Sobre la teoria de las senates analiticas. Revista de la U­nion Matematica Argentina, vol. XIV (1949), pag. 258.

28. Sobre algunos puntos de la teoria matematica de los circuitos Zineales. Revista de la Union Matematica Argentina, vol. XIV (195(j) , pag. 257-322. Este trabajo merec.io el Premia Nacio­nal de Ciencias (1950).

29. Teoremas limites para productos de variables aleatorias. (en colaboracion can Roque Scarfiello). Facultad de Ciencias Exactas, Fisicas y Naturales, Contribuciones Cientificas. Se­rie A, Matematica, vol. I, N° 1 (1950), pag. 3-22.

30. Sobre las funciones singulares de Schwinger. Revista de .la U nion Matematica Argentina, vol.XV (1951), pag. 78.

31. Sobre una representaci6n de la delta c6nica. Revista de la Q nion Matematica Argentina, vol.XV (1951), pag. 78.

32. Criterios de estabiZidad para circuitos lineales. Revista de la Union Matematica Argentina, vol. XV (1951), pag. 9.

33. La funci6n de Riemann como distribuci6n. (en colaboracion can J.J.Giambiagi). Revista de la Union Matematica Argentina. vol. XV (1953), pag. 212.

34. Forma can6nica de cuadripolos simetricos de impedancia de transferencia prefijada. Revista de la Union Matematica Ar­gentina, vol. XV (1953), pag. 215.

35. Aproximaci6n uniforme de n-polos arbitrarios por medio de n­polos racionales. Revista de la Union Matematica Argentina, vol. XV (1953), pag. 215.

xv

36.

37.

38.

39.

40.

41.

42.

43.

44.,

45.

46.

47.

48.

49.

so.

Sobre Zafunai6n de Green en Za eauaai6n de KZein-Gordon. Re' vista de 1a Union Matematica Argentina, vol. XV (1953), pag. 232.

Distribuaiones y funaiones anaZ!tiaas. Centro de Cooperacion Cient!fica.de 1a UNESCO para America Latina, Montevideo , (1952), pag. 92~105.

Definiai6n preaisa de partes finitas aon distintas hiperb6Zi­aas. Revista de 1a Union Matematica Argentina, vol. XIV (1953), pag. 43.

Sobre, Za integraZ, de Za derivada en4sima de Za deZta hiperb6-Ziaa. ,Revista de 1a Union Matematica Argentina, vol. XIV (195 /f), pag. 85.

Sobre Za multipUaaai6n de d'i'stribuaiones aausaZes. Revista de 1a Union Matematica Argentina, vol. XVIII (1958), pag. 166.

Produatos de distribuaiones de Feynman. Revista dela Union Matematica Argentina, vol. XVIII (1958), pag. 93.

Sobre aZgunas 7.ntegraZes divargentes de la eZeatrodinamiaa auantiaa. Segundo Symposium sobre "Algunos problemas matema­ticos que se estan estudiando en Latinoamerica". pag. 53-60, organiza'do por e1 Centro de Cooperacion Cientrfica de 1a UNE~ CO para America Latina. Villavicencio, Mendoza, 21-25 de ju­lio de 1954.

On some distributions of Quantum EZeatrodynamios. Proceedings of the International Congress of Mathematicians, Amsterdam, setiembre de 1954, vol. II, pag. 346.

Les parties finies des int4graZes de Riemann-WellZ et les Proaed4s de r4guZarisation. Seminaire de Theories Physiques (Seminaire De Broglie) de 1a Sorbonns; expose N° 4 (1954-1955), pag. 1-13. '

Les parties' finies des int4grales de Riemann-weyl et les Proa4d4s de r4guZarisation. Comptes rendus de l'Academie des Sciences de Paris, vol. 240 (1955), pag.' 499.

Sobre Za muZtipZiaaai6n de distribuaiones oausales y antioau­sales. (en co1aboracion con s. Vagi). Revista de 1a Union M.!. tematica Argentina, vol. XIX (1956), pag~ 32.

1 1 Nota sobre Za f6rmuZa i a = '2 a'. (en co1aboracion con R.

Scarfie110). Vo1umen de homenaje a Beppo Levi editado en 1a Union Matematica Argentina,J956, pag. 53-67.

Sobre aZgunas transformadas de Lapl.aae de distribuaiones. A£ tos de 1a X Jornada de 1a Union Matematica Argentina (Univer­sidad Naciona1 del Sur, Bah!a Blanca) (1957), pag. 39-41.

Sobre eZ produato de distribuaiones de Sahwinger. Revista de 1a Union Matematica Argentina, vol. XVIII (1958), pag. 177.

Un m4todo generaZ de s!ntesis de airauitos ZineaZes no disipa tivos no rea!proaos de matriz de dispersi6n prefijada. Revi~ ta de 1a Union Matematica Argentina, vol. XVIII (1959), pag. 44.

51. In tegraZes mu Ztip Zi cativas y s!ntesis de circuitos. Te r cer Simposio realizado por 1~ UNESCO sobre "Algunos problemas ma­tematicos que se estan estudiando en Latinoamerica", julio de 1959.

52. Propiedades en eZ aontorno de funciones ana'L!ticas. Fasc1cu-10 4 de la serie "Cursos y Seminarios de Matematica". Depar­tamento de Matematica, Facultad de Ciencias Exactas y Natura­les, Buenos Aires, 1959, pag. 1-151.

53. Sobre 'La s!ntesis de circuitos no disipativos pOl' medio de 'La l,as matrices de scattering. "Sesiones Matematicas" organiza­das por la Union Matematica Argentina con motivo del sesqui­centenario de la Revolucion de Mayo, Buenos Aires, setiembre, 1960.

54. A factorization theorem for scattering matrices. Internation al Congress of Mathematicians, Stockholm, 1962; Proceedings of short communications, pag. 186.

55. Anal,ytic regul,arization and the divergences of Quantum fiel,d theori€s. (en colaboracion con C.G.Bollini y ~J.Giambiagi); Nuovo Cimento, vol. 31, pag. 550-561, (1964).

56. On the reduction formul,a of Feinberg and Pais. (en colabo­racion con C.G.Bollini y J.J.Giambiagi); Journal of Mathemat­ical Physics, voL 6, (1965), pag. 165-166).

57. On some canonical, factorization formul,ae for scattering matrices~ with appZications to circuit synthesis. "Preprint" publicado en la Universidad de California, Berkeley, en se­tiembre de 1967. El ttabajo aparecera en el "Illinois Journal oJ. Mathematics':.

58. An e~tension of Be 'Levitch 's method of synthesis by factor­isation. Comunicacion presentada en la "First International­Conference on system sciences", reaiizada en Honolulu, Hawaii, durante los d'i:as 27, 28 y 29 de enero de 1968. Aparecera pu­blicada en los "Proceedings" de la Conferencia.

xvii

CONFERENCtAS Y ARTICULOS DE CARACTER NO MATEMATICO 0 DE DIVULGACION

1. La Universidad de Brown. el 1° de enero de 1940.

Articulo aparecido en "La Nacion"

2. ALgunos aspeatos de La vida universitaria nor1;eameriaana. Con ferencia pronunciada en la Facultad de Ciencias de la Univer= sidad del Litoral el 3 de setiembre de 1941.

3. La matematiaa y La Teaniaa moderna. Conferencia pronunciada en la Facultad de Ciencias Exactas, Fisicas y Naturales d~ Buenos Aires el 19 de setiembre de 1941 durante la realizacion de las Segundas Jornadas Matemlticas Argentinas.

4. Apliaaaiones de Za integral de StieZtjes. Conferencia pronun­ciada el 3 de setiembre de 1941 en 1a Facultad de Ciencias de 1a Universidad del Litoral.

5. dQue es Za estadistiaa y para que sirve? Conferenci. radiote lefonica propalada por Radio El Mundo el 3 de octubre de 1943.

6, La organizaai6n de la Cienaia. Conferencia radioteleronica propalada por Radio El Mundo el 8 de febrero de 1944.

7. Los estudios matematiaos en eL pais. Conferencia pronunciada el 17 de setiembre de 1945 en la Facultad de Ciencias Exactas Fisicas y Naturales de Buenos Aires, en la sesion inaugural de las Segundas Joruadas Matemlticas Argentinas.

8. Georges Valiron. "Ciencia e investigacion", vol. II (1946), pig. 394. Noticia biogrlfica-critica.

9. dEs Za matematiaa moderna demasiado exaata? "Ciencia. Iri~es tigacion", vol. II (1946), pig; 491.

10.

11.

12.

13.

14.

15.

16.

Notiaia hist6riao-aritiaa sobre el aonaepto de "partie-finie" (en colaboracion con R. Scarfiello). Trabajo presentado en la reunion 23a. de la Union Matemltica Argentina, realizada e1 10 de julio de 1947 en la Facultad de Ciencias ExaCtas , Fisicas y Naturales de Buenos Aires.

Las funaiones singulares de la F1:siaa. Conferencia pronunci~ da en la 13a. reunion de la Asociacion Fisica Argentina, rea­lizada en Buenos Aires, el 23 de mayo de 1949.

dQue es Za aibernetiaa? Conferencia pronunciada en la Socie dad Cientifica Argentina el 7 de setiembre de 1949.

ALgunos aspeatos matematiaos de Za teoria de Zas aomuniaaaio­nes. Conferencia pronunciada en el Instituto de Radio Inge­nieros el 21 de julio de 1950.

EZ medio sigZo de Za matematica. Conferencia pronunciada en e1 Colegio Libre de Estudios Superiores e1 4 de julio de 1951.

La previsi6n estadistiaa segun Wiener. Conferencia pronuncia da en 1a Fundacion "Roux" e1 5 de setiembre de 1951. -

Sobre Zas teorias matematiaas de Za informaai6n y de Za estra tegia. Conferencia pronunciada en 1a Escuela Superior T~cni= ca del Ministerio de Ej~rcito e1 5 de julio de 1952.

17. La teor-la de Za previsi6n muZtipZo de N. Wiener y sus relaaio nes' -a:on la teor-la de los n-polos pasivos. Ponencia .. presenta-= da en el .Se gundd· Co loq uioAJrge,n tino· de Es ta.d'ls tic!,-.•• reaii z'ado en Cordoba los dias 27-31 de octubre de 1953.

18. Sobr.e los fi Uros' .estadrls.ti.ao.s· de N .. Wiener. Confe-ren.ci,apr.2, nunciada en el "Ins t·i tut·e. ·of .. R.adio Engin.eers" .el 26: de j un;i.o de 1953.

19. Teor£.a Ma.temati.aa de laEstrategia. Conferencia pronunci.ada en .. la 'Escuela de Guerra Nav .. al el 2·2 de julio. de 1953. . .

, . , .. 20. Los infinitos de la F-lsiaa auantiaa. Conferencia pronuncia'da

el 23 de marzo de 1955 en la Real Academia de Ciencias Exac­tas, Fisicas y Naturales de Madrid.

21. La teor-la matematiaa de los juegos de von Neumann. Conferen­cia pronunciada en la Facultad de Ciencias de la Universidad de Sevilla, el 27 de marzo de 1955.

22. Distribuaiones y funaiones anaZ-ltiaas. CicIo de conferencias pronunciadas en la Universidad de Madrid en marzo de 1955.

23. Teor-la de Za apro~imaai6n y s-lntesis de airauitos lineales. CicIo de conferencias pronunciadas durante los meses de marzo y abril de 1955 en el Instituto de Calculo del Centro NacioA nal de Investigaciones Cientificas de Madrid.

24. Matematiaa y realidad. Conferencia radiotelefonica propalada el 10 de agosto de 1955 poi Radio del Estado.

25. Relations de Za theorie de Z'appro~imation des funations avea la synthese des systemes lineaires. Con£erencia pronunciada el 10 de marzo de 1955 en el "Institut Henry Poincar~" de la Sorbona.

26. Teor£a matematiaa de la estra~egia. Articulo publicado en "Mirador", N° 2, julio de 1957.

27. La vida aient£fiaa en Is rae Z. Conferencia pronunciada en la Facultad de Ciencias Exactas y Naturales de Buenos Aires e1 5 de noviembre de 1957.

28. Israel en la Cienaia. Conferencia pronunciada en 1a Sociedad Hebraica Argentina el 28 de noviembre de 1957.

29. Distributions aausaZes et distributions analytiques. Con fe­rencia pronunciada en el "Institut Henry Poincar~", de 1a SOL bona e1 26 de enero de 1959.

30. Einstein y Za Cienaia. Conferen~ia pronunciada en la Socie­dad Hebraica Argentina el 24 de julio de 19?8.

31. Cienaia y ·TeanoZog-la. Conferencia pronunciada e1 28 de abri1 de 1963 en 1a Sociedad Cient'lfica Argentina, bajo los auspi­cios del Consejo Naciona1 de Investigaciones Cient'lficas y T~c nicas.

32. Julio Rey Pastor. Conferencia pronunciada e1 20 de julio de 1963 en 1a Academia Naciona1-ae Ciencias Exactas, Fisic.as y Naturales.

xix

33. La Matematioa y ~as Cienaias F!siaas y Natupa~es. Conferen­cia pronunciada el 7 de mayo de 1964 en la Facultad de Cien­cias M€dicas.

34. La Matematiaa en Fpanaia. Conferencia pronunciada el 28 de mayo de 196$ en la Sociedad Cientifica Argentina.

35. La Matematiaa y nuestpa soaiedad tecno~6giaa. Conferencia pronunciada en Bogoti el 4 de diciembre de 1961 y publicada en el libro "Educaci6n matemitica en las Am€ricas", editado por $1 Teacher's College, Columbia University, 1962.

Revista de la Uni5n Matematica Argentina Volumen 25. 1970.

UNE PROPRIETE DES RACINES DE L'UNITE par J.Dieudonne

Ved~cado al P~o6e6o~ Albe~to Gonzalez Vom~nguez

1. Soient p un nombre premier. wl .w 2 •...• wr (r < p) des racines

p-emes de l'unite. deux a deux distinctes. et soient

r entiers. M. Morgenstern. assistant l la Faculte des Sciences

de Nice. a conjecture que Ie determinant

ml m2 m r wI wI wI

fi l m2 m r

'" w2 w2 w2

ml m2 m r w w w r r r

n'ftst jamais nul. Je me propose de donner une demonstration de

cette conjecture.

2. Considerons Ie polynome en x l .x 2 •...• x r

ml m2 m r xl xl Xl

ml m2 m r (1 ) LI(x l .x 2•··· .x r )

x 2 x 2 x2

ml xm2

m x x r

r r r

II est clair que", (x l .x 2 •...• x r ) est divisible par Ie determinant

de Vandermonde V(xl •...• x ) = TT(x.- xi) et que Ie quotient r i < j J

F(x l .x 2 •...• x r ) est un pOlynome a coefficients entiers. II s'agit

de prouver que F(w l .w 2 •...• wr ) # 0; or. les wk sont des puissances

d'une racine p-~me primitive ~ de l'unite. donc F(w l ,w 2 ' ...• wr )

s'ecrit comme pOlynome ~(~) de degre ~ p-l en ~. a coefficients

2

entieps; si lIon avait ~(;) = D, on aurait done, vu l'irreduetibi A _ (p-l p-2 -lite du polynome cyclotomique, ~(x) - A. x + x + ••• + 1) ,

ou A est un entiep; par suite ~(1) = FC1,1 , ... ,1) serait di.visi­

ble par p, et tout revient a voir que crest impossible.

3. Or, de fa~on generale, eonsiderons r fonctions d'une variable, f l , f 2 , ..• , fr ' indefiniment derivables; alors , pour

Xl < x2 <; ••• < xr reels on peut eerire Ie determinant

fl(X I ) f 2 (x l )

f l (x 2) f 2 (x 2)

f2 (x ) . . . f (x ) . r r r

comme produit du determinant de Vandermonde V(x l , ... ,xr ) et d'un determinant de la forme

£1 (;1) f 2(;I) f r (;I)

fJ. (;2) f2(E;2) f~(;2)

1!2!3! •.• (r-1)!

f(r-l)(E; ) 1 r f~r-l) (;r). f(r-l) (E; )

r r

0\'1 les E; j sont compris dans I I intervalle d I extremi tes Xl et Xn,( 1)

Appliquant cela au polynome (1) et faisant tendre les x. vers 1 , J

on ohtient

(l) Voir G.POLYA und G. SZEGO, Aufgaben und Lehpsatze aus dep Analysis, vol. II, p. 54 et 240 (Berlin (Springer), 1925).

(2) F(1,l, ... ,1)

1 !2!3! ... (r-l)!

3

m r

et comme r < p, il suffit de montrer que Ie determinant du second membre n'est pas divisible par p. Mais par combinaison de lignes. on voir aussitot que ce determinant n'est autre que Ie determinant de Vandermonde V(m1 ,m2 ••.•• m) = TT(m.-m.). qui ne peut ~tre di-

r i<j J 1.

visible par p.

Recibido en abril de 1970.

UNIVERSITE DE NICE Faculte des Sciences.

Revis ta de la Union Matematica Argentina Volumen 25. 1970.

ENRICHED SEMANTICS-STRUCTURE (META) ADJOINTNESS Eduardo J. Dubuc

Vedic.a.doa..f. PlLo6e.6olL A.f.belLto Gonzt1.f.ez Vominguez

INTRODUCTION. In this paper, which has a purely theoretical aim

and interest, we develop the rudiments of the general Structure­

Semantics (meta) adjointness of Categorical Algebra. We do so by means of two different and parallel techniques, one using the con

cept of Monads (often called Triples, sometimes Standard Constru£ tions, and some other times Triads), the other using the concept

of Theories. we then relate (specifically) these techniques and

prove them to be equivalent.

We do all this in the enriched context of a V-world, that is, our

categories are V-categories and our functors are V-functors, where V is a given (fixed) closed (symmetrical monoidal closed) category

V-Monads have already been considered in many places in the litera ture, [1] [3] [6] [8] and probably more. In [8] a Semantics-Structu­

re (meta) adjointness is established in which the Structure (meta)

functor is Only defined on V-functors which have a V-left adjoint. Here, in sections §1 and §2 we have reproduced parts of Chapter II

of [3] , where we developed the Semantics-Structure (meta) adjoin! ness by means of a technique relying heavily on the concept of Kan

extensions. Structure is defined on the broader domain consisting of those V-functors for which the (right) Kan extensions of them -

selves along the,mselves exist. The Semantics-Structure (meta) ad

jointness is given by (essentially) a direct instance of the ~d

jOintness of this Kan extension.

V-Theories have not been considered yet in the literature. We in­

troduce them here, and in doing so we have developed in detail the

case in which the V-category involved is the base category V. We did so because of certain peculiarities (due to the presence of a V-codense cogenerator in v"I'J which are not pr'esent in the more ge­

neral case. These peculiarities allow us to stress the similari­ties with the first and original treatment of the subject (at least

"in its modern form), conceived by Lawere ([7]) in his work on Al­

gebraic Theories in the category of sets. Here the concept of co~ tensors takes the role of products. A V-theory in V is a V-categ~

ry with the same objects as V and in which any object is a coten­sor of the unit object I. An algebra is then a cotensor preserving

V-functor into V. We develop in sections §3 and §4 a Semantics­Structure (meta) adjointness in this context.

6

In section §S we prove the equivalence referred to at the beginning of this introduction,and in doing so we take advantage of the (sim­ple) equivalence between the structure (meta) functors to deduce the equivalence of the two semantics. In this way we avoid the need for the more complicated theorems of V-triplability and charac terization of V-categories of algebras.

In section §6 we (briefly) indicate how to generalize these results to the general case of a V-theory in a V-category A, adopting in this case the V-versions of what have been considered as theories and algebras in [9] .

Throughout this paper (although it is not always necessary) we as­sume our base category V to be complete (all small inverse limits) and well powered. All the concepts and results (as well as the no­tation) of V-category theory used here can be found in [3]. All the logically illegitimate constructions, preceded here by the word (meta), become licit mathematical objects in any or the current foundations suited for category theory.

§1. Semantics of V-Monads. §2. The V-Monad Structure. §3. Semantics of V-Theories. §4. The V-Theory Structure. §S. Equivalence between the V-Monad and the V-Theory techniques

of producing a Semantics-Structure (meta) adjointness. §6. Remarks about V-theories in a general V-category A.

V-MONADS. Given a V-category A, recall that a V-monad in A is a V­endofunctor A ~ A together with a pair of V-natural transforma -tions TT ~ T and idA ~ T, II is associative and n is a left an right unit for II in the sense that the following diagrams commute:

TTT ~ TT T DT > TT T Tll > TT

jTll 1 II TT ~ T i~!' and i~jll

'l T

7

We write T = (T,p,n) and call P the multiplication and n the unit. A mozophi,sms of monads T .t T'is a V-natural transformation T 1 T'

such that the diagrams

and commute.

V-monads in A with morphisms of monads between them form a (meta) ca tegory . that we deno t%( A I .

§1. SEMANTICS OF V-MONADS.

Given a V-monad T = (T,p,n), a T-algebra is an object AEA together with a T-algebra structure, that is, a morphism TA!; A, associative and for which nA is a unit, in the sense that the diagrams:

TTA ~ TA

and commute.

We write A = (A,a) and call A the underlying object. _ f _ f

A morphism of algebras A + B is a map A + B in A such that the dia gram

commutes.

T-algebras and morphisms of algebras form a category . h f AT uT A T- Tf T . W1t a unctor ~,U A =A ,U = f. A 1S a

T T - - uT and U a V-functor by defining A (A,B) ~ A(A,B) to

AT provided

V-category be a V-equal-

a

izer of the pair of maps:

A(a,D) A(~,B) , A(TA,B)

\ f· B)

A(TA,TB)

uT is obviously V-faithful and we call it the forgetful functor. The followi~g proposition establishes the intuitive_fact that V­functors C ..¥ AT are the same thing as V-functors C ~ A together

s with a V-Batural T-algebra structure TS ~ S.

PROPOSITION 1.1 •. Given a V-functor C §'_A. S admits a l.i.6Ung into the T-aZgebras. that is, a V-functor C §. AT such that UTS = S. if and onZy if there is an ac.t:..i.on of T on S. that is, a V-naturaZ transformation TS ~ S such that the diagrams:

TTS ~ TS

I jlS Is and commute. v v TS => S

Proof. It is clear that in both cases we have the same data, i.e., a family of arrows TSC ~SC, C EC, and that the equations of T-al gebra for each one of the sC are exactly the equations of T'action for s. Consider now the diagram:

C(C,D)

~s (1) \ uT

. ~A(SC SD)

'~ \

A (sC D) A (TSC, TSD)

~ A (TSC, SD) A (0, sD)

9

S equalizes the two maps of diagram (2) (that is, s is V-natural) if and only if there is a map S making diagram (1) commutative (that is, if there is a V-functor structure for the function

sC C --+TSC -+ SC). This completes the proof. ~~.

REMARK. Since UT, being V-faithful, reflects V-naturality, it follows that V-natural transformations S 1 H are the same thing that V-natural transformations S 1 H such that the diagram

TS T<j> > TH

II s (1)

v <I>

S > H

~ h v

commutes.

AT id T T The identity V-functor -+ A is the lifting of U , and so there is an action TUT ~ uT, uA =. a.

Also, since TT ~ T is an action of T on T, there is a lifting of T into the T-algebras A ~ AT, UTFT = T, FTA = (TTA ~TA). It is

clear that uFT =~. One of the equations in the definition of an action is exactly diagram (1) above for u, and so there is a V-na­tural transformation FTUT ~ id, UTE = u, that, together with id ~ UTFT, establishes the fact that FT is V-left adjoint to UT.

The triangular equation is the other

equation in the definition of action, and

So we have just proven the following:

10

PROPOSITION 1.2. The V-funatop UT has a V-left adjoint FT and the T T T T .

V-monad (U F ,U e:F ,n) '!.s equal/;o T. ~.

We call the V-functor rT the free funator and a T-algebra of the T form F A a free algebra.

Given a morphism of monads T' .1 T it is trivial to see that T'UT <PUT .. TUT U > uT is an action of T' on uT , and so, there is

a V-functor, denoted A<P, which makes the triangle:

AT A<P T' ---'"'--+. A

"'\ / uP

A commutative.

Given a composite ~.<p, the V-functors A~'<P and A<P'A~ both cortespond

to the same action, and so~ they are equal. The assignment of

AT U: A to <P ~ a V-monad T and of A to a morphism of V-monads <I> is

then a contravariant (meta) functor betweenjb(A) and the (meta)

comma category (V-Cat,A):

G j&(A)oP m • (V-Cat,A)

the semantics (meta) functor.

If T .1 T' is a morphism of V-Monads and T'S ~ S is an action of Tt "'S " on S; the composite TS ~ T'S ~ S is an action of T on S, and it

is not difficult to check the following:

PROPOSITION 1.3. The one to one and onto correspondence

S ->6- (T) m

(Proposition 1.1) TS .... S

is natural in r with respect to morphisms of V-monads. (Where the

above arrow is understood to be a map in (V-Cat,A) and the above

double arrow an action of T on S).

11

§2. THE V-MONAD STRUCTURE

S Given a V-functor C ~ A, the right Kan extension of S along itself

RanS(S) A I A, if it exists, has a structure of V-monad given by:

S ..ii S

RanS(S)e: r ==1;6.==:> 0

and

RanS(S) ~> RanS(S)

RanS(S) S ~ S

(where r is the one to one and onto correspondence which defines o the right Kan extension).

We write TS = (RanS(S),p,n) and call it the oodensity V-monad. If it exists, we say that S admits a codensity V-monad. We say that S is stpongZy tpactabZe if, furthermore, RanS(S) is preserved by

the rellresentables A A(A,-~ v. (ef. [31, Proposition 1.4.3: If A

is cotensored, a right Kan extension with codomain A is preserved by the representables if and only if it is point-wise, that is, if and oniy if the Kan formula fo compute it as a point-wise end of cotensors in A can be used).

A complete proof of the fact that the unit and multiplication defi ned above for RanS(S) actually define a V-monad as well as of the next two propositions is to be found in [31.

PROPOSITION 2.1. Given any othep V-monad T in A. aotions of T on

Sand mopphisms of V-monads T + TS ooppespond to eaoh othep· undep

rot

TS > S

~.

G F PROPOSITION 2.2. If a V-funotop 5 + A has a V-Zeft adjoint A + 5,

12

id ! GF , FG ~ id, then it is strongly traatable and the aodensity

V-monad is (GF, GEF, 11).. Furthermore, RanG (G) is preserved by any

V-funator with domain A. ~.

THEOREM I. Given a V-funator C § A whiah admits a aodensity V-mo­

nad, for every V-monad T~A), there is, naturally in T, a one to

one and onto aorrespondenae between morphisms of V-monads T + TS

and V-funators C + AT making the triangle

aommutative, that

indiaate this by

C_AT

~/uT A

is, maps S +~ (T)

S + G (T) m

in (V-Cat.A).

Proof. Immediate from Propositions 1.3 and 2.1.

As usual, we

Let ~r(V-Cat,A) be the full (meta) sub-category of (V-Cat,A)

whose objects are the V-'functors admitting a codensity V-monad.

From propositions 1.2 and 2.2 we know that the semantics (meta)

functorCYm takes its values in s~r(V-Cat,A). The assignment of

TS~(A) to a V-functor C ~ A becomes then, by Theorem I, a con­

travariant (meta) functor, denotedc;m' in such a way that the one

to one and onto corresponce (in Theorem I) is also natural in S .

(;m is then a left adjoint to semantics, and it is called struature.

Given a V-functor C ~ A in s~r(V-Cat,A), the codensity V-monad

TS =~m(S) is the structure V-monad of S.

Notice that the (meta) adjunction:

S ---->- G: (T) m

13

is, essentially, just the one to one and onto correspondence which defines the right Kan extensions RanSCS),

It is iMmediate from ~ropositions 1.2 and 2.2 that the arrow T +~ ~CT) in~A). T + T T' is the equality. That is, the code~ m m U

sity V-monad of UT is T.

The

r Cid) is given by the action RanSCS) S ~ S.

The V-functor S is called the semanticaZ comparison V-functor of S. When S has a V-left adjoint we have:

PROPOSITION 2.2. Given any V-functor B Q A with. a V-Zeft adjoint

A ~ B. (&. n): F -I V G • the semantic comparison V-functor of G,

B G .TG· • b h . GFG G& G (h . G C . d)) -+ 1'\ 1.s· g1.ven y t e aat1.on => t at 1.S, & = ro 1 ,

and is unique making the foHowing two triangZes commutative:

G T B G -A

~/ /G A

(a simple proof of this fact is given in [3], Proposition 11.1.6).

Q.:b.!!.

V-THEOR I ES. By a V-theory in V we mean a pair (T,T), where T is a V-category whose objects are the objects of V(that we will write Vt when we think of them as belonging to T) and where

V(W,V) ! T(Vt,wt) is a V-functor structure making the identity on objects a cotensor preserving V-functor vJP 1 T.

We have then for each Vt E T, Vt = f(V,I t ) (where f(V,I t ) is the cotensor (in T) of V with It), and hence, the V-objects of mor -phisms into It determine the whole V-structure. of T. Specifically

..­,

14

we have T(Wt,Vt ) ~ V(V,T(Wt,r t )) (cotensoring isomorphism).

<I>

By a morphism of theories (T, 1) ... (T', T') we will understand a co-

tensor preserving V-functor T ! T' sending It into It';

valently, any V-functor T! T' making the diagram

T~T'

''\ / T' Vop

commutative.

or, equi-

V-theories in V with morphisms of theories between them form a (m~

ta) category that we denote~(V).

§3. SEMANTICS OF V-THEORIES.

Given a V-theory (T,T), a T-algebra is a cotensor preserving V­functor T ~ V. Since Vt ~ r(V,I t ), we have avt = V(V,a(I t )) and so a on objects is completely characterized by its value at It. Also; the composite Vop ~ V is cotensor preserving, and hence, since I is a V-codense cogenerator of VOP , it is representable: a·T = V(-,aCI t )) CcL [3], Theorem III.2.3). (a is cotensor pre­serving if and only if a ° T is cotensor preserving if and only if aoT is representable).

We can then redefine a T-algebra as being an object A E V togetheI with a T-algebra structure, that is, maps

TCVt ,wt) ~ V(V(V,A) ,V'CW,A)) gI.vmg a structure of V-functor T ~ V

to the function on objects VL~ vCV,A), and making the diagram:

T(Vt,wt) a) V(VCV,A),V(W,A))

Cl) ~ <yc-,A) commutative.

VCW,V)

We write A = CA,a) and call A the underlying object.

A morphism of algebras A ! B is a V-natural transformation a f > ~. It is completely determined by its value at It and hence we can re define a morphism of algebras as being a map A ! B making the dia­~rams:

15

T(Vt,wt ) 0 __________ -+ V(V(Vt,A),V(Wt,A))

I S I V (0, V (D,f))

V(V(D,f),D) V(V(Vt,A),V(Wt,B))

T-algebras and morphisms of algebras form a T

with a functor V(T) ~ V UTA = A , UTf

ry and UT a V-functor by defining v(T)(A,B)

It-projection of the (large) end:

commutative.

category V(T) provided

f. VeT) is a V-catego­

uT --+ V(A,B) to be the

That the above end exists can be seen as follows:

Consider the diagram: E

1 V (A,B) vcw./ ~.-)

V(V(W,A),V(W,B)) (1)v w V(V(V,A),V(V,B)) ,

V(T(V t ,Wt),V(V(V,A),VlW,B)))

j V CT.OJ

V (V (W,V),V (V(V ,A),V (W,B))

16

where the arrows fo f1 f2 and f3 are the maps which correspond by adjointness to:

t __ ~V~(_-L,~eW~_)~'~a~ __ -+, V(V(aWt,ewt),V(aVt,ewt))

. V(aV\-)·e t t t t ---->."'-'-~-<-....::...---+, V (V (aV ,ev ), V (aV ,ew ))

V(W,V) V(-,V(W,B))·V(-,A) , V(V(V(W,A),V(W,B)),V(V(V,A),V(W,B)))

V(W,V) V(V(V,A),-).V(~ V(V(V(V,A),V(V,B)),V(V(V,A),V(W,B)))

and where E is the intersection of all the equalizers of the two

maps in diagrams (l)V,W'

From diagram (1) (page 13) and the above definitions it is not di­

fficult to see that diagrams fa) and (b) commute. This, together with the equation V(A,B) = V(V(V,A),V(V,B)) (cf. [31 )

V

CV-Yoneda Lemma) easily implies that E = fvt V (aVt , evt) .

uT is the V-functor "evaluation at It", and from the above cons -truction it is obvious that it is V-faithful. We call it the for­getful functor.

T(Vt ) PROPOSITION 3.1. The T-algebras T \ ,-~ V are the values of a

T V-functor V ~ VCT) • V-left adjoint to UT,

Proof·

vCT) CT(Vt ,-) ,a) ~ aVt = VCV,A)

where the above CV-rratural in a) isomorphism is given by the V-Yo' neda lemma. Q.E.D

We call the V-functor FT the free functor and a T-algebra of the

form FTV = CTCVt,It),TCVt ,-)) a free algebra.

17

Given a morphism of theories fT' ,T') ! (T,T) , it is clear that for any T-algebra T ~ V , the composite T' t T ~ V is a T' -algebra.

T' From the universal property vf ends and the fact that U is V-faithful it is easy to see that this function between the objects

of VeT) and those of VCT ') has a (unique) structure of V-functor,

V~ ,.making the diagram

:ommutative. Again, it is completely straightforward to check the T

equation v~·~ = V~·V~ , and so, the assignment of VCT) ~ V to a

V-theory (T,T) and of V$ toa morphism of V-theories is a contrava

riant (meta) functor between{?(V) and the (meta) comma category (V-Cat,V) :

G t ---'---+. (V -Ca t, V)

the 8emantios (meta) functor.

14. THE V-THEORY STRUCTURE.

Given a V-functor C ~ V; we will say that it is tractable if for any pair of objects V,W E V, the end

Ie V(V(V,SC),V(W,SC)) exists in V.

,That is, if for any pair of objects V,W E V, the class of V-natu -

ral transformations between VCV,S( - )) and V(W,SC - )) is a set, and furthermore, it is the underlying set of an object of V, name­ly, the end displayed above.

There is no difficulty in checking that the objects of V together

18

with the a.bove end between them form a V-category, T S ' the clone of operations of S;

Ic V(V(V,SC),V(W,SC)) .

The collection of maps (which is a V-natural family):

V(W,V) V(-,SC), V(V(V,SC),V(W,SC))

lifts into the end, providing a structure of (contravariant) V­functor to the identity map between objects:

TS has a V-left adjoint [ putting W = I in the definition of tracta

ble, it follows that Ic V(V(V,SC),SC) = RanS(S) (V) (see Proposi -

tion 5.1) exists, then, for any other W,

V(W, f C V(V(V,SC),SC)) = Ts(Vt,Wt ) = TS(Vt,TS(W)) 1 and therefore

it preserves cotensors. W~ have then that the pair (TS,TS) is a VCtheory in V, V-theory which we call lithe structure of C § V ".

PROPOSITION 4.1. If a V-functor B Q V has a V-left adjoint V g B, then it is tractable and

Proof· IB V(V(V,GB),V(W,GB)) ~ IB V(B(FV,B),B(FW,B)) ~ B(FW,FV)

The second isomorphisms given by the V-Yoneda Lemma.

THEOREM II. Given a tractabZe V-functor C § V ~ there is a V-fun~

tor C § VeTS) making the triangZe

19

S (TS) C -> v \,(1)/ T

'\ IUS V

commutative

nd such that given any other V-theory (T,T) together with a V-fun£.

or C ¥ VeT) making the triangle

(Jommutativ.e, there is a unique morphism of theories T ! T S making the triangle

S (T S) C-V ~ (3) 1 v<P

.. ~ veT) commutative .

Proof. For any C E C define SC E VeTS) , SC = (SC,n e) where

Ts(Vt,wt ) ~ V(V(V,SC),V(W,SC)) is the C-projection of the end.

By definition of TS (page 17) ,

commutes, and so (SC,n C) is a Ts-algebra. The collection of maps

(which is a V~natural family)

C(C,C') ~ V(SC,SC') V(V,-l V(V(V,SC),V(V,SC')) = V(nc(Vt),nc,(Vt ))

lift into The end V(TS)(SC,SC'). providing a structure of V-functor

to S whi~ .. (in particular) makes triangle (1) commutative.

20

Given C ~ VeT) , GC =$C,yc) , then, there is a unique

commutes

(recall that aC was (by definition) the projection of the But the commutative diagrams (4) are exactly the equation that is, commutativity of triangle (3).

end). V4>.S = Gl

~.

Let 6tV-Cat,V) be the full (meta) sub-category of (V-Cat,V) whose objects are the tractable V-functors. From Propositions 3.1 and 4.1 we know that the semantics (meta) functorGYt takes its values

ill ~v-cat.V). The assignment of (TS,TS ) E~(V) to a V-functor

C ~ V becomes then, by Theorem II, a contravariant (meta) functor,

" denotedGt , left adjoint to semantics.

Gt k;(V)op ~ :t;;v-Cat,V) -

From the V-Yonada Lemma and Propositions 3.1 and 4.1 it is clear

that the arrow (T,T) ~~t~(T) in~(V) , T ~ T T ' is the equality U

(or rather, an isomorphism). That is, the clone of operations of

UT is T.

The arrow S --.C5,pt(S) in ~fv-cat,v). , c § VCTS) has been construe

ted in Theorem II and it is called the semantical comparison V-func tor.

21

§5. EQUIVALENCE BETWEEN THE V-MONAD AND THE V-THEORY TECHNIQUE

OF PRODUCING A SEMANTICS-STRUCTURE (META) - ADJOINTNESS.

First let us check tha.t the domain (meta) categories of the two structure (meta) functors coincide, that is, that they are both the same full (meta) sub-category of (V-Cat,V).

PROPOSITION 5.1. Given a V-funatop C § V. then: S admits a aoden­

sity V-monad if and onty if S is stpongty tpaatabte if and onty if S is tpaa tab te .

Ppoof. The first two statements are clearly equivalent since any _ __--1" igh-t----Kan- ex-teJl-s-i()n-w4-t-h-eedO'lltai-n-V--i-s---p1Jbrtwi.-s~-ar-t r ac note

implies strongly tractable is easily seen by putting Wt = I in the definition of tractable (page 16). The resulting end is just the Kan formula for pointwise computing of RanS(S), Vice-versa, assu~ ing that RanS(S) exists, since the representables preserve it, for every V E V, RanS(V(V,S( - ))) exists, and, being with codomain V, it is pointwise. Then, the'pontwise Kan formula shows that S is tractable. ~.

In order to relate the domain (meta) categories of the two seman­tics (meta) functors it is in order to de~ine the KteisZy V-cate­gory associated to a V-monad in V (cf [8]).

iecall that given a V-monad T (T,~,n), the objects of V with the following V-structure between them

(1) K (wt,vt) = V(W,TV)

T def

constitute a V-category,KT ' the Kleisly V-category of T.

V(W,V) V(D,nVl KT(Wt,vt ) give a structure of V-functor V

the identity between objects,which,just by definition has a V-right T

adjoint KT ~ V sending v t into TV. The adjunction isomorphism is

given by the equaZity (1). The V-monad associated to the adjoint

pair fT, uT = KT(I t ,-) is clearly T again.

The pair ((fT)OP,K~P) is obviously a V-theory in V, and there is no

difficulty in seeing that the passage,n,(V) ~ 6(V),

K~(T) '" ((fT)OP,K~P) is a (meta) functpr [the V-functor correspon­

ding to a morphism of V-monads T 1 T' is given by

22

On the other hand, given any V-theory (T,T), the V-functor t

T T(-,I l Vop is a V-left adjoint to Vop l T (the adiunction given

by the cotensoring isomorphisms T(Wt,Vt ) = T(Wt,f(V,I t » g,

g, V(V,T(Wt,I t ». Hence, we rediscover T by means of the formula

T - t T(-,I ).

The following formal manipulation,

V(V,T(f(W,It),I t ) g T(f(W,It),f(V,I t » = T(wt,Vt) , proves that

the Kleisly V-category associated to the V-monad de~ermined by the

V-adjoint pair a: f(-,I t ) -1V T(-,I t ) ~s the dual of T, while the

commutativity of the diagram below

[where nV

properly be called lithe Gelfand transformation")] shows that 'the de

finitiJn of fT produces in this case the V-functor T. Therefore we

wholly recover the starting V-theory (T,T). This implies that the

assignment of the V-monad T(f(-,It),It)to a given V-theory (T,T) is

actually a (meta) functor~(V)~f~(V) which together with the (m~

tal functor K~ establishes an equivalence of (meta) categories bet

ween (;(V) andj1o(V). (Notice that this V-monad sends an obj ect

WE V into the V-object of W-ary operations).

THEOREM III. There is an equivaLenae of (meta) aategorie~

23

between the (meta) aategories of V-Theories in V and of V-Monads

in V suah that the fo 1 Zowing diagram:

t;(V)Op ~

jlK lK~" G~nt~ . t;" ?:; (V-Cat V) ~ r sr '

~(V)op .~

aommutes up to naturaZ isomorphisms.

1)

2) ... G t

where the (meta) funators G9 G? are the respeative semantias-stru~ ture (meta) adjointness. .

Proof. The (meta) functors ~K and K~ have been defined and proven to be an equivalence in the considerations made before the state -ment of the Theorem

Since the semantics (meta) functors are adjoints to the structure (meta) functors (Theorems I and II), it will be enough to prove equations 1) That is: Given a tractable (equivalently, strongly tractable, Proposition 5.1) V-functor C ~ V:

a) The codensity V-monad of S is the V-monad associated by tK to the ClOne of operations of S. This is clear just by the defini -tions involved (the assignment Vt.~ RanS (S)V was seen to be a V-

I f d . . Vop T S T 1 7) eta J01nt to -+ S' see page .

b) The clone of operations of S is the dual of the Kleisly V-cate gory associated to the cod~nsity V-monad of S. Again, this is clear just by the definitions involved

(TS(Vt,Wt ) = Ic V(V(V,SC),V(W,SC)) = RanS(V(W,S(-))(V)

V(W,Rans(S) (V)) t t .

KT (W ,V )). S

24

These observations complete the proof of the theorem. ~.

Notice that equations 2) in the theorem just proven mean that

given any V-monad, the V-category of algebras is V-isomorphic to the V-category of algebras over the dual of its Kleisly V-category

and, vice-versa, given any V-theory, the V-category of algebras is

V-isomorphic to the V-category of algebras over the associated V­monad.

§6. REMARKS ABOUT V-THEORIES IN A GENERAL V-CATEGORY A.

We have observed that for any V-theory in V, Vop r T. the V-func­

tor T has a V-left adjo1nt. This is ultimately due to the fact that I E V is a V-codense cogenerator of VoP: Also, for the same

reason, given any T-algebra T ~ V, the composite Vop r T ~ V is a

representable V-functor (cf. [3] , Theorem III. 2.3).

We can define then a V-theory in A as a V-functor AOP r T, bijec­

tion in objects and having a V-left adjoint. Similarly, a T-alg~ bra as a V-functor T ~ V such that AOP r T ~ V is representable

(cf. [9]). All of section §3 of this paper can then be carried over

with no great difficulty. In particular, the T-algebras form a V­category and the forgetful V-functor (sending a into the represen­

ing object of a'T) has a V-left adjoint (which sends A E A into T(At ,-), which is a T-algebra since th~ composite T(At'-)'T is re-

v t v presented by T A E A (T the V-left adjoint to T)). We obtain in this way a Semantics (meta) functor which takes its values in the

(meta) sub-category of (V-Cat,A) of strongZy tractable V-functors. On the other hand, any strongly tractable V-functor C ~ A is trac­

table (but not vice-versa) (exactly the same proof given in Proposi tion 5.1 applies to this general case), and hence we can apply word

by word the Structure(meta)functor construction developed in section

§4 to strongly tractable V-functors C ~ A. In this case, the clone of T

operations of S, AOP ~ TS is such that TS has a V-left adjoint

(sending At E TS into RanS(S) (A)) and hence it is a V-theory in A

according to our definition above. We obtain in this way a Seman­tics-Structure (meta) adjointness between V-theories in A (having

a V-left adjoint) and strongly tractable V-functors into A, which,

is completely equivalent to the Semantics-Structure (meta) adjoin!

ness developed in sections §1 and §2. Theorem III with A in place of V holds exactly.

2S

If we do not require a V-theory in A to be such that AOP r T has a V-left adjoint, then some new kind of phenomena appears which makes the situation different than in the case of V-theories in V.

A T -algebra is defined in the same way, i. e., any V -functor T 5! V such that n'T is representable. T algebras form a V-category with a forgetful V-functor which now in general will not have a V-left adjoint (the V-functors T T(At.-), V are not T-algebras since T(At'-)'T is not r.epresentable any more). This forgetful V-functor, however, is still tractable (but not strongly .tractable), and we obtain a Semantics (meta) functor which takes its values ~n the (m~ ta) sub-category of (V-Cat,A) of tractable V-functors. The struc' ture (meta) functor construction (§4) applies exactly (in this case,

T the clone of operations AOP ~ TS will not have a V-left adjoint) and we obtain a Semantics-Structure (meta) adjointnes's between V­theories in A and tractable V-functors into A which contains the previous ones.

26

BIBLIOGRAPHY

[I] H. BUNGE. Re.i.a.t.i.ve. Func..toll. Cate.gOlL.iu and Cate.gOlL.ie..6 06 Ai.ge.­b~a.6. Journal of Algebra. Vol. II. Jan. 1969.

[2] B.J. DAY and G.M. KELLY. Enll..ic.he.d Func.toll. Cate.goll..ie..6. Lecture

Notes. Vol. 106. Springer-Verlag.

[3] E.J. DUBUC. Kan Ex.te.n.6.ion.6 .in EnII..ic.he.d Cate.gOlLY The.OILY. Lec­

ture Notes. Vol. 1~5. Springer-Verlag.

[~] G.M. KELLl. Adjunc.t.ion 6011. e.nll..ic.he.d c.ate.goll..ie..6. Lecture No -

test Vol. 106. Springer-Verlag.

[5] H. KLEISLY. Eve.II.Y Standall.d Con.6tll.uc.t.ion .i.6 Induc.e.d by a Pa.ill. 06 Adjo.int Func.toll..6. Proc. Amer. Math. Soc. Vol. 16 (1965).

[6] A. KOCK. On monad.6 .in .6ymme.tll..ic. mono.idai. c.i.o.6e.d c.ate.goll..ie..6. Aarhus Universitet. Preprint Series 1968-69. N° 13 (to appear

in J. Austr. Math. Soc.).

[7] P.W. LAWERE. Func.toll..iai. .6e.mant.ic..6 06 ai.ge.bll.a.ic. the.oll..ie..6. Di­

ssertation. Columbia U •• Summarized in Proc. Nat. Acad. Sci.

Vol. sa (1963).

[8] P.E.J. LINTON. Re.i.at.ive. 6unc.toll..iai. .6e.mant.ic..6: adjo.intne..6.6 1I.e.­.6ul.t.6. Lecture 'Notes Vol. 99. Springer-V·erlag.

[9] P.E.J. LINTON. An outi..ine. 06 6unc.toll..iai. .6e.mant.ic..6, Lecture N~

tes Vol. 80. Springer-Verlag.

Recibido en julio de 1970.

University of Illinois

Urbana

I I I I I I I I

, I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Revista de 1a Union Matematica Argentina Vo1umen 25, 1970.

IDEALS AND UNIVERSAL REPRESENTANTIONS OF CERTAIN C*-ALGEBRAS

Horacio Porta*

Ved~cado at P~o6e~a~ Atbe~to Gonz~tez Vomlnguez

INTRODUCTION. Let H be a Hilbert space and B(H) the C*-algebra of all bounded linear operators T: H + H. Consider the following

two problems:

A) Find all the closed two-sided ideals of B(H).

B) For each closed two sided ideal J C B(H), find all

the representations of the algebra B(H)/J in some

Hilbert space.

The first problem has been solved by B.Gramsch (11), (12) (see

also E. Luft (16)). The second has not yet been solved, even for

the case J =):OL See, however, [17, §22], [181, (20).

The solutions to A) obtained by Gramsch and Luft are based on a g~

neralization of compactness: the ideals being characterized in terms of the "degree of compactness" shared by their constituent 2-perators. The technique involves lengthy topological arguments. We remark that such generalizations of compactness abound: (21) ,

(8), (9) , (14) , (23) , (12). In this note we describe an alter­

nate approach to problem A) having only traditional notions of Hil bert space (projections, rank, etc.) as main ingredients, thereby

avoiding generalized compactness. The arguments are considerably

shortened.

Concerning problem B), except'when J is the only maximal ideal of

B(H), the dimension of the universal representation of B(H)/J is Zd

found to be equal to 2 where d = dim H Observe

that when J is the ideal of compact operators, theDe are faithful representations of B (H) / J of smaller dimens ion (20). This is pro­

bably true for all non-maximal J.

§1. PRELIMINARIES AND NOTATION.

We shall observe the standard terminology for Hilbert spaces, as

Supported by NSF Grant GP 20431.

za

used in [6]. All throughout, H will denote a fixed complex Hilber

space of dimension dim H = d = N6 ' where 6 is. some ordinal number 6 = 0,1, ••• etc. B(H) will denote the C*~algebra of all bounded Ii near operators T: H + H , C(H) and F(H) the two-sided ideals of B(H) of all compact operators and operators with finite tank, res­pectively. C(H) is closed in B(H) and F(H) dense in C(H). Greek letters a, a, y will denote ordinal numbers in the interval ["0, 6+1] , so that No " Na " NH1 • No misunderstanding should a­rise from a second use of a, in §3, to denot,e the Stone'-Cech COIl -

pactification ax of a topological space X. Let ~ E [0,6] , K Hil­bert space of dimension ,Na; we '.will denote by ma C"B(H) the set ma = {T} of all operators of the form T = QS, where S: H + K , Q: K + H are linear and bounded. Obviously IX " a implies ma C 118'

According to [19] or [1, § 5], ma is a two sided ideal ,of B (H), and if P E B(H) is a projection (= idempotent operator) with rank P=Na , then ([19,1.3] or [1,5.14]), ma = {TPT' ; T,T' E B(H),l;. It follow! from this characterization that T E ma if and only if the closed subspace generated by TH = {Tx ; x E H} has dimension at most Na ' and this at once implies that all rna are norm closed in B(H) (in fact, they, are also sequencially closed in the strong topology [5, § 3, N° 1] of B (H)) . If S is a set, Card S denotes the cardinal

power of S. We assume the generalized continuum hypothesis,

(ZNa = l"ia+1) although it is not used until Z. 7.

§2. IDEALS OF B(H).

Let J'be a two sided ideal of B(H). It ~s well-known that J is g~ nerated by the projections inJ ([5, Chap. '1" ,§1, Ex. 6] , [4] , [Z~]). Actually, the same proof gives a better result:

Z.1. LEMMA. Let J C B(H) be a two sided ideal. and T E J; then T

oan be approximated (in norm) by opera~ors of the form TP, where P is a ~ermitian projeotion and P E J.

Proof. (af. [16, Lemma 5.2] ): Set S T*T and let S = f~ AdPA , 0

be the spectral decomposition of S ;;;a. 0. FO,r E > ° , define K C H

1 • byK = P EH and let K denote the orthogonal', subspace. Then 1

a) K reduces Sand "sIK" "E ,b) for x E K , (Sx,x)> E(X,X).

It 'follows from b) thatllTxl1 ~ E l / 2 l1xUfot all x E C and therefore

29

TKl is closed and T: KI ... TKI is invertible. Let LE-B(H) satisfy

LT = identity on KI. Then, if P denotes the orthogonal projection

on K, we have LTP = P, whence PEJ, and ~'I-TPII2 ~ IITIKI12 = nSIKlloE;;&

the lemma follows.

2.2. REMARK. Assume that H is separable, and let JCB(H) be a

two sided ideal. If J contains a projection of infinite rank,

then J contains also the identity I: H ... H , and therefore J=B(H).

On the other hand, if all projections in J have finite rank, then,

by Lemma 2.1, J is contained in C(H). This shows an old result

due to J.W. Calkin [2) C(H) is the ~argest proper thlO sided i-

deal of B(H) (cf. [17, Chap. IV, §22, N° 1) ).

Consider a two-sided ideal J C B(H). We will associate to J an or­

dinal number h(J) and a two-sided ideal *J with some properties .

First. the set of ordinals {a E [0,0) ; ma CJ} , if not empty, is

an initial segment, and therefore an ordinal h(J) is well determi­

ned by the properties a) -loE;;h(J)oE;;o+l ; b) h(J) = -1 if and

only if J = {a} ; c) for J" to}, maCJ if and only if a<h(J).

If J = {O} , set *J = {O} ; if J " {O} • and h (J) = 0 , set

*J = F(H); finally, if h(J»O , set *J =u{ma ; a<h(J)}. It is

clear that for all J, *J is also a two-sided ideal and J ... h(J)

and J ... *J are mOllDtonic: J C K implies h (J) oE;; h (K) and *J C *K. AI·

so, it is easy to see that h(ma) = a+1.

2.3. LEMMA. For an J hle have *JCJcliJ.

Proof. If h (J) = 0, then (Remark 2.2) J C C (H) and therefore

*J = HH) = C [H) :J J.

Lemma 2.1, there is a

bed ~ > O. Clearly m _Q

and therefore T E *J.

Assume h (J) > 0 and take T E J; according to

projection PEJ with RT-TPII<&, for prescri­

= {TPT' ; T,T' EB(H)} CJ, if rank P = ~a '

Thus J C *J , as desired.

2.4. THEOREM ([11) , (16) ). The famiZy l of an aZosed thlo-si­

ded idea~s in B(H) is hlelZ ordered by set inalusion. An ideaZ in

l is Of the form m if and onZy if it has an immediate predecessor Q

tn l~ different from to}.

Ploof. We will show that the mapping h: l ... [-1,0+1) defined by

h: J -+ h(J) is an order isomorphism onto [-1, 0+1). First, Lemma

2.3 shows that h is one-to-one on closed ideals: h(J) = h(K) im­

plies *J = *K and therefore J = *J = *K = K. It was already ob-

30

served that h is mono.tonic: JCK implies h(J) ";;h(K).The converse

also. holds: h(J)";;h(K) implies *JC*K, whence J = *Jc*KCK. We

;how now that h is onto. Let Il E [-1,0+1] and define

h = U {m ; a < Il} , J = 31' Clearly *J C J1 and therefore a

h (J) ;;;'Il • Let P Il be a proj ection of rank !'Ie' Then 11 P e -Til;;;. 1 for

all TE ma.' for any a < Il. This is a general fact about closed i -

deals; if P is an idempotent and P does not belong to a closed

two sided ideal K, then IIp-TII;;;'l for all TEK. The proof is as

follows: if lip-Til = a<l , then II (P_T)nll";;an + 0 and (p--T)n=p-Sn ,

with S E K. Thus S + P and P <I. K, a contradiction. Hence Po n n "

does not belong to the closure of J 1 ,that ~s, to J and therefore

ma rt:. J. Hence h(J) ~fl, and so .h(J) = fl, proving that h(J) covers

[-1,0+1]. Finally, assume J has a predecessor KCJ. Then h(K)~a

and h(J) = a+l for some a. But also h(ma ) = a+l, so that by uni­

queness h(J) = h(ma ) implies J = mal as desired.

It is clear that the closed two-sided ideals of B(H) can be iden­

tified by their h(J), so that we may write J a to denote the ideal

J satisfying h(J) a. According to the proof of 2.4, we have

J a = closure U{m fl ; i3 < a}. Then Lemma 2.1 can be reworded as fol

lows:

2.5. TEJa if and only if there is a sequenae of aommuting her-

mitian projeations {Pn } suah that T lim TP n > and rank Pn <!'Ia for an n.

We shall prove also the following generalization of Rellich cri­

terion, due to E. Luft [16, Th. 5.2]

2.6. TE J a if and only if for eaah E > 0 there is a subspaae

HECH with codim HE<!'Ia suah that IITIH II<E. E

Proof. Assume this condition is satisfied, and let P be the or-1 n

thogonal projection with nullspace HE for E n Then rank Pn <

< !'Ia and IIT-TP II = IIT(I-P )11 = IITI H II..;; E, so that 2.5 applies n n E

and TEJa · The converse follows again from 2.5 taking

HE ker P for 1 n il";;E.

Now we consider the compactness condition used in [11] and (16]

to define Ja. :

31

2.7. TEJa if and o'!l1.y if for every E>O there is a set 5CH ~ith aardina1. po~er striat1.y 1.ess than M suah that for every

a x E H ~ith Ilx I <; 1 • there is s E S with IITx - s I < E (in other ~ords. 5 is an E-net for {Tx j Ilx K <; 1 }} •

Proof. Consider the case a> 1. Let T E J and {P } as in 2.5. ~ n

For given E > 0 , choose n large and set 5 =. {TP nX j Ix» <;1}. Now for x E H satisfying Hx II <; 1 if s = TP x we have RTx- s I = n = UTx-TP nX R < E. Obviously the cardinal power of 5 is not larger than ~1-rank P <~. If a = 1, rank P <;~ and 51 = {TP x;1x1l<;1}

nan 0 n is separable, so choose for 5 a countable set dense in 51' If a = 0, we already observed that J a C C (H), so that {Tx .; Ux U <; 1 } is relatively compact, and therefore totally bounded. In all ca ses, then, the "only·if" part of 2.7 is proved. Consider the "if" part: assume T satisfies the condition in 2.7, and let K be the closed subspace generated by S, P the orthogonal projection on K. For Oxll<;1 pick sE5 with UTx-sH<E. Then UPTx-Txy <; <; npTx-psll + OpS-TxH <;Upft UTx-sll + fts-Txll <2E , so PT tends to T. But the subspace K contains a dense subset of the same cardinal power as S (namely, the rational linear combinations of elements o·f S), and therefore rank P dim K = card S < ~a ' so that P , and PT, belong to J a . Thus T = lim PTEJa , as desired.

§3. UNIVERSAL REPRESENTATIONS.

We recall (see [10] or [17, Ch. IV,V]) that if A is a C*-algebra with identity e and involution x ~ x*, A can be faithfully repr~ sented as a closed *-subalgebra of B(HA) for certait Hilbert space HA. The description of HA is as follows: Consider the set L = {p} of all linear functionals p: A ~ C , where C deno­tes the complex numbers, that are positive, i.~., p(x*x)~ 0 for all x E A, and satisfy p (e) = 1. Such P will be called "states" of A. It can be seen that they are automatically continuou~ and that L is a convex subset of the dual of A as a Banach space. For pEL, a,bEA, define (a,b) = p(b*a); (a,b) is semi-bilinear (that is, linear in a and conjugate linear in b), (a,a) ~O and also l(a,b)12<;(a,a)(b,b) (Cauchy-Schwarz inequality). Factoring by the degeneracy set N = {al (a,a) = O} we obtain an inner pro -duct space A/N whose completion is a Hilbert space to be denoted by Hp' Corresponding to each a E A there is an operator apE B(Hp)

32

defined by extending a (x+N) = ax + N by continuity from A/N to H It is plain that II a H Ju aU and that a -> a is a representation 0/

P P A in H , i.e., a homomorphism of C*-algebras: (ab) = a b ,

P P P P (ha) = ha , (a*) = (a )* for a,bEA and A complex. The extreme

p P P P points of L will be called "pure states". When p ranges on the set of pure states we obtain a family {Hp} of Hilbert spaces and

representations a -> apE B(Hp)' However, different PI' P2 may de­termine equivalent representati,ons a -> api ' a -> ap2 in the sense

that for some invertible V: Hp -> H we have Va V-I = a , and I P2 PI P2

this is of course an equivalence each equivalence class we find a

tes. Then HA is defined as HA

relation. By selecting one p in determining subset C of pure sta­

LpECeHp and the representation

u' A ~ B(HA) defined by u(a) = LpECeap is called the universal. re­

presentation of A. We aim to compute dim HA when A = B(H)/J, for J a closed two-sided ideal of B(H). Observe that every two-sided ideal J of B(H) is a *-ideal in the sense that TEJ implies T*EJ [S,Chap. 1, §1, N° 11 and therefore the quotients B(H)/J are C*­

algebras when J is also closed. In §2 we proved that the family £ of closed two-sided ideals of B(H) is order isomorphic to the initial interval [-1,0+1 1 , where

~o = dim H. In particular J o is the largest proper two-sided i­deal of B(H).

3.1. THEOREM. 'Let H be a Hil.bert space of dimension ~o and J a

cl.osed two sided ideal. of B(H) different from the l.argest proper

two sided ideal J o' Then the universal representation of A =

= B(H)/J has dimension ~0+2'

The method of proof is suggested by a counting argument used in [1S] (and credited to I. Kaplansky) that shows that dim HA =

2~o = 2 when A = B(H) and H is separable infinite dimensional.

We need some preliminaries.

Let X be a discrete topological space of cardinal power ~v' and ex its Stone-Cech compactification. Thrn Card ex = ~v+2 ([7, Ch. 3, Problem L, (c)], [21] , [13]). Assume II is an ordinal number and II < v . Denote by Xll C ex the set Xll = U {S ; SeX and Card S .;; .;; ~ } , where S denotes the closure of S in ex.

II

3.2. LEMMA. For every II < v, Card (eX -~) = ~v+2 (where "-" deno­

tes set theoretical. difference).

33

P'I'oof. Choose 8 e X with Card 8';;; N. It is easy to see that S is Il

homeomorphic to e8 (in fact, the injection i: 8 ~ eX extends to a

mapping i': e8 ~ ex and the image i' (e8) is compact and contains 8

as a dense subset), so that Card S=NIl +2 . Thus Card XIl .;;; NIl + 2 • Card' W,

where W is the set of parts 8 of X with Card 8';;;N Obvio ~ Il

usly Card W';;; N Il';;;N 1 ' so that Card X ';;;N 2N 1 = N 1 ' whence v V+· Il Il+ V+ V+ Card(eX-X Il ) = NV+2 as desired.

P'I'oof oj" Theo'I'em 3.1. Let J = J a with a< o. 1hen h(J) = a<a+1 =

= h (rna) and therefcre J e rna ,; B (H) . Hence, if B = B (H) Ima, there

is a natural homomorphism onto A 7 B (where A = B(H)/J). We shall

show that there are NO+2 inequivalent pure states of B, a result

that carries over to A by the homomorphism A'7 B. This can be

done as follows: let X. be a discrete topological space with

Card X = No' and identify H with R.2(X) = {f: X ~ C; Ilf(x) 1 2 <+oO}.

Clearly R.""(X) = {b: X ~ C; 8uplb(x) I <+""} can be identified to a

subalgebra D of B(H): the operator corresponding to b being f(x) ~

~ b(x)f(x). D is then the algebra of diagonal operators, and from

a theorem of Krein [3, Ch. VI] or [17, Ch. V , §23 , N° 3, III],

every pure state of D can be extended to a pure state of B(H). CI~

arly R."" (X) can also be identified to -g(ex) , the Banach space of

all complex valued continuous functions on the compact space eX,

and each xE ex determines a positive functional p : b ~ b(x) , ~ x

where bE't&(SX) is the extension to eX of h-ER.""(X). It can be seen

that Px is a pure state for every XEeX ([13], [24]) and in fact,

these are all the pure states of R.""(X). Denote again by Px ~ pure

state of B(H) extending Px: R.""(X) = D 7 C. Assume now that

xE ex-xa ' and let PE D be the projection associated to the charac

teristic function b s of some subset 8ex with Card 8 = Na . It is

clear from x ~ S, that bs(x) = 0, or p (P) = 0, whence p (p*P) = x x = p (P) = ° and therefore Pp = 0, which implies Tp = ° for all x x x TEma ; this means that p induces a pure state'of B = B(H)/m,

x . a thus a pure state of A = B(Hi);lJ. This shows that there are

Card(eX-Xa) = NiH2' d;:ii£.£eren,t pure states of A. Clearly the mem­

b;ers. in an equi'l·alJ.e.n<l:.6lo class of representations are in one-to-one:

C0>l1'1iespo..n>cl1en.ce, with iillt.'le'Jt1!:iiiDJle O-peJta tors V E B (H), which means that

e.-ac.h elias.!> colittaLii1!li.£ at most Card B(H) representations. Now h'o,m

the liIllttrix repJtesemtathH1J of operators follows that Card B (H) . .;;;

S;; N.o+1 ' and therefore if C = {p} contains a pure state in each equj

valence class, we have Card C .No+1 ;;;. Card (eX-Xa) = No+2 ' whence

Card C;;;' No+2 . Thus, since HA = I pECEDHp' we have dim HA;;;' Card C p

~ No+2 . We need now estimates for dim HA and Card L (L = set of

34

all states). Clearly L C CB(H). so that Card L';;; l\ 0+1 = l'I0+ 2 . Also,

A/N, where N = (T ; p(T*T) = OJ is dense in H , so that dim H .;;; p p

.;;; Card A/N';;;Card B(H) = l'I8+1' Finally, dim HA ';;;LpEcdim Hp '"

.;;; Card C.l'I o+ 1 ';;; Card L.l'I o+ 1 .;;; ~+2l'10+1 = l'Io+2 ' We conclude that

dim HA = l'Io+2 ' as claimed.

3.3. REMARK. This proof does not actually depend on the conti·

nuum hypothesis when J = (OJ . Thus: "if dim H = d, the dimens7.on

d

of the universal representation of B(H) is equaZ to 22 ". can be

~Ct obtained without assuming that Z = ~Ct+1'

35

REFERENCES

(1] Earl BERKSON and Horacio PORTA, Rep~e~enzaz~on~ 06 SIX), J.

Funct.Anal. 3 (1969), pp. 1-34.

[2] J.W. CALKIN. Two-~~ded ~deal~ and eOKg~uenee~ ~n zhe ~~ng 06 bounded ope~azo~~ ~n H~lbe~z Spaee, Annals Math. 47 (1941),

pp. 839-873 (abstract in Bull. A.M.S., 46 (1940), p. 400

under the title A quoz~enz ~~ng ove~ zhe ~~ng 06 bounded op~ ~azo~~ ~n H~lbe~z ~paee, I).

[3] Mahlon M. DAY; No~med l~nea~ ~paee~, Springer-Verlag, Berlin

(1962) •

[4] Jacques DIXMIER, Appl~eaz~on~ dan~ le~ ann€aux d'op€~azeu~4, Compos. Math. 10 (1952), pP. 1-55.

[5] --------------- Le~ Alg~b~e~ d'opi~azeu~~ dan~ l'e~paee h~l

be~z~en, Gauthier-Villars, Paris (1957).

[6] Nelson DUNFORD and Jacob T. SCHWARTZ, L~nea~ ope~azo~~, I, II,

Interscience, New York (1958) and (1963).

[7] Ryszard Engelking, Ouzl~ne 06 Gene~al Topology, North-Holland­

Polish Sci. Publ. (1968).

[8) Zden~k FROLIK, Gene~al~zaz~on4 06 Compaez and L~ndel~6 4paee4, (~n ~u~4~an), Czech. Math. J., 9 (1959),pp.172-217 (Eng. Sum.)

[9] Istvan S. GAL, On zhe zheo~y 06 (m,n)-eompaez zopolog~eal ~paee~, Pacif. J. Math. 8 (1958), pp. 721-734.

{10] Israel M. GELFAND and Mark A. NAIMARK, On zhe ~mbedd~ng 06 no~med ~~ng~ ~nzo zhe ~~ng 06 ope~azo~4 ~n H~lbe~t ~paee,

Mat. Sbornik, 12 (1943), 197-213.

Ill] Bernhard GRAMSCH, E~ne Ideal~z~ukzu~ Sanaeh4he~ Ope~ato~­

algeb~en, J. Reine u. Angew. Math., 225 (1967), pp. 97-115.

(12] ----------------, Abge~ehlo~4ene Ideal4 ~n Ope~az~~algeb~en zopqlog~~ck~ Vekto~~~ume. J. Reine u. Angew. Math. 226

(1966), pp. 88-102.

-[13] Felix HAUSDORFF, Ube~ zwei Sazze von G. F~ehzenholzz und L.

Kanzo~ov~teh, Studia Math. 6 (1936), pp. 18-19.

36

[14] John R. ISBELL, Rema.JI.lu. on .6pac.e.606 £.alLge c.alLdina£. numbelL, Czech. Math. J., 14 (1964), pp. 383-385.

[15] Richara KADISON and Isadore M. SINGER, Ex.tc.n.6 (o~/.,~ tlu PL~lLe

S.ta.te.6, Amer. J. "".th. 81 (1959), pp. 383-400.

[16] Erhard LUV I~ .two-.6ided c.£.o.6ed idea£..6 D6 .the a£.geblL~ 06 bounded'11 pelLa.tolL.6 06 a Hi£.belL.t .6pac.e, Tech. Rep.,

Univ. British Columbia (May 1967).

[17] Mark A. NAIMARK, No~med Ring.6, P. Nordhoff N.V., Gromingen

(1964).

[18] Horacio PORTA and Jacob T. SCHWARTZ, ReplLe.6en.ta.tion.6 06 .the a£.geblLa 06 a£.£. opelLa.tolL.6 in Hi£.belL.t Spac.e, and lLe£.a.ted Ana­£.y.tic. Func..tion A£.geblLa.6, Comm. Pure Appl. Math. 20 (1967),

pp. 457-492.

[19] Horacio PORTA, Id~aux bi£.a.telLe.6 de .tlLan.660ILma.tion.6 £''<'n~ai­

lLe.6 c.on.tinue.6, Comptes Rendus Acad. Sci. Paris, 264 (1967),

pp. 95-96.

[20] ------------- A no.te on homomo!Lph'<'.6m.6 06 Ope!La.to!L A£.geb!La.6, ColI. Math. 20 (1969), pp. 117-119.

[21] Bedrich POSPISIL, On b.<.c.ompac..t .6pac.u, l'ubl. Fac. Sci. Univ.

Masaryk, N° 270 (1939), pp. 1-16.

[22] ----------------, RemalL~ on b.<.c.ompac..t .6pac.e.6, Ann. Math.

38 (1937), pp. 845-846.

[23] M.K. SINGAL and Shashi Prabha ARYA, On m-palLac.ompac..t .6pac.e6, Math. Ann. 181 (1969), pp. 119-133.

[24] Marshall STONE, App£..<.c.a.t.<.on.6 06 .the IheolLY 06 Boo£.ean !L.<.ng6 .to genelLa£. .topo£.ogy , Trans. A.M.S. 41 (1937), pp. 375-481.

[25] F.B. WRIGHT, A !Leduc..tion 601L a£.geblLa.6 06 6.<.ni.te .type, Ann.

Math. 60 (1954), pp. 560-570.

Recibidu en julio de 1970.

urbana r 11 inois

Revista de la Dni6n Matemiiica Argenti.a Volumen 25, 1970.

UNIQUENESS OF DISTRIBUTldNS by A. P. Calderon*

Vedie~do 4l P~o6e~o~ Albe~to Gonz4lez Vomlnguez

1. INTRODUCTION. Let f be a distribution on an open set 0 in Rn

and ~(~) an infinitely differentiable function with support in 1~1<1 and integral equal to 1. Let ~t(~) = t-n.(~/t? andconsi­der the function

(1 ) t>O

i.e. the convolution of f and ~t' which is well defined for ~ at distance larger than t from the complement 0' of O. We are inte­rested in the-extent 'to which lim F(~,t) determines f. If 0 coin-

t .... o

cides with Rn we, may replace tHe condition on the support of • by one less restrictive but sufficient to insure the existence of F(~,t). For example,if f is the Fourier transform of a function g and the integral of Igl over a sphere of radius 2' does not grow faster than a fixed power of 2', we can take ~ to be the Fourier

transform of e-I~I and F(~,t) becomes the Abel means of the Fourier integral of g. Thus in this case our problem becomes that of uni­queness of Abel summable Fourier integrals. We shall consider ~wo modes of approach of the point (~,t) to the hyperplane t ~ 0, name­ly, non-tangential and normal. Results on non-tangential limits are relatively' simple and require no conditions on f. Furthermore', they have interesting applications to the theory of partial diffe­rential equations. On the other hand, results on norm.al limits re­quire some restrictions on f. The ones we present here are closely related to the work of V. Shapiro (see [3)).

2. STATEMENT OF RESULTS.

We shall always assume that f and • are real an we shall associate with them the functions

F(~) = lim F(y,t) , F*{~) lim F(y,t) , t>O , t .... 0 , I~-YI <t

* This research was partly supported by NSF GP - 23563

38

where the upper limit F is taken through aZZ positive values of t

and the lower limit F* only through a fixed but otherwise arbitrary

sequence tending to zero. These are non-tangential limits. We al­

so consider the normal limits

f * (x) = lim F (x, t) , [(x) = lim F (x ,t) , t > 0 , t ->- 0

where, again, f is taken through all positive values of t and f*

through an arbitrary but fixed sequence tending to zero.

THEOREM 1. Suppose that 4> (x) ;;. 0 has support in I x I";; 1 and inte -

graZ equaZ to 1. Suppose that h is ZocaZZy integrabZe in 0 and v

is a measure which is finite on compact subsets of 0 and such that

F(x) ;;'h(x) aZmost everywhere in 0 and either F*(x) >_00 or

(2) I F (y , t) I - F (y , t) = 0 [( v * 4> t) (y)] , Ix-y I ..;; t

as t tends to zero through the sequence defining F*, for aZZ x in

O. Then f-h coincides with a measure in O.

Here, and throughout this paper, "measure" means a non-negative

measure, and to avoid tedious repetitions we shall always assume

that a measure in an open set is finite on its compact subsets.

THEOREM 2. If in the preceding theorem we have 4> (0) > 0, h (x) 0

aZmost everywhere and [(x) ..;; 0 everywhere in 0, then f o.

THEOREM 3. The assertions of the preceding theorems remain vaZid

if condition (2) is repZaced by the foZZowing: the set of points

where F*(x) ~ -00 is a countabZe union of sets E such that if x t denotes the characteristic function of the set of points at dis­

tance Zess than't from E then

remains bounded as t tends to zero, and fol' I x-y I ..;; t and x E E,

F(y,t)ta tends to zero uniformZy as t tends to zel'O thl'ough the

sequence defining F*. The number a may depend on the set E.

This result has interesting applications to the theory of linear

partial differential equations. They are extensions to general e­

quations or systems of familiar facts about analytiC functions such

as the theorem of Besicovit.:h (see [2] , chapter V, tho 5.3) or the

39

theorem of Looman-Mencho£f. Although the result stated below, when

specialized to the case of the Cauchy-Riemann equations, neither

implies nor is implied by the theorem of Looman-Menchoff, it is of

the same general character. For some recent results in the same

direction see also [4J. In order to state our results we must re­

call some generalized notions of differentiability introduced in

[11. If h is a locally integrable f~nction in an open set 0 we

shall say that h belongs to T! (x 0) , a;;' -n , if there exists a po­

lynomial P of degree less than or equal to a, P = 0 if a < 0 , such that

where X t (x) is the· ::d", "'.'ristic function of the sphere of radius

t with center at tlw nri",in, remains bound.ed as t .... O.Ifonthe other

harid this expression 'cdds to zero as t .... 0 , we say that h be -

longs to t1(x O)' If h belongs to t1(X O) for all Xo in a set E .a a

with the co~fficients of the corresponding polynomials bounded in

E and the prec~ding expression tending to zero uniformly, we say

that h belongs to t!(E). When h belongs to t!(x o) the coeffi -

cients of P are uniquely determined and one defines the generali­

zed derivatives of h of orders less than or equ~l to a at Xo as

the corresponding derivatives of P at the origin. Thus if L is a

differential operator of order less than or equal to a, (Lh)(x o) can be defined accordingly. These notions can be generalized,in

the obvious way to the case of vector-valued functions.

THEOREM 4., Let L be a system of linear partial differential opera­

tors of order m with coefficients of terms of order k in Cm- k . Su£

pose the vector-valued function h belongs to t~(x) for almost all

x in an open set 0 and satisfies the equation L h = 0 there. Sup­

pose that at the remaining points x either h belongs to T!(x) or

else x be longs t'o a coun tab le union of se ts E such that hE t 1 (E), m-a

o<a = aE";;n, and, if xt(x) denotes the characteristic function of

the set of points at distance less than t from E,

is finite and remains bounded as t .... O. Then h is a weak soZution

of the system L h = O.

We may complete this statement with the observation that if L is e­

lliptic determined or overdetermined and has infinitely differentia

40

ble coefficients then h coincides almost everywhere with an infini tely differentiable function.

We pass now to the results on normal limits. They are contained in the following

THEOREM s. Le~ f be a distribution with aompaat support and ¢(x)

an infiniteZy differen~iabZe funation suah that <I>(x) = n(lxl) where

net) is a non-deareasing funation of t suah that n{k) (t)

O(t- n - k - E ) as t ~~. Suppose that ~he Fourier transform f of f s~ tisfies the foZZowing aondition

a) J If(z) I dz = a (r2) as r ~ ~ Izl<r

Let the funation h be ZoaaZZy integrabZe in the open set 0 and sup­

pose that F(x) ;;'h{x) aZmost everywhere and iJx) > -00 everywhere in

O. Then f-h aoinaides with a measure ~n u.

The aonaZusion remains vaZid if the aondition that [(x) > -~ is re­

pZaaed by the foZZowing weaker one. There exist a aZosed non-dense

subset C of 0, a measure v in 0 and a non-inareasing funation

A{t»l in (0,1) with A(t) -;. ~ as t -;. O. suah t.ha~. if X denote

the aharaateristia funation of the sphere Ixl ~1 , then

at every x of C, and if x is a point of O-C suah that f(x)

then

(3)

ft ).(t) [IF(x,t)1 - F{x,t)] dt<~ and o _

IF(x.!;)1 - F(x,t) = a [(v*<P t ) (x)] as t -;. 0

-THEOREM 6. Let f and <P be as in the preaeding theorem. Suppose , that the Fourier transform f of f satisfies the aondition that

r=q=l.

Let h be a loaaZZy integrabZe funation and v a measure in an open

41

set O. Suppose that F(x) ;;'h(:x:) a'Z.most everYIAlhere in 0 and either

!J:x:) > -00 or

as t .... 0 for an :x: in O. Then f-h ooinoides IAlith a measure in O.

THEOREM 7. Suppose that under the assumptions of either of the "two

preceding theorems IAle have h(:x:) = 0 and tJ:x:) <0 eve1'yIAlhere in 0

Then f = 0 in O.

We note that condition a) in theorem 5 is closely related to the condition on the coefficients in the theorem on uniqueness of Abel .summable Fourier series of Verblunsky-Shapiro (see [3]). We shall see that, as in the case of Fourier series, it cannot be renlaced by the weaker condition 0(1'2.). This will be shown in the last se'=. tion where we also give an example illustrating the limits of pos­sible improvements of theorem 6 by exhtbiting an f such that

fez) = oClzl-(n-3)/2) as Izl +- 00 and that F(:x:,t) .... 0 as t ... 0 for

all x.

3. We start with some lemmas which will be used in the proof of our results.

LEMMA 1. Let F", be defined by 'Z.etting t tend to zero through a se­

quence S. Suppose that F",(:x:) > -00 for al.l x in an open set O. Let

C be cZosed and such that en 0 is non-empty. Then there exists an

open subset 0 1 of 0 with cn 0 1 non-empty and such that F(y.t) ;>-N>

> -'" in the set Iy-xl..;;t. xECno l • tES.

Proof. Let £(x) = inf F(y,t) , Iy-xl";;t , tES. Then £ is upper semicontinuolls and everywhere finite in O. Consider t'he sets {£(x);>-k ,xECno} , I<. ='1,2, •... They are relatively closed in cnO and their union is cno. Since cno is of the second cat~ gory in itseJf, one of these sets contains a non-empty relatively open subset C no 1 of C no. This prove~ the lemma.

LEMMA 2. Suppose that !Jx) > -'" for al.l x in an open set O. Let C

be cZosed and such that C n 0 is non-empty. Then there exists an

open subset 0 1 of 0 IAlith cnol non-empty and such that F(x.t);>-N>

42

> -co for x i~ C n O~.

The proof of this is almost identical to that of the preceding lemma and is left to the reader.

LEMMA 3. Let f .be a distribution in an open set 0 and", >0· have

support in Ix I < 1. Suppose that F(x. t) >-N >-co for x EO. t ES and

t < Il. and Let F (x) > 0 aLmost everywhere in O. Then f coincides

with a measure in O.

Proof. Let ",(x) = ~(-x) and z;(x»O be infinitely differentiable and supported in O. Then

-Since F(x,t»-N for xEO and tES, and since 1;*"'t and all its de-rivatives converge uniformly as t + 0, if we let t +0 through swe will have

Thus the distribution f+N is such that (f+N) (r;;) > 0 for every 1; > 0

with support in 0 and therefore coincides with a measure ~l in O. Let now g be the Radon-Nikodym derivative of ~l with respect to Le besgue measure and v i tss ingular part. Then

Now, at almost all Xo the derivative U~·v(St)/IStl , where St is

the sphere with center at Xo and radius t, exists and is equal to zero, and at such points the first integral above tends to zero as

x + .x o and t + O.with Ix-x o I < t. On the other hand, at every Le­besgue point Xo of g the second integral above tends to g(xo) as x + Xo and t + 0 with Ix-xo I < t. Thus we have

g ex) = F ex) + N;;' N

almost everywhere in O. This shows that ~l -Ndx is still non-nega­tive whence it follows that f coincides with the measure

~ = \l 1 - Ndx in O.

LEMMA 4. Let f and", be as in theorem 5. Suppose that F(x.t) >-N

43

for x in an open set 0 and tES and let F(x);;;'O almost everywhere

in 0. Then f coincides with a measure in 0.

Proof. The argument used in proving the preceding lemma applies

to the present case with only minor changes. We first observe

that on ,account of the properties of ~, if g is a distribution

with compact support then g*~t + 0 as t + 0 in the complement of

the support of g. Then, as in the previous lemma, we show that

f+N = ~j in 0. Given an open subset 01 of 0 and an infinitely dif

ferentiable function n which is equal to 1 in ~and vanishes out­side 0 we will have that

( n ~ j *~ t) (x) - F (x , t) + 0

as t + 0 for all x in OJ' and arguing with n~j as we did above

with ~j it will follow that n~j - Ndx is non-negative in OJ. Since

OJ is an arbitrary open set regularly contained in 0, the desi~ed result will follow.

LEMMA S. Let f and ~ be as in theorem 5. Suppose that f+N coinci

des with a measure in an open seJ ° where F (x) ;;;. 0 holds almost eve

r'ywhere. Then f itself coincides with a measure in 0.

This was shown in the second part of the proof of the preceding

Lemma.

Proof of theorem 1. At first we shall assume that h = 0 and v = 0,

i.e. that F(x);;;'O almost everywhere and F*(x» -'" everywhere in 0.

Then according to lemma 1 every open subset of ° contains a neigh­borhood where F ex, t) is bounded below uniformly for t E S and where

consequently according to lemma 3, f coincides with a measure. Now,

if f coincides with a measure in every set of a family of open

sets, it coincides with a measure in its union. Thus there exists

a maximal open subset OJ of ° where f coincides with a measure.

Now, suppose that OJ is a proper subset of 0. Then according to

lemma 1 there exists an open set °2 , OJ CO 2 cO, containing OJ pr~

Derly and a number N such that for yE0 2 -O j , Ix-yl<;'t, tES we

have F(x,t);:;'-N. Let now 03 be the set of points of 02 at distance

greater than E from its complement. If E is sufficiently small,

then 03 contains points not in OJ. Consider now F(x,t) with x E03

and tES, t< E. If x is at distance greater than t from the com­

plement of OJ' since ~t is non-negative and has support in Ixl < t

44

and since f coincid~s with a measure in 01 we have

If the. distance between q; and the complement of 01 is less than or equal to t. then there exists a y in 02-01 with /q;-y / <; t and we have F(q;.t»-N. Consequently F(q;.t) is bounded below in Os

for t E Sand t < e and according to lemma 3. f coincides with a measure in Os' But Os is not contained in 01' and thus 01 is not maximal. a contradiction. Hence 01 coincides with 0, and the the orem is established in this special case.

In the general case. given a large integer Nand e > 0 we let g be the distribution defined by g = f-hN + e\l. where hN(q;) = hero) if

h (q;) <; Nand hN(q;) = N if h (q;) > N. Then. as readily seen, we have G(q;) > 0 almost everywhere in ° and G", (q;) > -III everywhere in 0.

Hence g coincifles with a measure in ° and for every testing func­tion n. n > 0 • we have

g(n)· fen) - J hNn dq; + e J n d\l;;;'O

Letting N + + ... and e + 0, we find that

f (n) - J h n dq; > 0

whence the desired conclusion follows.

Proof of theorem 2. Since h (q;) > 0 almost everywhere 'in ° , acco!. ding to theorem lour distribution f coincides with a measure ~

in 0. Suppose that ~ ~ O. Then there exists at least one point q;o such that, if St denotes the sphere with center at q;o and ra­dius t. lim t-n ~ (St) > 0 as t + O. Now, if 4> (q;) > e for /q;1 < e , since + > 0 , we have F (q;o ,t) ;;;. t -n e~ (StE)' and consequently !(q;o) > 0 • a contradiction. Hence we must have ~ = 0, and our a~ sertion is established.

Proof of theorem 3. We start with the observation that if F", is redefined by making t tend to zero through an arbitrary subseque!!. ce of the one originally used to define F"" the hypotheses of the theorem will still be satisfied and the conclusion will be proved if we .how that a proper choice of the subsequence will imply the existence of a measure \I satisfying (2). Let

4S

where ! x-y! ..;; t , x E Em and t tends to zero through the sequence de fining F*. Let us select a subsequence tk of this such that

Lk E;/2C t k )";;Mm<oo for all m. If x~Cx) denotes the characteris­

tic function of the set of points at distance less than Zt from

Em ' let

Consider now the function

-(X () \ 1/2(t ) Z-m m t m M-1N- 1

g x = Lk,m. Em k Xtk(x) k m rn

Then if xEEm ' !x-y!";;t = tk we have

m X t (Z) ~ t (Y - z) dz

k k

which shows that for xEU Em ' !x-y!";;t k we have

as tk + O. Since g is clearly an integrable function, its indefi nite integral gives the desired measure v.

Proof of theorem 4. For simplicity we shall restrict ourselves to the case of a single differential operator of the form

Furthermore, and without loss of generality, we may assume that the coefficients as belong to cm-I 131 in the closure of O. Let now Q be a polynomial and R = h-Q. Consider the distribution f = Lh and let <p be as in theorem 1. Then

The first term in the last expression is dominated by a multiple

46

of a bound for the coefficients of Q, and for the ~cond we have

~ c t-n-mJ IR(y) Idy Ix-y I <2t

1 where c is a constant. Thus if h belongs to t m_a (E) , a > 0 , and

Xo E E , by setting Q(x) = P(x-x o) , we see that for Ix-xo I ~t

as t -+ 0, and consequently F(x,t) t Cl -+ 0 as t -" 0 with Ix-xol~t

uniformly for Xo E E.

we s'ee that F (x, t) remains bounded or tends to zero as t -+ 0 with

I x -x 0 I ~ t. Thus, according to theorem 3, f coincides with a measu

re in O. Since the same conclusion holds for -f, it follows that

f = 0 in 0 and thus h is a weak solution of the system Lh = 0 in O.

The proof of theorems 5 and 6 will require a few more lemmas.

LEMMA 6. If f is a distribution with compact support C, there

exists a g with compact support such that ~g-f is infiniteZy dif­

ferentiabZ~ and vanishes in a neighborhood of the support of f.

Proof. Let ~ (x) be an infinitely differentiable spherically sym­

metric function with support in Ixl < 1 and integral equal to 1

Let Z(x) be the fundamental solution of Laplace's equation and

h = l - (l *~). Then h = 0 - I; , where 0 denotes Dirac I s delta

function, and since l (x) is harmonic in Ix I > 0 and I; (x) is sphar!.

cally symmetric and has support in I.x I < 1, the mean value property

of harmonic functions implies that hex) = 0 for Ixl > 1. Consider

now the function gl = (f*I;)*Z . Evidently, gl is infinitely dif­

ferentiable and gl = if*l;) has compact support. Now let g =

(h *1) + ljIg 1 ' where ljI E C~ and ljI = 1 in a neighborhood of C

~ g ~ (hd) + ~ (ljIg ) 1

f - ~[(1-ljI)g 1 • 1

47

Thus, ~g - f = 81(1-~)gl I vanishes in the complement of the sup­port of (1-~),. i.e. in a neighborhood of C. Furthermore, since 1/1

and gl are infinitely differentiable, the same holds for dg - f.

LEMMA 7. Let f and ~ be as in theorem 5. Then (f*~t)(x) ... 0 as

t ... O. uniformly in the aomplement of any neighborhood of the sUR

port of f.

Proof. According to our assumptions the derivatives of order k

of Hx) are of the orde.r Ixl-n-k-e as Ixl ... "". Thus, as t ... 0 , ~t and all its derivatives tend unifo.rmly to zero in Ix I > ~ > 0 • This clearly implies the assertion of the lemma.

LEMMA 8. Let n(.t) be as in theorem 5 and let 1/I(x) = -ni(lxl) Ixl-1. Then if g is a distribution with eompaat support t-2(g*~t)(x) ... 0

as t ... O. uniformly outside any neighborhood of the support of g.

Proof; Again, according to our assumptions, ~ and its derivative~ of order k are of the orders Ixl-n-2-e and Ixl-n~k_2-e respective­ly as Ixl ...... This implies that t-2~t(x) and all its derivatives converge to zero uniformly in I x I > ~ > d, ·whence the des ired concl!! sion follows .

. LEMMA 9. Let ~ be as in theorem 5 and ~ as in lemma 8. Let g be

a distribution with aompaet support, f 8 g •

F(x.t)

Then

Proof. One merely has to verify that

which implies that

tF (x ,t)

48

LEMMA 10. Let g be a distribution with compact support. Suppose

that g coincides with a function g in an open set O. Let g be uR

per semicontinuous in 0 and lim (g1cl/lt) (x) = g (x) ;;;, -a> for an x in t .... o

0, where IjJ is the function in lemma 8 normalized so as to have i~

tegral equal to 1. Then if f = 6g and f(:x:) > 0 in 0, g is subhar­

monic in O.

Proof. We must prove that if xoE 0 and S is a sphere contained in 0 with center at Xo then

whe~e lSI is the surface area of S and do is the area element. To show this we let B be the closed ball with boundary Sand gl = g in Band gl = 0 in the complement B' of B. Let h be continuous in B harmonic in the interior of B and vanish in B'. Suppose

that gl <;'h on S. Since gl-h is upper semicontinuous in B, it ta­kes a maximum M at a point Xl in B. Suppose that Xl is in the in terior of B, at distance e: > 0 from B'. Since g = 'if = gl in the interior of B, if tl < t and t is sufficiently small, according to iemmas 8 and 9 and the fact that [(xl) > 0 we will have

and letting tl .... 0 we obtain

(g 1 * IjJ t) (X 1) - g (X 1) ;;;, ~ i (X 1 ) t 2 + 0 ( t 2) •

On the other hand, since IjJ is "pherically symmetric and h is har­monic in the interior of B, we have

= J [hey) - h(x l )] IjJt(x l -y)dy = 0 (t 2 )

Ixl-yl>E

Finally, since 1jJ;;;'0 and gl- h<;'M in B and vanishes in B', we have

M-M J IjJ (:x: -y)dy = M +o(t 2 ). B' t 1

49

Combining these estimates we obtain

" 2 2 M + 4' fCred t + oCt)

that is

which is impossible, since fCre 1 ) > O. Thus the maximum occurs on S, and since il <;h on S, we have il -h <;0 in B. Thus

Since this holds for any h with h;;.{j on S, the desired conclusion fOllows.

LEMMA 11. Let g be a dist1'ibu-tion bJith "ompaat support and 1/1 the

fun"tion in Zemma 8 normalized so as to have integral equal to 1.

Suppose that as t .... 0, (g*W t ) (re) ..: i(re) ;;'_00 for aZZ x in an open

£let 0, bJhe1'e the fun"tio,n g(x) is upper semiaontinuous and loaaZly

integ1'abZe in O. Let f '" b.g and F(x);;'O almost eve1'ybJhe1'e and

fJre) > -00 eve1'ybJbe1'e in 0'. Then if g aoinaides bJith g in 0, f aoin

aides bJith a measure in O.

Proof. We use the well known fact that an upper semicontinuous locally integrable function g is subharmonic in an open set 0 if

and only if ~ coincides with a measure there. Let 01 be the la£ gest open subset of 0 in which f coincides with a measure. Then, since.,.;;' 0, lemma 7 implies that !.Cre);;' 0 in 0 1 , If 0 1 is a proper­subset of 0, then, according to lemma 2, there exists an open sub­set O2 of 0, containing 0 1 as a proper subset, such that!. > -N < 0 in O2 -0 1 , Let xCre) be the characteristic function of the set O2

and let gl g + :n X Cre) Ire 12. Then from lemma 8 and the fact

f + N it follows that gl satisfies the conditions of lemma 10 in O2 and therefore it coincides with a subharmonic func­tion anc b.g 1 '" f + N coincides with a measure in O2 , But then, since F(x);;' 0 almost "everywhere in 0, by lemma 5 we conclude that f coincides with a m,easure in O2 , This contradicts the assumed

so

maximality of °1 , Thus 01 cannot be a proper subset of u and the proof of the lemma is complete.

LEMMA 12, Let g be a distribution with compact support and ret

f = tog satisfy condition a) in' theorem 5. Let z;{x) ;;;'0 be spheri-

cally symmetric, infinitely differentiable, with support in Ixl < 1,

and such that f 1;; dx = f lj! dx , where lj! is as in lemma 8. Then

as t ->- 0, uniformly in x and y, provided that ix-v I <,t.

Proof. On account of the properties of ~ and lj! we have the ine­

quali ties

where c is a constant. Conseque'ltly, ,~- ix-y! <'t, then

r ~ ,- - [-2-1i(x.z) o-2'ITi(y.z)1~,(tz)d" I,;;: ! g Lz J e - '" ~ _-

On the other hand, since ~ and ~ have bounded flrst order deriva­

tives and 2(0) = ~(O)

I (g*!;;t) (x) - (g*lj!t)Cx) I = I r g(?) e- 2rri (x.z) [((tz) - i1(tz)] dzl <, J

and it will suffice to show that the two last integrals tend to

zero as t ->- O. To see this let

r Ig(?)llzl , I z I <u

II :: E (u)

Then a) implies that E(U) ->- 0 as u ~ ~ and

- 1 r t

t J o

_I u

t - I

dt- i ) + t r du) du J 0

51

which tends to zero as t + O. On the other hand

and this also tends to zero as t + 0 , and the lemma is establi­

shed.

LEMMA 13. Undep the aonditions of the ppeaeding lemma we also

have

as t + 0 , unifopmly in x and y, ppovided that Ix-y I <,t.

This is an immediate consequence of the preceding lemma which was

incidentally established in the course of its proof.

LEMMA 14. Let h(x)ELP(Rn ), p>1, n>2, and let

g(x) = J h(y) Ix-y I-n + tx dy , n > txp > 2

if the integpal is absolutely aonvepgent, g(x) = -00 othepwise

Then if

the funation k(x) = g(x) - E kj(x) is uppep semiaontinuous fop

evepy E, E > o.

Ppoof. Since, as readily seen, Ixl-n+ tx is integrable to the power

p/(p-1) in the complement of any neighborhood of the origin, the

contribution to the integrals in the lemma from Ixl ~N is continu­

ous in Ix I < N. Thus it will suffice to prove the lemma under the

assumption that h has support in Ixl <N. Let q = p/(p-1) and

p = I (n-tx) - (n-2)/plq. Then p<n and Holder's inequality gives

Jlh(y)llx-yl- n+ tx dy <,kj(x)l/p [J Ix-yl-P dY ] l/q I y I <N

This shows that if k j (x) is finite then the integral defining g is

absolutely convergent. Let us denote now by I] (x,t) and 12 (x,t)

S2

the integral defining g extended over Ixl < t and Ixl:>t respecti~ vely, and de.fine similarly J 1 (x,t) and J 2 (x,t) with the integral expressing kl (x). Then, from Holder's inequality again, we obtain

Let now Xo be a point such that kl (x o) < 00. Then, if IXo~xl·tIZ, we have 2Ix-yl:> Ixo-yl for Ix-yl ~t and, consequently the inte -grands of 12 (x,t) and J 2 (x,t) are dominated by a multiple of the int.egrands of 12 (x O '0) and J 2 (xo'0) respectively. But, since kl (x o ) is finite, the last two integrals are absolutely conver -gent, and this implies that

as x ~ xO' On the other hand we have

< s~p [ a slip t(n-~}/q - E S ] _ (alpE)l/(p-l) t(n-~)/p

whence it follows that

lim [I 1 (x. t ) - E J 1 (x. t) ] < 0 t .... o

Combining this with our previous result we obtain

lim k (x) = lim [I2 (x , t ) - e: J 2 (x , t ) ] + lim [II (x, t) - e: J 1 (x, t)] <

as x .... xo' Suppose now that kl (x o) ~ 00. Then since, as readily seen, kl (x) is lower semicontinuous, we have lim kl (x) = 00 as x .... x o •

and since

it follows that

lim k(x) = lim [g(x) - e: kdx )] x~xo

and the proof of the lemma is complete.

53

Proof of theorem 5. We start observing that, without loss of ge­

nerality,·we can make some additional simplifyi'ng assumptions. In

the first place, lemma 6 shows that with only a harmless altera -

tion of f we may assume that f = 6g , where g has compact support.

Furthermore, by restricting our attention to subsets of 0 if nece

ssary, we may also assume that h is integrable and v totally fini

te in O. Evidently, it will suffice to prove our theorem in the

case when h is bounded above, and subtracting from h an appropri~

te infinitely differentiable function with compact support we can

further reduce the proof to the case h';;; O.

In our proof we shall need some auxiliary functions and distribu­

tions we now introduce. We extend h and v to all of Rn by set­

ting h(z) = 0 outside 0 and v = 0 on every set not intersecting O. We choose an arbitrary positive number c and applying lemma 6 we

let g2 be a distribution with compact support such that

6g 2 - (cv - h) is infinitely differentiable and has support at

distance not less than 1 from O. We set 6g 2 = f2 ' fl = f + f2 '

gl = g + g2 and, as before, we let F(z,t} = U*<Pt)(z) , and defi­

ne similarly FI(z,t) and F2 (z,t).

We shall first prove some properties of the functions F and F]

Let us begin by showing that t 2 F(z,t) ~ 0 as t • 0 , uniformly in

z. Let 6(8) be defined by

f lfez) I dz , I z I .. 8

Then condition a) in our theorem implies that <I(s) ,0 as U' ~'.

Since .(x) is integrable and has lntegrable derivatives of all o~

ders, its Fourier transform;; satisfies the inequality I;(z) I .;;; .;;; a (1+lzl)-3 with some constant c. Consequently

Ijrz) 11~(tz) I dz .;;;

.;;; ct' I'D (1+8t)-3 do (8)s2 = 3ct 3 J"'<5 (8)s2 (1+8t)-"ds

o 0

f '" 2 - 4 3c <5 (8/t)8 (1+8) ds

o

and the last integral evidently tends to zero as t > O.

Next we shall show that FI (x) ~o almost everywhere and £1 (z) > -~,

54

everywhere in O. On account of the fact that f2 - (EV - h) is an

infinitely differentiable function with support disjoint from 0, we have

F (x,t) 2

as t + 0 for every x in O. Now, if x is a Lebesgue point of hand

y ,.. x and t + 0 with Ill-x I ';;'t we have lim (h*$t) (y) = hex) and can

sequently, since (v*$t) (y) ;;;'0,

F (x) = lim F (y,t);;;. lim F(y,t) + lim F (y,t);;;' F(x) - h(x);;;' 0 I I 2

On the other hand, since h';;'O , we have F 2 (x,t) ;;;',;(v*$t) (x) + 0(1)

as t .... o. This evidently implies that .fl (x) ;;;'.f(x) everywhere in O~ Now, if x is a point of 0 not in C and such that .f (x) = -00 ,

then from condition (3) in our theorem it follows that .fl(x) ;;;'0.

If x is a point of C then lim t 2 (v*x t ) (x) > 0 and, as we saw above, t+o

lim t 2F(x,t) = O. But since evidently $t(x);;;'o Xot(x) for some t+o positive 0 and 0, we have lim t 2 F 2 (x,t) > 0 , and consequently

t+o lim F (x,t) = +00 , that is, f (x) = +00 Thus in all cases we t+o I -1

have .fl (x) > -00 , as we wished to show.

Let us turn now to the distributions g. Let ~{x) be like the func

tion $ in our theorem, but having support in Ixl < 1. Let 0/ and ~

be related to $ and ~ respectively as in lemma 8 and normalized so

as to have integrals equal to 1. Set G(x,t) = (g*~t)(x) and defi­

ne similarly G1 and G2 • We shall show that lim GI(x,t) = il(x) t+o

exists everywhere in 0, is upper semicontinuous and locally inte-

~rable and coincides with the distribution gl in O. Since

Fl(x);;;'O a.e. and .fl(x) > -00 everywhere in 0, it will follow by Ie!!!

rna 11 that fl coincides with a measure in O. Since fl=f+sv-h in 0; and since s can be taken arbitrarily small, this in turn will

imply that f-h coincides with a measure in 0, which is the asser­

tion of the theorem.

We begin with some observations. By lemma 9 we have

G (x, t ) 2

G (x ,t ) 1

o J t 2 S (f * ~ ) (x) ds , e > 0 tIS

55

Then, since ~;;;.O , if x is a point in an open set in which f coin­

cides with a measure, and d is the distance from x,to the comple­

ment of the set, G(x,t) is a non-decreasing function of t for

O<t<:.d. A similar remark applies to G1 and Gz . Thus', since f2

coincides with EV -h ina sufficiently large open set containing

0, Gz(x,t) is a non-decreasing function of t for 0<t<:.1 and all

x in O. About the function G(x,t) we remark that, according to

lemma 12, it can be replaced by (g*W t ) (x) with an error which is a bounded function of x and tends uniformly to zero as t ~ o.

, To show the existence of gl we observe that G 1 = G + G2 • Since

G2 (x,t) decreases as t~O, it has a limit, finite or infinite for

all x in O.

On the other hand, if x is a point of 0 not in C, by lemmas 12

and 9 we have

G (x, t) - a Jl s F(x,s) ds + 0(1) t

where c> 0, and since by condition (3) in our theorem the first

integral in the last expression has a finite limit as t ~ 0, we

conclude t~at, as t • 0, G(x,t) also has a limit, finite or infi­

nite. Combining this with our previous observation we conclude

that gl (x) exists fo. ,,1 x not in C. Finally, if x is a point

of C, since obvious:) .,');;;'a Xot(x) for sonie positive 15 and a,

we have

(f ,* ~t) (x) ;;;. c (c *x <I t) (x) ;;;. a It - 2 0<t<:.1

Consequently

G (x, t) I

G (x ,t) + G (x, t) 2

fir I --(g'~WI) (;x:) + G (x,1) - e t" r(x,s)ds - C J U Cf '" <p )(x)ds+o (1)

2 t .? S

<:. (g*~Jl)(x) + G2(x,1) - c J>,-l[F(X,S)SZ + allds + 0(1)

where, again, C'> O. Since, as we saw, F(x,,'t)s2 -. ° as s • 0, the

la~t expre5sion tends to -~ as t ~ 0, and consequently gl (x) =

S6

We proceed now to show that gl has the required properties and coincides with iTl in 0. At first we shall assume that the set C is empty. Let 01 be the largest subset of ° with the property that g 1 is upper semicontinuous, lo::a:'.:.y integrable and coincides with iTl in.0 1, and suppose that 01 is a proper subset of O. Con­sider the function

f).(s) s llF(x,s) 1 - F(x,s)l ds o

which evidently is lower semicontinuous in O. Since C is empty, on account of condition (3) in our theorem, this function is fini te everywhere in 0. Thus, as in the proof of lemma 1, we conclu­de that there exists an open set O2 , containing 01 as a proper subset, such that the integral above is bounded, say by N, in

1\ - °1 ,

Let now x be a point in 0 1 at distance not larger than s from

02 - °1 , and let s<t<1. Then if y is a point of 02 - 01 and Ix-y I <s , by lemma 12 we have

(4)

where aCt) tends to zero as t + O. On the other hand, by lemma 9,

ftu = a F(y,u) du 8

a> 0

and since let) is a decreasing function of t,

(g*\)IsHy) - (g*\)It)(y) < a ruIIF(y,ull - F(y,u)1 du < B

and combining these inequalities we find that

G (x ,B) < G (x , t ) + 2 e (t) + a N l (t ) - 1

and, since G2 (x,t) is a non-decreasing function of t, this, in turn, implies that

57

which holds for a";s<t";l , a being the distance of x from O2-0 1,

Now for a> s, since gl = gl is upper semicontinuous in 0 1 and therefore 11 = ~gl coincides with a measure in 0 1 , G 1 (x,s) is a non-decreasing function of s, and thus (5) is seen to hold also for 0 <s < t"; 1 and all x in O2 , Now letting s tend to zero we ob tain

Suppose now that x -+ Xo ' xoE O2 , Then, since the righthand side of the pr.eceding inequality is a continuous function of x we find that

lim x-+x

o

and since a(t) -+ 0 and A(t) -+ ~ as t -+ 0 , letting t tend to zero we obtain

lim gl(x)"; gl(X o) x-+x

o

which is the desired upper semicontinuity of gl in O2 ,

Let now z;(x);;;'O be infinitely differentiable with compact support

contained in O2 , Since, as t -+ 0 , z;*~t converges uniformly with all its derivatives to 1;, we have

lim \ G1 (x,t) Z;(x) dx = lim gl (I;*~t) = gl (1;) t-+o t-+o

On the other hand, on account of (5), G1 (x,t) is bounded above for 0 < tor;;;; 1 and x in any compact subset of O2 , Thus, since

lim G1 (x,t) = gl (x) , by Fatou's lemma we have t+o

j g 1 (x) I; (x) ax ;;;. lim \ G 1 (x, t) dx) dx t-+o

and thus gl is seen to be locally integrable in O2 , But then mu! tiplying (6) by z;(x) , integrating and letting t tend to zero, we get the preceding inequality reversed, Thus we have equality and

58

for all 1;. Thus gl ·is upper semicontinuous, locally integrable

and coincides with gl in °2 , which contains 01 as a proper subset,

in contradiction with the assumed maximality of °1 , Hence 01 must coincide with 0, and, as observed earlier, this proves the theorem

in the case when C is empty.

Let us pass now to the case when C is non-empty. We shall show that gl is upper semicontinuous, locally integrable and coincides

with gl in 0, and then the desired conclusion will follow as bef£ re. Let 01 be again the largest open subset ofO on which gl has

these properties. Since, as we know now, 01 must contain ° - C

we have ° -ole C. Consider the function inf t 2 (V*Xt) ex), t

0< -t E;; 1. Because t n x t is the characteristic function of the clo-

sed sphere I x I E;; t, this function is upper semicontinuous and, ac­cording to our hypotheses, positive at every point of C. Conse­

quently, as in the proof of lemma 1, it follows that there exists

and open subset 02 of 0, containing 01 as a proper subset, such

that inf t 2 (v*Xt)(x);;'o> 0 for all x in 02 - °1 , Set now t

1

(J.) (x) = J (V * x t) (x) d t o

Evidently, this is an integrable function. If d(x) denotes the

distance between x and the set 02- 01 ' 2d(x)E;;sE;;1 and y is a 5 point in 02 - °1 such that I x -y I E;; '4 d (x), then as readily seen

-2 S

where the a are positive constants. Thus, for d(x)E;;1/2 ,we

have

which shows that w (x) = +00 in 02 - 01

gue measure zero, and that d(x)-l is

of ° 2 - ° l' Let now d (x) E;; s < t E;; 1" • beginning of the proof, t 2 F(x,t) is

, which therefore has Lebe~ integrable in a neighborhood Then since, as we saw at the

bounded, by lemma 9 we have

t 1

I (g*<t> ) (x) - (g*<t> )(x)l=aIJ uF(x,u)dulE;;o J u-1du =a log d(x)~1 t s s d (x)

59

and this, combined with (4) and the fatt that G2 (x,t) is a non-de creasing function of t gives

(7) G 1 (x ,s) ..;; G 1 (x , t ) + 29 (t) - 1

+ a log d(x)

which is analogous to (5), and which for the same reasons as in the case of (5), holds also for O<s<t";;l and d(x)";;1. Letting 8 tend to zero we obtain

(8) ~(x) ..;; G1 (x ,t) + 29 (1;)+ a log d(x) -1 •

This shows that gl(x) -+ _00 as d(x) -+ 0 , and since gl{x) for x E C, it follows that gl is upper semicontinuous in O2 • To

prove that il is locally integrable and coincides with gl in O2

we argue with (7) and (8) as we did in the preceding case with (5) and (6), keeping in mind that log d(x)-1 is integrable in a neighborhood of O2 - 0 1 " This will contradict the assumed maxima

lity of 0 1 , showing that 0 1 must coincide wIth 0, as we wished to show.

Proof of theorem 6. As in the case of the preceding theorem, we may assume that f = /:;g, where 9 has compact support, that h (x) ..;; 0

and is defined and integrable in all of Rn , and that v is defined on all Borel subsets of Rn and is totally finite.

We shall assume first that n> 2 and q> 1. Let

A A 2 (1 s) A 2s l(z) = g(z)lzl - = -4rr2 f(z)lzl-

-1 where s is such that r";;s<q and (n-2sq)/(n-2)<q. Since 9 has A A

compact support, 9 and l are continuous and bounded near the ori-A 2 _ A -2 A

gin. Since f(z)(l+lzl ) r is in Lq and s~r, f(z)lzl sand l(z)

are integrable to the q-th power in Izl < 1. Thus Z(z) is in Lq

and its inverse Fourier transform lex) is in LP, P = ql(q-1).

Let now cr = 2(1-s) and a a constant such that the Fourier trans -form of c Ixl-n+cr coincides with Izl -cr. Let g(x) be defined by

l (y) Ix-y I-n +cr dy

if the int~gral is absolutely convergent, or g(x) = -00 otherwise. Since (n-2sq)}(n-2) < q and sq < lone verifies readily that n>pa> 2 and pen-a) >n, so that Ixl- n+a is integrable to the q-th

60

power in Ixl > 1 and the integral above is absolutely convergent for

almost all rand g(x) is locally integrable. Furthermore, our distribution g coincides with the function g. In fact, if 1; (x)

is an infinitely differentiable function with compact support, we have

J g(x) ~(x) dx = a J zex) J ~(y) Ix-y rn+a dy dx

Since the Fourier transform of lal-a is alxl-n +a , the inner inte

gral above is the Fourier transform of 1;(a) la I-a, 1; here being

the inverse Fourier transform of 1;. On the other hand the convo­lution ~(x) * Ixl- n +a evidently belongs to Lq and therefore, by

Plancherel's theorem, we have

J g(x) ~(x) dx = a J lex) J ~(y) Ix_yl-n+ a dy dx

J A a = l (a) 1;(a) I a I - da J ~ (a) z;(a) da

which shows that the Fourier transform of g coincides with g, that is, g coincides with g.

Let now E be an arbitrary positive number and 1;(1£) an infinitely

differentiable function with compact support which equals 1 in O. Let gl (x) ~ -00 be defined by

gl (x) = g(x) - a z;(x) E I ley) I Ix-y I dy + J p -n+2

(9)

where -a I x l- n+ 2 , a> 0 is the fundamental solution 0 f Laplace's e­

quation. Evidently gl is locally integrable and the distribution

f = ~gl coincides with

in 0. As we shall see, fl has the property that Fl (x) ~O almost

everywhere and il (x) > _00 everywhere in 0, and gl satisfies the co~ ditions of lemma 11. Thus it will foJlow that fl coincides with a

measure in 0, ann since this will hold regardless of the value of E we will conclude that f-h also coincides with a measure in O,and

our theorem will be established.

61

On account of lemma 7 we have

uniformly in any compact subset of O. From this it follows that

11 ~f· Now if at the point x we have 1(x) : _00 , then, according to our hypotheses, F(x,t) = 0 [(v* \) (x)] and consequently

lim F(x,t) + e:(v*<P t ) (x) ~u, and since h';;;O, it fo11ows that t-+o il (x) ~ 0, and thus we have 11 (x) > -00 in a11 cases. On the other hand, if x is a Lebesgue point of the function hand t -+ 0 and

y -+ x with Iy-xl ';;;t, we have (h*<Pt)(y) -+ hex) and therefore PI (x) ~P(x) - h (x). Thus we have PI lx) ~O almost everywhere in O.

Let us turn now to the function gl' Evidently, since h ';;;0, the

last two terms on the right of (9) are upper semicontinuous func­tions of x, and according to lemma 14 the sum of the two first is

upper semicontinuous in O. Thus gl is upper semicontinuous in 0, and there remains only to show thatg 1 has the property that

cgl*1/it) (x) -+ gl (x) as t -+ 0 for a11 x in u. To see this let 0> fl> -n arid consider the convolution 1/i t * Ix IS. Evidently

sup 1/i t * Ixl S is finite everywhere but at the origin. Further t

more, as readily verified, it is homogeneous of degree S and sph~ rically symmetric. Thus we have

Thus if ~ is a signed measure and the convolution ~ * Ixl S is abs solutely convergent at the point Xo then by the dominated conver­

gence theorem we have

lim [(~ *lxl i3 ) t-+o

= (~ * I x I 6) (x ) o

We use this observation to calculate the limit of (g *1/it)' For 1 '

this purpose we convolve the righthand side of (9) with 1/it' Accord ing to Lemma 8, in calculating the limit of the resulting expres­sion at points of 0 we may drop the function ~(x) and we obtain a

sum of terms of the form ~ * Ix IS. Thus if at a point x a11 in'tE. grals of the righthand side of (9), including the one defining

g(x), are absolutely convergent we have lim (g *1/it) (x) = g (x) t-+o 1 1

62

If on the other hand one of the integrals is divergent, then 'gl(X) = _00 , and the desired result follows from the upper semicon

tinuity of gl' This concludes the proof of the theorem in the case n > 2 , q > 1

In the case nO;;; 2 or q = l' = 1 the distribution g, f = t:. g , co inc,!.

des with a continuous function. In fact, if q > 1 we have

and since 2(1-r)p = 2(1-r)ql(q-1) = (2q-2rq)/(q-1) > 2 the last in­

tegral above is convergent. But, because g is continuous, this i~

plies that it is integrable and that g coincides with a continuous function. If on the other hand q = l' = 1

J ~ J ~ 2 Ig (z) I dz = If(z) liz 1-Izl>1 Izl>1 .

dz < 00

and, again, g coincides with a continuous function, and the rest of the proof consists in applying lemma 11 to the function

gl(X) = g(x) + t;(x) J E: 4>(x-y) dv - t;(x) J hey) 4>(x-y) dy

where 4>(x) is the fundamental solution of Laplace's equation. The

argument is identical with the one used above and need not be re­peated here.

Proof theorem 7. Theorems 5 and 6 having been established, theo­

rem 7 can be proved by using lemma 7 and the argument used in the proof of theorem 2. The details are left to the reader.

4,

In this last section we shall give two examples of distributions

f for which (f*~t) tends everywhere to zero with t. The first is

the analogue of the series En sin nx which !s known to he every­where Abel summable to zero. In this case f barely violates cond,!.

tion a) of theorem 5 but is unbounded. In the second example,

63

which is more complicated, fez) is of the order Izl- (n-3)/2 as

Izl -+ 00 , and thus is bounded for n = 3 and tends to zero as

Izl -+ 00 if n ~4. We remind the reader that if ;(z) = e- Izl , then A

(f*CPt) is the Abel means of the Fourier integral of f. Thus the

Fourier integrals of the functions f are Abel summable to zero

everywhere.

Let x = (u,x) , u ER , xERn-1 , and w(x) be infinitely differen­

tiable and have compact support. Let f be the distribution defi­

ned by

f(~) = I ( au ~) (O,x) w(x) dx

Then f has support in u o and, setting z = (v,a) , we have

A

fez) 2rriv w(a)

whence it follows that

Now, since f has support in u 0, we have

u " 0 t -+ 0

and if u 0, on account of the spherical symmetry of CPt we have

and, consequently (f*CPt)(O,x) t -+ O.

o u = v o

In our second exampl"e we shall assume that n ~ 2. Let S be the

sphere Ixl = 1 and let f be defined by

f(~) = Is [~+ 2(n-1)-1 av ~l do

where do stands for the elem'ent of are,a of Sand ddV denotes normal

outer differentiation. To calculate f we merely replace ~ by

64

,2I1'i(lI:.a) in the preceding integral. Setting Ixl = r, lal = p,

(e.a) .. rpcos s we have do = (J (sin s)n-2 ds and our integral b!!.

:omes

rr e 21Tip cos s (sin s)n-2 ds +

° -1 J1T + hi (n-1) P

0,

'and replacing cos s by t and integrating these integrals by parts

we see that fez) O(lzl-(n-3)/2) as Izl ~ 00

Let now ~(x) = n(lxl), where n(t)(k) = O(t- n- k - E ) as t ~ 00 and let

us calculate lim (f"'~t) (x) as t ~ O. Since the support of f is 5,

by lemma 7, this limit is zero if x f. S. If xES we have

Since, as readily seen, if Ixl = 1 then

a 1 a; Ix-yl = Ix-yl- [1 - (x.y)l

attd'thus we have

a -n 'I 1/ -n -1 I I 1 a;t n(x-y t) =t n'(x-y It) Ix-yl- [1 - (x.y)l.

Setting Ix-y I = s = 2 sin a/2 we obtain

1 - (x .y) = 1 - cos a = s2/2

(J sn-2 ~(s)ds

and substituting in (11) we get

2 2

J -n n-2 -If -n-l n-l (f",4>t) (x)=, t n(slt)s <Ii(s)ds+(n-l) t n'(slt)s <Ii (s)ds o 0

-n, -n-l Since t n(s/t) and t n' (sit) tend to zero as t ~ 0, uniformly

in s;> 1, replacing the upper limit of integration by 1 and inte -

grating by parts the second integral we obtain

65

(f*cf>t)(X) = _(n_1)-1 r t- n s n - 1n(slt) <I1'(s) ds + 0(1) o

But, evidently, 1<11' (s) I .;;; a s in 0';;; s.;;; 1 and consequently the last integral is dominated by

as t + 0 , and we find that (f*cf>t) (x) + 0 everywhere as t + O.

REFERENCES

(I) CALDERON A.P. and ZYGMUND A., Loeal p~ope~t~e~ 06 ~olut~on~ 06 ell~pt~e pa~t~al d~66e~ent~al equat~on~, Studia Math.

20 (1961), 171-225.

[2) SAKS S., Theo~y 06 the ~nteg~al, Monografie Matematyczne,

vol. VII, Warsaw, 1937.

(3) SHAPIRO V.L., Canto~-type un~quene~a 06 mult~ple t~~gonome­'t~~e ~~teg~ala, Pacific J. Math. 5 (1955), 607-622.

(4) -----------, Removable aeta 60~ po~ntw~ae aolut~ona 06 the gene~al~zed Cauehy-R~emann equat~ona, Ann. of Math. 92

(1970), 82-101.

[5) VERBLUNSKY S., On t~~gonomet~~e ~nteg~ala and ha~mon~e 6une­t~ona, Proc. London Math. Soc. 38 (1935), 1-48.

University of Chi~ago.

Recibido en setiembre de 1970.

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

I I I I I I I I I

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\ I I I I I I I I

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\ \ I

\ \ I I I I I I I I

\ I I

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\ \ \ \ I I I I I I I I I I I I I

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Revista de la Union Matematica Argentina Volumen 25, 1970.

SOME 'COMMENTS ON THE SPECTRAL THEOREM

by M.H. ~tone*

Ve.d..i.eada at PlLa 6e.<!> OIL Atbe.lLta Ganzoite.zVarnlnglle.z

The spectral theorem for self-adjoint operators in a Hilbert space

(with real, complex, or quat ern ionic scalars) generalizes the cla~ sical theorems on the canonical reduction of quadratic or hermi -

tian forms and their matrices. Usually two steps are needed, the first passing from finite-dimensional spaces to bounded operators

in general spaces, the second from bounded operators to unbounded. There has always been a certain interest (see [1], [2], [3], [4], [5])

in carrying out this generalization by "pure" Hilbert space me -thods - that is to say, by using only intrinsic algebraic and geo­

metric properties of abstract Hilbert space without recourse to special theorems drawn from classical analysis. For bounded oper~ tors the spectral theorem was treated in this spirit by F. Riesz

[1] and by Lengyel and Stone [2], for unbounded operators by Y. Y.

Tseng [3]. The present paper, while closely related to Tseng's,

expounds a variant of his approach that may appear somewhat sim -pIer and may shed some additional light on the techniques required.

All methods for treating the case of an unbounded self-adjoint op~

rator A involve the discussion of certain related bounded opera -tors. Most of them also use the spectral theorem for the bounded

case, either explicitly or implicitly. Here we shall assume the

bounded case, as treated in [2], and apply it to one of the opera­tors appearing in the characteristic !\latrix of A (see [6]) in such a way as to settle the unbounded case. We shall not assume any

knowledge of [6], but shall develop on the spot the essential pro­

perties of the elements of the characteristic matrix for A.

As commutativity of operators is continually stressed in our arg~

ments, we must recall that a bounded linear operator D commutes

with the self-adjoint operator A if and only if it maps the domain

of A into itself and AD is an extension of DA. The set of all op~ rators\D commuting with A is called the commutant of A, while the

set of all bounded linear operators commuting with every member of the commutant is called the second commutant of A. Clearly, if D

is in the commutant of A, then so is its adjoint D*: for, if x and

* The preparation of this paper was partially supported by a grant from the National Science Foundation, GP-20856.

68

y are vec~ors in the domain of A, we have (Ax)(D*y) = (DAx)y = = (ADX)y = x (D*Ay) ; and then, since A is self-adjoint, D*y must be in the domain of A and satisfy the relation AD*y = D*Ay. Simi larly, the second commutant contains both D and D* if it contains either.

In order to state the spectral theorem and present its proof it will be convenient to introduce the following

DEFINITION 1. A ppojection P splits a seZf-adjoint opepatop A at

A, -""<X<+ .... if and only if

(1) P commutes ~ith A.

(2) if x is.·a vectop in the domain of A and in the pange of p. then (Ax)x";;;XUxIl 2 •

(3) if x is a vectop in the domain of A and in the 2

pange of I-P. then xllxU "(Ax)x ~ith equaUty

hoZding if and only if x = O.

Here we note that the ranges of P and I-P are mutually orthogonal subspaces and that every vector x is the sum of components Px and (I-P)x in these two subspaces respectively, in just one way. The commutativity required in (1) shows that x is in the domain of A if and only if its two components are. Commutativity shows fur -ther that A acts on each of these subspaces as a self-adjoint op~ rator therein and that the behavior of A is completely determined by what it does there, in accordance with the equations Ax = = APx + A(I-P)x = PAx + (I-P)Ax where APx = PAx and A(I-P)x ... = (I-P)Ax. The concept of splitting demands in addition a certain quantil:ative behavior (semi-boundedness) in each of these subspa­ces, as described by (2) and (3) respectively.

We shall now state the spectral theorem, in two parts.

THEOREM la. (Spectral Theorem, Analytic Part.) If A is a self­adjoint opepatop. then thepe exists fop each peal X. _00 < X < +00 •

a unique ppojection Ex spUtting A at X. The ppojections E). nece!!,

sapiZy have the folZo~ing ppopepties:

(1) Ex is in the second commutant of A. as ~ell as

in the commutant.

69

(2) EAElJ = E \I

where \I = min(A,lJ).

(3) lim EHe:x EAx strong~y when e: > O. e: .... o

(4) lim EAx 0 8trong~y. A-~_~ co

(5) lim EXx x 8trong~y. A .... +'"

We recall that a family of projections satisfying (2) to (5) above is called a speatra~ fami~y or a aanoniaa~ reso~ution of the iden­

tity.With this terminology we state the second part of the spec -tral theorem as follows.

THEOREM 1b. (Spectral Theorem, Synthetic Part). If EA' -'" < A < +"'. is a speatra~ fami~y of projeations. then there exists a unique

se~f-adjoint operator A suah that EA spUts A at A.

Here we shall prove only Theorem 1a. The proof of Theorem 1b, as is well-known, depends on the construction of A as a limit of Rie mann-Stieltjes sums n-l'

r A~+I(EA - EA ) k .. O . k+l k

, and the verification

of the splitting property for EA,

The proof of Theorem 1a depends in the last analysis on the follow ing specialization.

THEOREM 2. (Splitting Theorem). If A is a seLf-adjoint operator.

there exists a projeatio~ E that sp~its A at 0 and is in the se -

aond aommutant of A ••

Indeed, we shall begin by proving

THEOREM 3. The SpUtting Theorem impUes the Speatra~Theorem.

Ana~ytia Part.

The proof will be presented as a series of lemmas and theorems.

LEMMA 1. A projeation P sp~its A at A if and onLy if it spLits

A-AI at O.

Proof. This proof will be left to the reader.

COROLLARY 1. The SpUtting Theorem impUes that t'hel'e exists a

70

projection El in the second commutant of A such that El spLits A at l, _00 < l < +00

LEMMA 2. If P~ and Ql are commuting projections that spLit A at

~ and A respeotiveLy, then ~ ~ A impLies P Q, = P and l = ~ im-~ 1\ ~

Proof. Since P and I-Q, are commuting projections, the intersec ~ 1\ -

tion of their ranges is a subspace with R = Rl~ = P~ (I-Ql) as its projection. Thus for arbitrary x the vector y = Rx is in the ra£

ges of P and I-Q,. Since P and Q, split A at ~ and at l respec ~ 1\ ~ 1\ -

tively, we have AUyU 2 ~ (Ay)y ~ ~UyU2 with equality on the left

if and only if y = O. Thus ~ ~ A implies AU yll2 = (Ay)y and hence

y = O. It follows that Rx = 0 or P Q, = Q, P = P. When A = W , ~ 1\ 1\ ~ ~

we can interchange P ~ and QA' obtaining QA = P ~ QA = QA P ~ = P ~

COROLLARY 2.1. The SpUtting Theorem impUes that, if PA spUts

A at A, then P is unique and is in the second commutant of A.

Proof. Let EA be the projection of which the existence is assert ed by the Splitting Theorem; and let PA split A at A. We verify

that PA and EA commute. In ract, PA commutes with A and El is in the second commutant of A, so that this is obvious. In Lemma 2

we can now take A = ~ , QA = El and conclude .that PA = EA'

COROLLARY 2.2. The splitting Theorem impLies aLL statements com­

bined in the SpectraL Theorem, AnaLytic Part, except those concern

ing properties (3) , (4) , (5).

Proof. The existence of a splitting family is given by Corollary 1. Its uniqueness and its inclusion in the second commutant of A

are guaranteed by Corollaries 1 and 2.1. Property (2) is then e­

vident from Lemma 2.

THEOREM 4. If A ~ ~ , the projection F = FA~ = E~- EA = E~(I-El) commutes with A and has range Lying in the dC'main of A. For aLl

. 2 2 Y 1-n the range of F A lIyll ~ (Ay)y .;;; ~ UyU with equaUty on the

Zeft if and onZy if y = o.

Proof. Apart from the notation, all of this theorem except for the assertion that the range of F lies in the domain of A is

proved in the discussion of Lemma 2: it is only necessary to take

71

P = E and Q, = E" there, Now if x is in the domain of A so is j.l j.l A

Y - Fx. Since UyH2 = (Fx)x and (Ay)y - (AFx)x we have "(Fx)x <

< (AFx)x < j.l (Fx)x or, equivalently, 0 < ((A-AI)Fx)x < (j.l-") (Fx)x <

< (j.l-A) H xH 2. Thus the operator H - F (A-" 1) has the same domain

as A and satisfies the relations (Hx)z - x(Hz) and 0 < (Hx)x <

< (j.l-A)HXH 2 for all x and z in the domain of A. A standard use

of polarization shows that HHxH < ~-")UxH for all x in the domain

of A. If Y i~ an arbitrary vector in the range of F, there is a

sequence zn in the domain of A converging strongly to y. Thus

Yn - FZn and HZn are Cauchy sequences converging to Fy - Y and z*

respectively, Hence for all x in the domain of A we have

(Ax)y - lim (Ax)Yn - lim (FAx) zn n+ oo n+ oo

lim ((H+" F)x) zn lim x(Hzn ) + lim x (AYn) n+ oo n+ oo n+ oo

x(z* + " y)

Since A is self-adjoint, we conclude that y is in the domain of A

(and Ay z* + "y).

LEMMA 3. E" has property (3).

2

Proof. If v < j.l we have UEj.lx-EVxll - ((Ej.l-Ev)x)x = (Ej.lx)x-(Evx)x,

so that (Ej.lx)x is a monotone increasing function of j.l with real

values between 0 and UxU 2 • As a function of x it is quadratic.

When" < j.l the function q(x) - lim ((E - E,,)x)x exists and is also j.l->" j.l

quadratic with real values between 0 and HxU 2 • Hence there exists

a bounded self-adjoint operator F such that (Fx)y = lim((Ej.l-E,,)x)y. j.l+"

Thus if " < v < j.l we have

(FX)((Ej.l- E\)y) = lim ((E v - E,,)x)((Ej.l- E,,)Y) v+"

lim ((E v - E,,)x)y - (Fx)y . v+"

Hence (Ej.l- E,,)Fx - Fx , so that Fx is in the range of Ej.l- EA, Thus 2

by theorem 4 we see that Fx is in the domain of A with AUFxU <

< (AFx)x < j.lHFxH 2 ,where the equality holds on the left if and

only if Fx - o. we obtain "HFxU 2

2 lim U E x - EA xU j.l+A j.l

If we let j.l tend to A in this double inequality

(AFx)x an'd hence Fx - 0, It follows that

lim ((Ej.l - E,,)x)x = (Fx)x = 0 . j.l+"

Putting IJ

72

A+e: , e: > 0 , we conclude that lim IIEA+e:X - EAXII e:-+o

LEMMA 4. EA ha$ property (4).

o .

Proof. Let x be an arbitrary vector and e: an arbitrary positive real number. We can then take y in the domain of A so that

lIx-yll ~ t e: Let A <-Z.UAyll/e:. Then EAy is in the domain of A

and (AEAy)y ~ dEAyU Z. Hence IA 1 UEAyllZ ~ (-AEAy)y = (EAy) (-Ay) ~

~ IIEAyUllAyl1 and REAyl1 ~IIAyU/IAI ~ t e:. Finally

1 UEAxU ~ UEAyU + UEA(x-y)U ~ I E + Ux-yU ~ e: , as was to be proved.

LEMMA 5. EA has property (5).

Proof. The discussion is similar to that of Lemma 4. For given

x and e: we choose y as before and A so that A > 2UAyU/e: We

observe that (I-EA)y is in the domain of A and that

1 1 Since Ux-yU ~ I e: and U(I-EA)yU ~ UAyU/A ~ Ie:, we conclude that

Ux-EAxil ~ e:

We have thus established Theorem 3 and reduced the proof of Theo­

rem 1a to the proof of Theorem 2, the Splitting Theorem. For the latter we need to introduce the bounded self-adjoint operators B

and C that occur in the characteristic matrix

of the sel£-adj oint operator A (see [6]). It is then easily shown that the projection E supplied for C by the Splitting Theorem ser­

ves also as the desired splitting projection for A. Thus the Splitting Theorem for bounded self-adjoint operators is seen to

imply the theorem for all self-adjoint operators. With this moti­vation we turn to the discussion of the operators Band C.

Following von Neumann [7], we study the graphs of the relations

y = Ax , -Ay = x in the Hilbert space of ordered vector-pairs

(x,y) with the scalar product (xl 'Yl) (xz,Yz) = x1x Z + YlYZ. The

73

graph of A or the graph of the equation y = Ax is the set

MA = {(x,y) ; y = Ax}. Similarly the inverse graph of the oper~

tor -A or the graph of the equation -Aw = z is the set

NA = {(z,w) ; -Aw z} The orthogonality of elements (x,y) and

(z,w) chosen from these sets is expressed by the equation

xz + yw = x(-Aw) + (Ax)w = 0 or (Ax)w = x (Aw) . The fact that

(z,w) is orthogonal to every element of MA is expressed by the

statement that xz + yw =xz + (Ax)w = 0 for all x in the domain

of A; and the latter statement is valid for self-adjoint A if and

only if w is in the domain of A and Aw = -z, that is, if and only

if (z,w) is in NA. Thus when A is self-adjoint, NA is the ortho­

gonal complement Mr of MA. Similarly, MA is the orthogonal com­

plement of NA. Thus MA aHd NA are both closed linear subsets, or

subspaces, of the Hilbert space of vector pairs. We denote by

P = PA the projection of the latter on MA , t·he graph of A. Now

the operators Band C are defined as the composite mappings x + z

and x + w , respectively, read off from the diagram

z p t

x + (x,O) + (z,w) \, w

Since each arrow in the diagram represents a bounded linear map -

ing from source to target, Band C are bounded linear mappings or

operators with the original Hilbert space as source and target.

Since P is the projection on MA ' the projection on NA is I-P

The equation (x,O) = P(x,O) + (I-P)(x,O) shows that P(x,O) =

= (Bx,Cx) is in MA and that (I-P) (x,O) = (x-Bx,-Cx) is in IJ A

Thus Bx is in the domain of A and Cx = ABx, while -Cx is in the

domain of -A and (-A)(-Cx) = x - Bx or Bx + ACx = x. It follows

that Bx is in the domain of A2 (which is the same as that of I+A2)

and A2(Bx) = A(AB)x = ACx , (I+A2)Bx = Bx + ACx = x. We have

thus proved

LEMMA 6. Band C have ranges contained in the domains of I + A2

and A respectively. The operators A. B. C satisfy the identities

(1) C AB (2) B + AC = I

LEMMA 7. B is a self-adjoint operator with self-adjoint inverse

1+ A2.

Proof. By Lemma 6 (3) we see that Bx o implies x O. Hence B

74

has an inverse, of which I+A2 must be an extension. Now if y is

an arbitrary vector in the domain of I+A2 , we put z=y-B(I+A 2)y,

noting that (I+A2)z = O. Thus IIzll2 + IlAzl1 2 = ((I+A2)z)z = 0 ,

z = 0 , y = B(I+A2)y, and y is in the range of B.Hence the range

of B is the dOmain of I+A2 and the two operators are inverses of

one another. Now (Bx)y = (Bx) ((I+A2)By) = ((I+A 2)Bx)By = x(By)

for all x and y because A is self-adjoint. To show that I+A2 is,

like B, self-adjoint, Yet y and y* be such that ((I+A 2)x)y = xy*

for all x in the domain of I+A2 Here we can put x = BZ,obtain­

ing zy = (Bz)y* for all z. It follows that By* = Y because B is

self-adjoint. Hence y is in the domain of I+A2 and (I+A2)y = y*.

Accordingly, I+A2 is self-adjoint.

We turn now to some commutativity properties of A, B, C.

LEMMA 8. Band C commute with A and with each other. Consequen~

ly C is self-adjoint, as are A and B.

Proof. If x is in the domain of A, the equation ACx = Bx - x

shows that ACx is also in the domain of A and that A2Cx = ABx-Ax =

= Cx-Ax. Thus Ax = (I+A 2 )Cx and BAx = Cx = ABx. Thus B commutes

with A. It now follows that Cx = BAx is in the domain of A and

that ACx = A2Bx = ABAx = CAx because B commutes with A. Hence C

commutes with A. Now for all z we have BCz = BABz ABBz = CBz

because B commutes with A. Hence Band C commute. Finally we

observe that C = AB implies that C* is an extension of B*A*

= BA C AB. Thus C and C* coincide on the domain of A and must be

identical by continuity, since the domain of A is everywhere

dense. Thus C is self-adjoint.

LEMMA 9. A bounded linear operator D commutes with A if and only

if it commutes with both Band C.

Proof. If D commutes with A we have DCz = DABz = ADBz , Dz

DBz + DACz = DBz + ADCz = DBz + A2DBz = (I+A2)DBz. Hence BDz

DBz and Band D commute. We .then have from the first equation

DCz = ADBz = ABDz CDz, so that C and D also commute. On the

other hand, if D commutes with Band C and x is in the domain of

A, we have CDx = DCx = DABx = DB~x = BDAx and hence Dx = B(Dx) +

+ CAC) (Dx) = BDx + ABDAx. Thus Dx is in the domain of A. We now

have BADx = CDx = DCx = BDAx and hence ADx = DAx.

We are now ready to prove our principal result.

75

THEOREM 5. The projeation E supplied by the Splitting Theorem

for the bounded self-adjoint operator C serves as the operator re

qui red in order to validate the Splitting Theorem for A.

Proof; We have to show that E splits A at 0 and is in the second commutant of A. E is in the second commutant of C and therefore commutes with Band C, both of which commute with C by Lemma 8. Hence E commutes with A, by Lemma 9. If D commutes with A, it also commutes with C, by Lemma 9. Hence it commutes with E, be­cause E is in the second commutant of C. Thus E is seen to be in the second commutant of A. To show that E splits A at 0, we take x in the domain of A and note that Ax = (I+A 2 )BAx = (I+A 2 )ABx =

- 2 = (I+A )Cx by Lemmas 7 and 8. Now if x is in the range of E so is Ax because EAx = AEx = Ax. We therefore have (Ax)x = = ((I+A 2)Cx)x = (Cx)x + (ACx)(Ax) = (Cx)x + (CAx)(Ax) ~ 0 , be-cause E splits C at O. Similarly, when x is in the range of I -E we see that Ax is in the range of I-E. We then have (Ax)x (Cx)x + (CAx) (Ax) ~ 0 with equality if and only if (Cx)x = 0 and hence if and only ifx = O.

The proof of the Spectral Theorem, Analytic Part, is thus comple! ed by reference to the paper of Lengyel and Stone [2], where it is shown by "pure" methods that the Splitting Theorem holds for every bounded self-adjoint operator.

REFERENCES

[1] F. RIESZ, Acta Szeged, 5 (1930), 19-54.

[2] LENGYEL and STONE, AnnatJ.> 06 Mathemat-i.cJ.>, 37 (1936), 853-864.

[3] Y.Y. TSENG, Sc-i.ence Repo~tJ.> 06 the Nat-i.onat TJ.>-i.nghua Un-i.ve~-J.> -i.ty, A, 3 (1935), 113 - ! 2 5 .

[4] S.J. BERNAU, Jou~nat 06 the AUJ.>t~at-i.an Mathematicat Soc-i.ety, 8 (1968), 17-36.

[5] A. WOUK, S.l.A.M. Rev-i.ew, 8 (1966), 100-102.

[6] M.H. STONE, Jou~nat 06 the lnd-i.an Mathemat-i.cat Soc-i.ety, 15 (1951), 155-192.

[7] J. von NEUMANN, AnnatJ.> 06 Mathemat-i.cJ.>, 33 (1932), 294-310.

Recibido en setiembre de 1970.

The University of Massachusetts at Amherst

Revista de la Union Matematica Argentina Volumen 25, 1970.

A GEOMETRIC OBSERVATION ABOUT LINEAR PARTIAL DIFFERENTIAL OPERATORS

W. Ambrose

Ved~cado at p406e~o4 Atbe4to Gonz~tez Vom~nguez

We consider a linear partial differential operator on an open U C RP of the form

(1.1) a"~ Acp = I a "---"-

1"I:;:k" au" (cp E COO (U))

o

where the a "

a" are complex-valued functions defined on U an the

aU" are the usual partial derivatives of order a ,where" ("I'···'''p)

and 1,,1 = Ii"~. In this paper we make two observations:

1) The symbol of A~ nowadays usually considered as a function on

the cotangent bundle, can be considered, instead, as a function

on a subset of G (RP+2), where G (RP+2) is the bundle of p-planes P P

at points of RP+2. The symbol can be defined naturally here, by

geometric considerations, without using coordinates (tho it does

use the decomposition of RP+2 as RP x R2). If one then puts a

coordinate system on G (RP+2), the coordinate expression of this P

function becomes the usual expression for the symbol.

2) The most general definition of a partial differential equa­

tion (equation , as distinct from operator) seems to be as a sub­set of a higher order Grassman bundle. We indicate, again geome­

trically, how an operator of the form (1.1) gives rise to a par­

tial differential equation in this sense. Furthermore, it gives rise to a sequence of such equations, of orders 1 to k; the j'th

equation being of order j. The equation of order k is (1.1) con­sidered as a differential equation; the equation of order 1 is

the characteristic equation (the zeros of the symbol).

We wish to consider the A of (1.1) as a k-th order vector field on U, i.e. as a map which assigns to each XEU a k-th order

complex tangent vector to RP at x. Then we wish to show how such a k-th order vector field gives rise to functions on certain sub-

sets of G1 ,c(UxR1), ... ,G k ,c(UxR1), where Gl,cCUxR1 ) is the set of P P P

complex i-th order p-planes at points of UXRl. This is to be done

separately at each point of U so what we wish to show is that a

k-th order complex tangent vector at x E RP gives rise to some com­plex valued functions defined on certain i-th order p-spaces in

RP+l.

Since RP is a real manifold and we are referring to complex tan -

gent vectors and p-spaces we briefly discuss complex tangent vec­tors to a real manifold. Let M be a p-dimensional COO real mani­

fold and mE M. We now define M~, the complex tangent space to M

at m. We could define MC to be just the complexification M ~ C m m of the usual real tangent space Mm. However we prefer to define

~ directly, as follows. Let Rm be the complex local ring of M at m, i.e. the elements of Rm are the germs of complex valued COO

functions at m. Let 1m be the maximal ideal of Rm' so 1m con­sists of the germs that vanish at m. For each non-negative inte­ger k we define the complex linear space M;'c to be the dual space

of R /Ik+1 . We call elements of Mmk,c k-th order tangent vectors m m

at m and call M~'c the k-th order tangent space at m. Alternati­

vely a k-th order tangent vector may be considered as a linear k+l function on Rm that vanishes on 1m . Because every f E ~ can be

uniquely expressed as f = fa + f1 where fa is constant and f1 Elm

it is easily seen

t = to + tl where

that every

t f = cf a a

tEM;'C can be uniquely expressed as

(c E C, independent of f) and tl is

zero on constants. It is clear that Mk,c C Mk+1,c and that the m m

Mk. k c usual real k - th order tangent space m cons lStS of those t E Mm'

such that t is real whenever f is real valued. And Mk,c = m

= Mk e iMk. Also, a linear partial differential operator defined m m

on U ~ RP is essentially the same thing as a map which assigns to

each xEU an element of (RP)k,c. For this reason we call it a x

k-th order complex vector field on U.

Now we define the Grassman manifold k,cG (M) of complex k-th order P

p-spaces over the real d-dimensional manifold M. This is done es-

sentially as for the real case but since we need below the expli­cit relation with the real case we give some details here. We shall write RO , 10 for the local ring and maximal ideal formed

m m 00

from the real valued C functions at m. We define a p-ideal in

79

~ to be an ideal I in Rm for which there exists a set of genera­

tors fn+l, ... ,fd such that dfp+l, ... ,dfd at m are linearly inde-

pendent over C. Then we define a z-th order p-ideal at m to be

any ideal in Rm of the form I + I~+l , where I is any p-ideal in

Rm' We define a z-th order complex p-spaoe at m

space of an I 1(1 + rz+l) where I is any p-ideal m m

define k,cG (M) to be the ~et of all z-th order p '" made into a real C manifold, and a bundle ~ver

as any

at m.

complex M, with

dual

We now

p-spaces, the follow

ing differentiable structure. Let xl, ... ,xd be any coordinate system of M with a cubic domain Q. Let N be the submanifold of M defined by

N = [m E Q I xp+l (m) = ••• Xd (m) = 0]

and let p be the associated projection of N into Q. L~~ zQ all %-~h order complex p-spaces (m,P) such that mEQ and P

= (Im/(I + I~+l))*, where P is any p-ideal having a set of gener~ tors o£ the form

where the hp+l, ..• ,hd are C'" functions on N. We then define the

collection of functions w~ , 'w; , "w; for a = ("'1"" ,,,,p) ,

1",1 ~ z , p+1 ~ r ~d , by

w~ (m,P)

'wa(m,P) = Re w"'(m,P) , r r

w'" (m,P) r

aah r

ax'" (m)

'rhe set of all such (w~ 'w l' r

"w } make k,cC; (M) into a real COO r p

!!lI~lit'.1! i. and a bundle ave!" >f.

We note that the usual kG (M) is a submanifold of k,,:(. (M) Lon,ist p p'

ing of all the (m,P) such that the c~responding p-ideal has a set of real-valued generators. These are the elements (in the domain of such a coordinate system) for which all "w'" = O. We

r call elements of kG (M) real z-th order p-spaces.

p

80

We now define, for each ye R, an applicatien t , of a ,Subset of y

k,cG (RP+l) ~ kG (RP+2). This is the application that carries P P

each xe RP+l ~ (x,y) e RP+2 and carries the complex p-plane whose

coordinates (relative to the w~, '",rJ. 1 ' "wrJ. 1 ' - obtained from 1 P+ p+

the usual coordinate system of RP+l) to the element of G (RP+2) p ,

whose coordinates are these same numbers, i.e.

o w. 1

w~ 1

The geometric construction in going from a linear partial differential op'erator to a partial differential equation, or to the symbol of the operator, is just the projection of a k-th oraer vector into a plane P, skew to both factors, followed by a projection into the second factor. Then in case the second factor is Rl the resulting vector, which is a function of P, can be described by its coefficients of orders 0 to k (there is one coefficient for each of these orders when the second factor is R1). so we have k+1 functions of P (depending on the initial vector field), and these include the symbol and the partial dif­ferential equation. In the case where the second factor is of dimension greater than 1, we obtain vector-valued functions of P. We now describe this geometric process more precisely.

Let M and N be real C· manifolds of dimensions p and q, and let d = p+q. Let m be a point of M and n a point of N. We shall speak of vectors tangent to M or to N at (m,n), meaning vectors tangent to the submanifolds M x (n) or (m) x N ~ by tangent vector we shall always mean complex tangent vectors. Let

k,CGp(m,n) be the set of k-th order complex p-spaces of M x N at

(m n)' let k,cM be the space of k-th order tangent vectors , , (m, n)

to M (really to M x (n) at (m,n), and define k,c N( ) similarly. m,n We have natural projections, that we denote by p and cr of

k,C(MxN)(m,n) into k,CM(m,n) and of k,C(MxN)(m,n) into k,CN(m,n)

p and cr are the k-th order differentials of the natural

projections of M x N into M x (n) and (m) x N (and we could

generalize to the case where M and N are only "k-th order factors"

of some manifold Q). From p we define a subset E of k,CGp(m,n)

by: E consists of all elements of k,cG (m n) on which p is non-p ,

81

singular. We then define a map V of E x k·~(m.n) into k,CN(m.n)

by:ifPEE and tEk,cM( ') and if t' is the unique element of P m.n

such that Pt' = t then

V(p.t) at'

We now write a formula for this V. in terms of coordinate systems. or equivalently. we wish coordinate expressions for V. in case M = RP and N = Rq • in terms of the usual coordinates of Eucli-

I

dean space. We hence assume now that M = RP • N = Rq and let u1 ••.•• us De the usual coordinate system of RS . Then. as usual. V(P.t) can be expressed as

V (P. t)

where the vS-are complex valued functions which we wish to find explici t Iy in terms of the coordinates of P and of t. Since V(P.t) is clearly linear in t the main thing will be to compute the functions vSa defined by

aa (P • -)

aua

and lal.;;; k •

lsi.;;; k. And we now wish to. express these vSa in terms of the

above coordinates {we:' • 'wa '''wa } of p. we shall write 1. r' r '

w~= 'w; + i"w~ and express the vSa in terms of the w~ .

First. for each a = (al •...• ap ) and S = (Sp+I •...• Sd) with

lal> lsi we define PSa to be the unique polynomial such that.

for all functions gp+I •...• gd in C· at m •

aa S+l Sd (m) [(gp+l - gp+l (m)) p ... (gd - gd (m))

aUa

P Sa ( ...• a l1g

r (m) •••. ) . au ll

82

Hence if N = the number of n = (n1, ... ,n p) such that 0 ~ n ~ a then Plla is a polynomial in (d-p)N variables. It is clear that this defines unique polynomials Plla . It will be important that the variables here are labellea with the subscripts (n,r) as ahov

The formula we desire to prove for ~he vSa is:

(1 .2)

and we nQW make the calculations. to prove (1.2). We first seek

aa the coordinates of that t! E P such that pt I = -- That t~

will have the form

(a) t' a

~ a aua ~

aY Lyaya -y-\.m)

au

~. ~ thru all y

we have

(b) V Sa (II) a(O, S),a

Hence we are interested in the 8ya for which Yl = ••••

But for the moment we consider general y. We write

From (a) we have the usual formula:

t' (U - U(m,n))Y a

(g* ~(m))(u - U(m,n))Y Ua

a _d_(m) (eU - U (m,n)) y • g) aua

o .

83

We write y = y' + y" where y' = (Yl' ... ,yp'O, ... ) and

y" (O, ... ,O,Y p+1 ' ..• 'Yd). So the above becomes

a a '_tf

-a(m) [ (U-U(m))Y (g-g(m))Y 1 au

(c) = L

We recall the fact, easily proved by induction,

(d)

(where a

(e)

n! 0 an if

a.! a 0:,- y t II

---'::.!-- ---, (m) ((g- g (m)) Y ) (a-y')! aua- y

If we take y = (D,S), so y' 0, y" y, this gives (1.2).

Now we specialize to the case where q= 1 (d=p+1). so the

vSa ' PSa ' aSa becomes vo.a,···.vk •a • po,a.···.Pk.a. ao.a.···,ak.a·

In this case we give the explicit expression for-the P_ by J .a

iterating the Leibnitz product rule. We have

j -2 j -1 j-l a! a a- n h an - n han

1, j-2 j-l), j-l, ---1··· j-2 j-l---y::T (a-n ) .... (n -n ·n . aua- n aUn -n aUn

1 j -1 . where this sum is taken over all choices of n , ... ,n such that

Applied at m to h = g-g (m), all undifferentiated terms vanish so

(1 .3) P. ( ... ,w n, ••• ) J,'"

84

where this sum is taken over all n1, •.. ,n j - 1 such that,

'" > n1 > n2 > ••• > nj - 1 > O. We note, in particular, that all

non-zero terms of Pj,'" are products of at least j w's, and

contain no wn with Inl > k-j+1. Hence

j-l n wp + 1

where this sum is taken over all n1, ... ,n j - 1 such that

'" > n1 > ••• > nj - 1 > O. If j=k and this is oJ 0 then we must

have 1",1 = k, and all these products of wp+1's are the same, all

being equal to

Furthermore, when j = k 1",1, k!/"'! is the number of such 1 k-l sequences n , ... , n (as is shown by an easy induction) so

Hence

(1 .4)

6 '" (w p ) P

p+l if I", I k

which is the usual formula for the symbol of (1.1).

As given by (1.2) and (1.3) the v j ,'" are functions on k,CGp(M) but

since all non-zero terms in (1.3) contain only wn with

85

Inl < k-j+1 we see that v. is the lift of a function defined on J ,ct

k-j+l,CGp(M) , hence the same is true of v(P ~ a a ) 'Llctl<k ct --ct - aU

and this function defines the j-th partial differential equation

associated with A, to which we referred in the introduction. In

particular, the k-th equation, given by (1.4), is defined on

1, c G (M). p

Centro de Investigaciones del I.P.N. Mexico

Mass. Inst. of ~echnology, Cambridge. Mass., U.S.A.

Recibido en agosto de 1970.

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Revista de 1a Uni6n Matematica Argentina Volumen 25, 1970.

UNIFORM APPROXIMATION TO BOUNDED ANALYTIC FUNCTIONS T. W. Gamelin and John Garnett

Vedicado al P~o6e4o~ Albe~to Gonz4lez Vomlnguez

Let ~ denote the open unit disc in the ~omplex plane C, and let H-(~) denote the algebra of bounded analytic functions on ~~ We wish to prove the following theorem, which was proved in the case that E is open by A. Stray [51.

THEOREM 1. Let f e H-(~) , and ~et E be a subset of b~ such that

f e~tends continuousLy to each point of E. Then there is a se­

que nCR fn e H-(~) such that each fn e~tends to be ana~ytic on some

neighborhood of E, and fn converges uniform~y to f on ~.

For E a subset of b~ , let Hi denote the subalgebra of H-(~) of functions' which extend continuously to each point of E. The the­orem asserts that the functions in H-(~) which extend analytically to a neighborhood of E are dense in Hi. Combining the theorem with Carleson's corona theorem, we obtain the following corollary, which is due to Detraz [21.

COROLLARY. The open unit disc ~ is dense in the ma~imaL idea~

space of Hi .

Proof of the main theorem. We proceed now directly to the proofs. The symbols C~, C1 ' •••• will all denote universal constants. All norms will be ~upremum norms.

LEMMA 1. Let Q .be a closed subset of b~, let W be an open subset

of C at a positiv~ distance fromQ, and let € > O. Let f be a

bounded Borel function on C, such that f is analytic on~. Sup­

pose there is !l continuous function u in a neighborhood of Q such

that

I fez) - u(z) 1 <d

for aZL z e ~ whioh are near Q. Then there is a bounded Borel

function h s~ch that

88

(i) h is analytia on an open set aontaining t:. UQ.

(ii) h exteMds analytiaalZy aaross any ara on bt:. aaross whiah f extends analytiaally.

(iii) f-h is analytia on Wand satisfies If-hi < E there.

(iv) If(z)-h(z) I < C1d for all z E t:..

Proof. For 6 > 0, the open c-neighborhood of Q will be denoted by Qfl5). By hypothesis, we can choose Co > 0 so small that Q (co) does not meet W, that u is defined on Q(c o)' and that If(z)-~(z) I < d for z E Q(15 0) n t:.. Since u is uniformly continuous in a neighborhood of Q,we can shrink Co so that also !u(z)-u(l;)! < d for all z,1,; E Q(c o) satisfying IZ-1;1 < ZOo .

'Let r be the union of the arcs on bt:. across which f extends analy! ically. There is then an open set U containing r such that !f(z)-u(z)! < d for all z E Q(c o) nu. Let F be the function which coincides with u on Q(c o)\ (t:. UU), and which coincides with f else­where; Then F is a bounded Borel function which satisfies

(A) IF(z)-F(I,;)I < 3d whenever Z,I,; E Q(c o), Iz-t;1 < 215 0 •

~ince'F coincides with f on t:., on W, and in a neighborhood of r, l.t will suffice to obtain the conclusions of the lemma, with f replaced by F.

Now we are in a position to use Vitushkin's scheme for approxima­tion, as developed for instance in Chapter VIII of [3], or in [6]. Because we are working on the unit circle, we can employ the ver­sion of this technique matching only one coefficient of the appr~ priat.e Laurent expansions (cf. [61, V. 4) . The details are as fol­lows.

For a fixed 15 satisfying

t:.k = {I z - zk I < c} , zk E

gk supported on t:.k such

o < 0 < 00 • choose discs Q , which cover Q, and choose functions that 0 ~ gk ~ 1 , L gk = 1 in a neigh-

I ag_k I borhood of Q ~ 4/15 , and no point z is contained in more az .

than C2 of the discs t:. k • If

F(I,;) - F(z) ag k dx dy , t; E C , I,; - z a z

89

then Gk is a bounded Borel function, Gk is analytic wherever F is

analytic, Gk is analytil= off Ak , and Gk(ao) = O. Moreover, F-r Gil. is analytic on the interior of the se~ on which r gk assumes th~

value 1. In particular, F - L Gk is analytic in a neighborhood of Q. The condition (*) can be used to estimate Gk , yielding the bound

Suppose the expansion of Gk near ao is given by

By Schwarz's lemma we have

Now the analytic capacity of the connected open set Ak\6 is at least one fourth its diameter. Hence we can find a continuous function Hk on C such that Hk is analytic off a compact subset of

Ak\ 6 ,

(**)

for z E Ak . As Hk has been defined so that Gk - Hk has a double zero at 00, the estimate (**) persists for all z E C.

Now we define

Since F - L Gk is, analytic wherever F is analytic, and each Hk is

90

analytic in a neighborhood of ~, the function h is analytic on fi

and extends analytically across r. Moreover, h is

neighborhood of Q, so that (i) and (ii) are valid. is analytic off Q(o), F - h is analytic on W. To

proof, it suffices now to obtain the estimates in

analytic in a Since F - h

complete the

(iii) and (iv).

To verify (iii), fix z E Wand consider F(z)-h(z) = L[Gk(z)-Hk(z)l. Since no point lies in more than C2 discs fi k and each fi k meets bfi, there is a grand total of at most 2nC2/o discs fi k . Thus by (**)

(***)

Taking 0 much smaller than dist(W,Q(oo))' we get IF - hi < £ on W.

To verify (iv), we first observe that F - h = L[G k - Hkl is ana­lytic off Q(o), so that it suffices to obtain the estimate

for z E Q(o). So fix a point z E Q(o). Let M(m) be the number

of discs fi k whose centers satisfy mo .;; I z - zk I < (m+1) o. Since no point z is contained in more than C2 discs, there will be a constant Cs such that

if o .;; m .;; 1/0

providing 0 is sufficiently small. (Here we use the geometry of the unit circle, and the fact that z is close to the unit circle)

Using the estimate IGk(z) - Hk(z) I .;; C4d for the at most C2 indi-

ces k for which Iz - zk l < 0, the estimate (**) for those k for

which mo .;; Iz - zk l < (m+1)o and 1 .;;k.;;1/o , and the same esti-

mate used to obtain (***) for those k for which Iz - zk I ;;;.1 , we

find that

That completes the proof.

91

LEMMA 2. Let f E H""(ll) , and '"let E be a subset of bll. Suppose

thepe is an open set U aontaining E, and a funation u defined and

aontinuous on U. suah that If(z) ·u(z) 1 < d lop an z E U n ll. Then

thepe is h E H""(ll) suah that h extends to be ana'"lytia in a neigh­

bOPhood of E, and

sup I£(z) - h(z) 1 0;;; Cod "Ell

Ppoof. By replacing E by U n bll , we can assume that E is rela­

tively open in bll. Then we can write E ( U Qn) U (U Rn) , where

Ql' Q2"" are ,p'airwise disjoint closed intervals, Rl' R2 , ... are

pairwise disjoint closed intervals, each Qn joins the endpoints

of two of the Rk's, and each Rn joins the endpoints of two of the

Qk's. Then we can choose on > 0 so that the on-neighborhoods of

the Qn's are pairwise disjaint.

Starting with ~O ~ f, we construct by induction a sequence of Bo­rel functions $' such that

n

(i) ~n is analytic on ll, and ~n is analytic on a neighbor­hood of Qn ,

(ii) ~n - ~n~l is analytic off the on-neighborhood of Qn

and satisfies I'" '" 1 < d/Z n there. "'n - "'n-l

Indeed, having chosen ~n-l' we note that on the part of linear Qn we have

< d/Z n- 1 + ,., + d/Z + d < Zd

so that Lemma 1 will provide the desired function ¢n'

For each z, 1 cp. (z) - ~. 1 (z) 1 < d/Z j for all but at most one index J J-

j, while always I~. - cp. 11 < ZCld. Hence the ¢. converge point-J J - J

92

wise to a function ~ satisfying

The convergence is uniform on any compact set at a positive dis­tance from lim Qn = bE , so that ~ is analytic on h. Since

• j - <I> j -1 is analytic on the on -neighborhood of Qn for j ." n ,

while ~n - ~n-l is ana,lytic in a neighborhood of Qn ' ~ - f will

also be analytic in a neighborhood of each Qn'

Now we perform .essentia11y the same construction on the Rn' s,

being careful to retain analyticity across the Qn's. Choose En>O so that the En-neighborhoods of the Rn's are disjoint. Starting with $0 ~ constrUFt by induction a sequence $n such that

(i) $n is analytic on a neighborhood of h U Rn

(ii) $n is analytic across the arcs of bh across which .n-1 is analytic.

(iii) $n - $n-I is analytic off the En-neighborhood of R n

and satisfies I$n - $n-1 1 < d/2n there.

(iv) h n - ·n-1 11 < C7d.

This is again possible by Lemma 1. As before we see that the 'n converge to a function h. uniformly on sets at a posittve distance from bE, such that h E H~(hl, h extends analytically across each Qn and across each Rn , and Ih - $1 < (C 7 + 1)d. Then h is analyt-

ic across E, and Ih - fl < (C 7 + 2C I + 2)d, so that h is the re­quired function.

COROLLARY. Let f E H~ (h) • 1..et E be a subset of M • and 1..et

d > O. Suppose that for eaah Z E E. the diameter of the a1..u8te~

set of f at Z is 1..ess than d; Then there is h E H~(h) suah that h

extends to be ana1..ytia in a neighborhood of E. and

sup If(z) - h(z)1 < COd zEh

93

Proof. As the diameter of the cluster set of f at z E b6 is an upper semicontinuous function of z we can replace E by a larger open set. It is now easy to construct a continuous function sat­isfying the hypotheses of Lemma 2.

Proof of Theorem 1. If f extends continuously to each point of E, then we can take the d of the preceding corollary to be arbitra­

rily small. The resulting h's will approximate f uniformly on 6, and they will be analytic on E.

Proof of the CoroZZary to Theorem 1. To show that 6 is dense in the maximal ideal space of H; , one must show that if

fl, ... ,fn E H; satisfy Ifll + ••. + Ifni> 6 > 0 on 6, then there

are gl, ... ,gnE H; satisfying L fjgj = 1. In fact, it suffices

to show this for fl, ... ,fn lying in any dense subalgebra of H; ,

so that by Theorem 1 we can assume thatf l , ... ,fn extend analytical

ly to a neighborhood of E. Then there is a simply connected open

set U ~ 6 U E such that f l , ... ,fn are bounded on U and satisfy

If I > 6/2 there. n

By Carleson's theorem, applied to U,

there are bounded analytic functions gl, ..• ,gn on U satisfying

L fjgj = 1. Since the gj 's belong to H; , they are the required

functions.

CONCLUDING REMARKS. For a subset E of b6, let L; denote the uni­

form closure of the functions in L~(de) which extend continuously

to an open set containing E. Then L; consists of the functions

in L~ which are constant on each "fiber" of the maximal ideal

space of L~ lying over points of E. If we identify functions in

H~(6J with their radial boundary values, we can regard H~(6) as a

subalgebra of L ~ (de) . Under this identification, H~ E

becomes a

subalgebra of L; • In fact, H; = H~ n L~ E

, and H; is a logmodular

subalgebra of L~ E (cL Detraz [2] ) •

For f E L~(de), we define as usual the distance from f to L; by

94

and we define d(f.H;) similarly. Lemma 2 can be restated as fol­lows.

THEOREM 2. There is a universal. constant Co such that !or aZ'Z. E C b~ and al.l. f E H~(~) •

We hope to study the smallest possible constant Co in another paper.

ACKNOWLEDGMENTS. The authors would like to acknowledge the partial support of the National Science Foundation Grant *GP-11475 in the preparation of this manuscript. The first-named author would like to acknowledge the partial support of the Alfred P. Sloan Foundation.

REFERENCES

[1] L. CARLESON. Th~ co~ona zh~o~~m, Proceedings of the Fifteenth Scandinavian Congress (Oalo. 1968), Springer Lecture Notes in Mathematics, Vol. 119, pp. 121-132~

[2] J. D.ETRAZ, Ezud~ du ~ p~Cz~~ d' alg e.blL~~ d~ 6 oncUon~ analyi:.i­quu ~u~ l~ d.i~qu~ un.izl, C.R. Acad. Sci. Paris 269 (1969), 833-835.

[3] T.W. GAMELIN, Un.i601Lm alg~blLa4, Prentice-Hall, 1969.

~] E.A. HEARD and J.H. WELLS, An .inz~lLpolaZ.ion plLobl~m 601L ~ub­alg~b~a~ 06 H~,·Pacific J. Math. 28 (1969), 543-553.

(5] A. STRAY, An app~olC..imaz.ion zh~OIL~m 60~ ~ubalg~b~a~ 06 H~. Pa cific J. Math., to appear.

[6] A.G. VITUSHKIN, Analyz.ic capac.izy 06 ~~z~ and plLobl~m~ .in a~ plLox.imaz.ion zh~o~y, Uspehi Mat. Nauk 22 (1967), 141-199. Russian Math. Surveys 22 (1967), 139-200.

Recibido en agosto de 1970.

University of California Los Angeles

Revista de 1a Union Matematica Argentina Vo1umen 25, 1970.

PROBABILIDADES SOBRE CUERPOS CONVEXOS Y CILINDROS por L.A.Santa16

Vedieado al P406e~o4 Albe4to Gonz~lez Vomlngue,

SUMMARY. H. GIGER and H. HADWIGER [2] and R.E. MILES [3] have

recently considered different questions related to lattices of figures (convex bodies, r-flats or convex cylinders)

These lattices are ass~ed generated by N independent which intersect a fixed sphere S of radius R as Rand - in such a way that N/R tends to a positive constant, density of the lattice. In this paper we prove:

in E . n

figures N tends to called the

a) The result does not change if instead of the sphere S we con­sider a convex body of arbitrary shape, which expands to the

whole space En; this is a consequence of our Lemma 2.

b) This result is applied to Theorem 1 (distribution function

(3.4) of the number of cylinders of a lattice which are intersect

ed by a convex body Ko placed at random in space), which is essen tially due to MILES [3] with different proof.

c) Theorem 2 refers to lattices of convex cylinders i~ E3 cross­ed by an arbitrary convex cylinder and we find the distribution

function (4.6) of the number of intersected cylinders.

1. INTRODUCCION Y FOiMULAS FUNDAMENTALES. Todo cuerpo convexo

Kn del espacio euclideano n-dimensional En tiene asignadas sus proyeaaiones medias Wi(Kn ) (i=1,2, ... ,n-l). Estos invariantes fueron introducidos por Minkowski y reciben distintos nombres: en

aleman se llaman Quermassintegrale, en frances travers exterieurs,

en ingles, a veces mean aross seational measures y tambien

Quermassintegrals. Un estudio de los mismos puede verse en la

clasica obra de BONNESEN-FENCHEL [1]

Los valores extremos de las proyecciones medias son

(1 . 1 ) WO(Kn) = V = volumen de K n

nW 1 (Kn) = F area de K , n

nWn_1 (Kn) B = norma 0 anchura media de K n

96

volumen de la esfera unidad de En'

o sea,

K n

donde r es la funcion "gamma" que tiene la propiedad de recurren­cia r(h+1) = hr(h).

Para una esfera de radio R de En ' las proyecciones medias valen

(1 .2) Wi (esfera)

Para n 3 , es

(1 .3) W2 = M/3 W3 = (4/3)'11"

donde M es la anchura media de K3 , que si aK3 es de clase C2 coin :ide con la integral de curvatura media de K3

Para el plano, n = 2, es

(1 .4)

donde f es el area de la figura convexa plana y u su perimetro.

Sea Ln_p un subespacio lineal de dimension n-p de En y sea Kn_p

un cuerpo convexo ~ontenido en L . Supondremos siempre que se n-p trata de cuerpos convexos acotados, de manera que los invariantes W. son todos finitos.

1.

DEFINICION 1. Llamaremos aiZindro Z de seccion recta K p n-p al

conjunto de L en los n-p

los subespacios lineales Lp que son ortogonales a puntos de K n-p

El numero p = 1,2, ... ,n-l se llama el rango del cilindro Z y es p

igual a la dimension de sus generatrices.

DEFINICION 2. Llamaremos proyeaaiones medias W~(Z ) de Z a las 1. p P

proyecciones medias Wi de la seccion recta Kn_p considerada como

cuerpo convexo de L n-p

El acento un cuerpo

En' Por

97

indica, precisamente, que K n-p debe considerarse como convexo de L , no como un cuerpo convexo aplastado de n-p tanto, para W~(Z ) solo hay las posibilidades i ~ 1,2, •.

1 P .. ,n-p , siendo W~o(Z ) el volumen de la seccion recta de Z y

p p W (Z)-n-p p K n-p

Supongamos ahora un conjunto de cilindros congruentes con Z. La p

densidad para medir conjuntos de cilindros congruentes, invarian-te con respecto al grupo de los movimientos de En' es

(1 .5) dZ p

... dL 1\ dK [0] p n-p

donde dL p es la densidad paTa subespacios lineales de dimension

p, referente a un subespacio generatriz de Z y dK [0] es la p n-p densidad cinematica en el espacio ortogonal Ln_p alrededor del

punto 0 en que se cortan Ln_p y Lp' Laflecha sobre Lp indica que este subespacio debe considerarse orientado. Esta densidad se encuentra en [4] para n = 3 , P = 1 Y la generalizacion a E

n no ofrece dificultades (ver tambien MILES [3]).

Con la densidad (1.5) la llamada formula fundamentaJ para cilin­dros, que da la medida del conjunto de cilindros Zp que tienen punto comun con un cuerpo convexo fijo K se escribe (ver ~] para n = 3)

(1 .6)

n-1 Li =p-1 [ n-p )

i-p+1 W·+ 1 (K) W~ 1 .(Z ) 1 n- -1 p

Como ejemplos de esta formula, consideremos los casos posibles de

E3 •

1. n = 3 p 1 . Resul ta

(1. 7) 21T(rrF + uM + 4rrf)

2. n = 3 P

(1 .8) 2M ·f 4rra.

En este ultimo caso, Zl es una banda de pIanos paralelos a distan

cia a.

98

2. DDS LEMAS .. En los trabajos referentes a conjuntos de cuerpos convexos 0 de subespacios lineales 0 de cilindros distribuidos al al azar, con ley uniforme, en En' 10 que se hace siempre es:

a) Considerar un conjunto de N cuerpos convexos 0 espacios linea les 0 cil indros que cor tan a un cuerpo convexo fij 0 dado K;

b) Suponer luego que K crece de tamafio hasta cubrir todo En' al mismo tiempo que tambien N crece, de manera que entre N y una ca­racteristica de K (el volumen, el area 0 cualquiera de las WiCK)) exista una relaci6n constante # 0, ~ que se llama la densidad en En de las figuras gepmetricas consideradas.

En general, por simplicidad, se toma que K sea una esfera, pero cabe la duda de si, partiendo de otra familia de cuerpos convexos, se hubiera llegado 0 no al mismo resultado. Es decir, queda la duda de saber si se puede hablar de una "red de figuras" (cuerpos convexos, r-espacios 0 cilindros) 0 si debe especificarse la man~ ra como esta red fue generada a partir de un cuerpo convexo K que CTece a todo el espacio.

En los casas considerados por GIGER-HADWIGER [2] y MILES [3] el resultado es independiente de K y por tanto esta justificado que esos autores tomen una n-esfera cuyo radio crece hasta infinito. Pero falta la demostraci6n, que vamos a dar aqui. Para ello nece sitamos dos lemas.

LEMA 1. Sea R eZ radio de Za esfera maxima aontenida en un

auerpo aonvexo K de En. VaZe entonaes Za desiguaZdad

(2.1)

Demostraaion. Puesto que Wo(K) es el volumen de K, llamando h a la funci6n de apoyo respecto del centro de la esfera maxima conte nida en K y do el elemento de area en el punto de contacto, es

(2.2) - f h do . n oK

Pero en cualquier direcci6n es R ~ h, de donde, teniendo en cuenta

que WI (K) = Fin, resulta (2.1).

99

LEMA 2. Sea K(t).una familia de cuerpos convexos de En (O.,;;;t .,;;;"')

tales que:

ii) Para cualquier punto P de En' existe un tp taL

que, para todo t > tp es P E K(t).

La ultima condicion indica que K(t) tiende a lZenar todo eZ espa­

cio para t + "'. Con estas condiciones es

(2.3) lim Wi+l (K(t))

° t+'" WiCKet))

para i = 0,1, ... ,n-1, cuaZquie1'a que sea K(t).

Demost1'acion: Pongamos w~n) en vez de W.(K) para poner de mani-1. 1.

fiesto la dimensi6n ndel espacio que contiene K. Supongamos que

proyectamos ortogonalmente K sobre un hiperplano Ln_1 y sea dOn_1

e1 elemento de area de la esfera unidad de En correspondiente a

la direcci6n de proyecci6n. Obtenemos asi el cuerpo convexo pro­

yectado Kn_1 .cuyas proyecciones medias representamos por w~n-l) .

Dentro de L 1 proyectamos K 1 ortogonalmente sobre un L Z segun n- n- ~

la direcci6n definida por dOn_Z' obteniendo un nuevo cuerpo con-

vexo cuyas proyecciones medias representamps por W~n-2) Proce-

diendo sucesivamente, para cada r, q (r ~q) vale la siguiente

f6rmula (generalizaci6n de una cIasica f6rmula de KUBOTA) (ver [5]),

(2.4) Wen) r

n-q

nO Z· .. O 1 n- n-q-J w(n-q) dO 1 A ... A (0

r-q n- n-q

donde 0. indica el area de la esfera i dimensional, 0 sea esta te 1.

lacionada con los Ki por la igualdad 0i_l = i Ki

Apliquemos (2.4) a r = r , q = r y a r = r+1 , q r. Resul ta

(2.5) Wen) n-r

J w(n-r) dO A ... A dO

r nO n-Z" .On_r_l 0 n-l n-r

Wen) n-r

J w(n-r) dO A ... A dO

r+l nO n-Z" .On_r_l 1 n-l n-r (2.6)

100

La desigua1dad (2.1) da

(2.7) w(n-r) ~ R w(n-r) o 1

donde R es e1 radio de 1a esfera maxima, de dimension n-r, conte­nida en K n-r

Si R es e1 radio de 1a maxima esfera n-dimensional contenida en n

K. es Rn ~ R Y por tanto vale tambien

W(n-r), R w(n-r) o ..... n I

De aqui y de (2.5), (2.6) resulta

w(n) ~ R w(n) r n r+l

y puesto que Rn ~ ~ cuando K(t) crece hasta 11enar todo e1 espa­cio (o sea, cuando t ~ ~), resulta (2.3).

EZ caso n = 3. Consideremos en particular el caso del espacio or dinario de 3 dimensiones. Las relaciones (2.3) se escriben

(2.8)

siendo la ultima trivial.

.J:!..... ~ 0 F

Cabe considerar que sucede con los cocientes F3/ 2/V , FM/V , M3/V, para t ~ ~ Las clasicas desigualdades isoperimetricas dan

(2 9) F 3 / 2 --~611i

V ~~12 11

V

En cambio no existe una a.cotacion .superior.. Consideremospore­jemplo un paralelepipedo recto cuya base sea un cuadrado de lado a y 1a altura sea b. Es

F = 2 a 2 + 4 a b ,M = 211 a + 11 b •

Supongamos que a + ~ y b = A a (A constante). Resu1ta

101

F3 / 2 .. (2+4h)3/2 M F (2+4h) (2+h)1T M3 (21T+h) 3

V h V h V h

Y estes valores pueden ser tan grandes como se quiera tomando h

suficientemente pequeno.

3. CUERPOS CONVEXOS EN UNA RED DE CILINDROS.

Sea K = K(t) un cuerpo convexo de En y supongamos que contiene en su Interior otro cuerpo convexo Ko. Se dan N cilindros Zp al azar (vale decir, con la densidad (1.5)), independientes, que cor tan a K(t). ~Cu§l es la probabilidad de que exactamente r de e-

1105 cor ten a KO?

La solucion es inmediata, a saber,

(3.1)

siendo

(3.2) p

p = (N) pr (l_p)N-r r r

m (Z ; Z n Ko # 1/» p p

donde el numerador y el denominador est§n dados por la formula (1 .6) .

Supongamos ahora que K(t) sea una familia de cuerpos convexos en las condiciones del Lema 2 y hag amos t .... 00 , al mismo tiempo que N .... 00 , de manera tal que

(3.3) N Wi) (Z ) ---=_ .... p'-- .... D

W (K) p

siendo D una constante positiva, que llamaremos la densidad de los cilindros Z en E .

p n

Sustituyendo la expresi6n (1.6) en (3.2), haciendo t .... 00 y tenie!!,

do en cuenta (2.3) y tambien que W~_p(Zp) = Kn _p , resulta, en el llmi te

(3.4)

donde

(3.5)

P* r r!

n-1 L

i=p-1

Por consiguiente:

102

D i. J exp (- D ~ J K W (Z ) K W (Z ) n-p 0 p n-p 0 p

TEOREMA 1. Dada en En una red de aiZindros aonvexos Zp aoZoaados

aZ azar e independientemente unos de otros aon densidad D, eZ nu­

mero de aiZindros aortados por un auerpo aonvexo de' prueba KO ao­

loaado aZ azar en eZ espaaio, sigue una distribuaion de Poisson

de parametro

D ;:

En consecuencia, el num.ero medio de cilindros que son cortados

por Ko es

Eer) A

4. REDES DE CILINDROS EN E3

Queremos considerar ahora el caso en que, en vez de Ko' tenemos

otro cilindro Z , es decir, queremos buscar la distribuci6n del q .

numero de intersecciones de un cilindro de prueba Z con los ci-.

q

lindros de una red que cubre el espacio con densidad D. Los re-

sultados son un poco complicados para el caso general de En por

10 cual nos vamos a limitar al caso de E3 y de cilindros convexos

propiamente dichos.

Sea K un cuerpo convexo fijo de area F y curvatura media (0

proyecci6nmedia) M. Sea Zo un cilindro que corta a K y sea fo el

area y u o el perimetro de la secci6n recta de ZOo Exceptuando p~

siciones del contorno que no van a influir en el limite, podemos

103

suponer que la intersecci6n Zo nK es un cilindro limitado,de 'rea 2fo+auo+£ y proyecci6n media (0 curvatura media) lIa+(1I/2)u o+£,sie!! do ! la longitud de un segmento de generatriz contenido en K e in­dicando con £ cantidades (diferentes) tales que efa ... 0 para a'" CD •

La medida de los cilindros ZI (cuya secci6n recta tenga por 'rea fl y perimetro u 1) que cortan a Zo n K, segun (1.7) vale

La medida de los ZI que cortan a K es

(4.2)

y por tanto: l.a probabil.idad de que ZI' supuesto dado al. aaar aon

la densidad uniforme dZ 1 , aorte a Zo n K sabiendo que aorta a K vaZe

(4.3) p

Si suponemos N cilindros congruentes con ZI' independientes, que ~ortan a K, la probabilidad de que r de ellos corten a Zo n K es­U dada por

P (W) pr (1 _ p)N-r r r

Supongamos ahora que K es una esfera de centro un punto fijo 0, cuyo radio R crece hacia infini to, al mismo tiempo que el mlmero N de cilindros que cortan a K tambien crece de manera tal que

(4.4)

siendo D una constante positiva. Diremos entonces que se tiene en E3 una red de cilindros ZI de densidad D. Recordemos que para una esfera de radio R es F = 411 R2 , M = 411 R.

Si h es la distancia de ° al cilindro Zo ' es (a/2)2 y por tanto

104

(4.5) a = + E

es decir, se cump1e aiR ~ 1, cua1quiera que sea 1a posIcIon fija

de 0 (0 sea, cua1quiera que sea h). Sustituyendo (4.4) y (4.5) en 1a expresi6n de P r y haciendo N ~ - resu1ta, en e1 limite

(4.6)

Es decir:

p* r

r!

TEOREMA 2. Supuesta en el. espaaio E3 una red de aiLindros aon­

vexos 21 distribuidos at azar aon densidad D, el. numero de ail.in­

dros que son interseaados pOI' un ail.indro aonvexo 20, sigue una

distribuaion de Poisson de parametro

Se observa que solamente intervienen los perimetros de las seccio

nes rectas de los ci1indros, no las areas.

BIBILIOGRAFIA

[1] BONNESEN, T. - FENCHEL, W., Theo~ie de~ Ronvexen Ko~pe~, Ergebnisse der Mathematik, Berlrn, 1934.

[2] GIGER, H. and HADWIGER, H., Uebe~ T~e66zah!wah~~chein!ich­Rei~en im Ei~Ro~pe~6e!d, Zeits. fur wahrscheinlichkeitsthe0-rie 10, 1968, 329-334.

[3] MILaS, R.E., Poi~~on 6lat~ in Euc!idean Space~, Pa~t 11: Ho mogeneou~ Poi~~on 6!at~ and the comp!ementa~y theo~em, (en prensa). '

[4] SANTALO, L.A., Integ~a!geomet~ie 5, Uebe~ da~ Izinema~i~che Ma~~ im Raum, Hermann, Parrs, 1936.

[5] SANTALO, L. A., Su~ !a me~u~e de,~ e~pace~ !iniai~e~ qui cou­pen~ un co~p~ convexe et p~ob!~me~ qui ~'y ~a~tachent, Co­lloque sur les questions de realite en Geometrie, Liege, 1955, 177-190.

Facultad de Ciencias Exactas y Naturales. Buenos Aires.

Recibido en setiembre de 1970.

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Revista de la Union Matematica Argentina Volumen 25, 1970.

A REMARK ON SIDON SETS R. Kaufman

Vedicado at P406e~o4 Atbe4to Gonz4tez Vomtnguez

Let A = (A k); be an increasing sequence of positive numbers and E

a compact set of real numbers. Then A is a Sidon set for E pro­vided an inequality

o > 0 constant

iA t holds for all polynomials Lake k with frequencies in A. It is natural to study sets E with the property that there is a Sidon

~et A for E subject to a growth condition; the most familiar con­dition is log Ak = O(k) [4, p. 223]. Following the method of Hel

son and Kahane [11, it is proved in [5] that if T > 1 and Haus­dorff dimension E > 0 13,11], there exists a Sidon set A for E

fulfilling Ak+1 < TA k .

It seems very difficult to decide whether this condition on a Si­

don set for E forces E to have positive dimension; concerning a

related problem a fina.1 answer is obtained by Ivasev-Musatov (2].

In this note we prove that the theorem stated above is best-pos­sible in a certain direction.

Let h(u) be a continuous increasing function on [0,+00) and h(O)=O,

and let us write E E (h) provided there is a Borel probability

measure ~ concentrated in E such that ~(I) = O{h(III))for all in­tervals I. A theorem of Frostman [3, p. 27] shows that dim E > 0 if and only if E E (uc ) for a c > O.

THEOREM 1 .

Then we aan

A for E aan

THEOREM 2.

and a system

tains E.

Suppose that for every a > 0, ua = o(h(u)) (u + 0) aonstruat a aompaat set E E (h) so that no Sidon set

fuLfiZL log Ak = O'(k).

Let there exist, for eaah R > 1, and integer N > R, (Im)!=1 of N intervaLs of Length N- R whose union ao~

Then no Sidon set A for E aan fuZfiLZ log Ak O(k)

106

Theorem will be derived afterwards from Theorem 2. To prove

Theorem 2 we suppose on the contrary that for any integer M ~ 1

iA t L ± e k satis-and any choice of signs ± the polynomial pet)

fies max I p (t) I ~ 8M , while 8 log Ak ".; k for each k. Let then E

R~ 48- 2 and N be the integer specified in the hypotheses; next .LR

l R 8 let M be defined by AM ".; N 2 < A whence l+M> log N M+l , 2

Choose any a E m t (1 ".; m ".; N) and observe that

m

max I p (t) I ".; max Ip(am)1 + N-Rmax I p' I E m

< max Ip(am)1 + MAM N- R m

Thus max Ip(am)1 ~ 8M-W RMAM ~ 1 8M. 2 m

To complete the proof we choose the signs ± as the Rademacher func

tions ¢l(x)' ... '¢M(x) on (0,1); we write P instead of dx for Le­besgue measure, and p(t;x) to indicate the dependence on x. We

have only to prove that for large N

"

and this is a consequence of

P{lp(t;x)1 ~ -00 < t < 00

For any y > 0

1 1My2 f exp ylRe p(t;x)ldx"'; 2(cos hy)M ".; 2e 2 o

and similarly for the imaginary part. Therefore for any b > 0 we

obtain

1My2 _1 by p{lp(t;x)1 > b} ".; 4e 2 e 2

4 exp _l b2W 1 4 (for the best value of y > 0)

4 _.l 82M when b 1 exp 16 "20 M.

107

Using the inequality M+1 > Ra log N we obtain 2

1 Ra3 p{ Ip(t;x) I > b} .;;; CN-TI

This proves Theorem 2.

The deduction of Theorem of the facts in [3, I,II].

tive numbers decreasing to

from Theorem 2 is an easy consequence

Let r = (r j )7 be a sequence of posi­

o and Er the set of all sums

Ii=1 ± r 1···r j Then ~ has the property specified in Theorem 2.

Moreover, if h is the function defined in Theorem 1, there is a

sequence r such that 2- j = o(h(r 1 ... r.)) and now E E (h). J r

REFERENCES

[I] H. HELSON and J. P. KAHANE, A Fou~~e~ method ~n d~ophant~ne p~ob.tem.6, J. Analyse Math. 15 (1965), 245-262.

[2] 0.5. Iva;ev-Musatov, M~.6et.6 and h-mea.6u~e.6, Mat. Zemetki 3 (1968), 441-447. (Russian).

[3] J.-P. KAHANE and R. SALEM, En.6emb.te.6 Pa~6a~t.6 et Se~~e.6 T~~gonomet~~que.6, Hermann, Paris, 1963.

[4] J.-P. KAHANE, Gene~a.t~.6at~on d'un theo~eme de S. Be~n.6te~n, Bull. Soc. Math. de France 85 (1957), 221-230.

[5] R. KAUFMAN, A ~andom method 6o~ .taeuna~y .6e~~e.6, J. Analyse Math. 22 (1969), 171-175.

Recibido en febrero de 1970.

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Revista de la Union Matematica Argentina Volumen 25, 1970 •

.oN UNIQUENESS C.oNDITI.oNS F.oR THE INITIAL VALUE PR.oBLEM F.oR THE DIFFERENTIAL EQUATION y' = f(x,y)

by S. C. Chu and J. B. Diaz

Ved~cado at P~o6e~o~ Albe~to Gonz~lez Vomlnguez

Consider the initial value problem

(1 ) y' f(x,y) x E (O,a)

yeO) YO

with f a real valued function defined on the (open) strip S: (O,a) x (-oo,oo} , where "a" is a positive constant. Notice that no continuity of any sort is assumed about f. By "a solution to problem (1) on the interval [O,a)" is meant, throughout the pres­ent paper,a real valued function y, defined and continuous on the "half-closed" interv<l1 0 OS;; x < a , possessing a finite deriv­ative y' (x) on the open interval 0 < x < a , and which satisfies the ordinary differential equation y' (x) = f(x,y(x)) on the open

interval 0 < x < a, and also obeys the initial condition y(O)=yo. In this exposition, we are concerned with the uniqueness of the solution of (1) on [O,a), with f satisfying "one sided" versions of some of the various well known "two sided" conditions, such as Lipschitz, Nagumo [1], Osgood (2), Krasnosel'skii-Krein (3), and Moyer (4) - the last one is a general condition which, in a sense, includes the previous ones, as well as others. Further­more, we wish to present a unified, direct, and, at the same time, elementary approach to the proofs of these uniqueness theorems. The only tool we will use here is the elementary mean value theo" rem of the differential calculus. Since we wish to stress the method, rather than the generality, of the results, we will, for simplicity, focus our attention on the case of a single equation. The method presented can, however, be easily adapted to the more general case of systems. of ordinary differential equations, for example (compare the discussion in the book of W. Walter (5)).

Use of this elementary method was made in the paper of Diaz and Walter (6); however, there it was applied only to the cases of the two sides Lipschitz and Nagumo conditions, and in a way which does not appear to be immediately applicable to other conditions on f.

110

In section 2 of the present paper, we will redo these two cases,

in a yet simpler form, in one sided versions of the Lipschitz and Nagumo conditions (see Theorems 1 and 2); ~nd then extend this

method to prove theorems involving one sided versions of the con­ditions of Osgood, Krasnosel'skii-Kreiri, and Moyet.

Theorem 5 contains Theorem 1 (one sided Lipschitz) and Theorem 4

(one sided Osgood)as special cases, while Theorem 6 contains Theo rem 2 (one sided Nagumo)as a special case. Thus, Theorem 5 ap­

pears as that one may call "a prototype of the one sided Lips­chttz-Osgood uniqueness theorem", while Theorem 6 appears as what

one may call "a prototype of the one sided Nagumo uniqueness theo rem".

It may appear, at first glance, {see, for example, J.H. George [7] ,

who states this opinion relative to the work of Moyer [4]) that

all the uniqueness theorems in question may be obtained as "spe­cial cases of Okamura's theorem". But this first impression is not correct, because Okamura [8J always assumes explicitly that

the function f(x,y) is continuous in (x,y), while no explicit as­sumption about the continuity of f(x,y) is made here.

a. LIPSCHITZ CONDITION. /'

THEOREM 1. Let f satisfy a one sided Lipsc.hitz condition. with

Lipschitz constant L ~ 0, on the strip S, that is

f(x,y) - f(x,z) ~ L(y-z)

for 0 < x < a , -~ < z < y < +00. Then the probZem (1) has at

most one soZution on the intervaZ [O,a).

Proof. Suppose not; then there exist two solutions y(x) and z(x),

and two numbers Xo ' Xl ' with the property that 0 ~ Xo < Xl < a, y(xo) = z(xo) , while y(x) .,. z(x) for Xo < x ~ xl' (This is read

ily seen~ Since y .,. Z , there is a number Xl' with 0 < Xl < a such that y(x l ) .,. z(x l ); further, since the difference function y-z is

continuous on [0 ,xl)' just take Xo to be that zero of y- z on

[O,X I ) which is. closest to Xl)'

~

Without loss, merely by interchanging the roles of y and z, if

need be, we can assume that y(x) > z(x) for Xo < x ~ Xl .

111

Hence, by the mean value theorem, applied to th~ product function

e-Lx [y (x) - z (x)) on the closed interval [xo ,xII, there exists a number x, with Xo < i < xl' such that

+ f(i,y(i))- f(x,z(i))} < 0

where the very last inequality follows from the one sided Lips­chitz condition. This contradiction completes the proof.

REMARK 1. It is to be noticed that the special case of the last theorem,when the Lipschitz constant L = 0, is of independent interest (here, the one sided Lipschitz condition reduces to the requirement that the function fis nondecreasing in its second argument, when its first argument is fixed). This uniqueness result, for "monotone" f, is not a special case, for L = 0, of the usual uniqueness theorem for the two sided Lipschitz condition:

If(x,y) - f(x,z) I < Lly-zl

for 0 < x < a , -~ < y < +~ , -~ < z < +=

WARNING. It is very tempting to say that "f satisfies a one sided Lipschitz condition on the strip 5" means that:

f(x,y) - f(x,z) < Lly-zl

for O<x<a,-~<y<+~,-=<z<+~, a condition which differs only slightly, but in a very essential way, from the one sided condi­tion formulated above in Theorem 1. There is a pitfall here, which should be appreciated, for the condition just written can be easily shown to be precisely equivalent to the "two sided" Lipschitz condition written previously, as can be realized merely by interchanging the roles of y and z. A similar word of caution applies when seeking to obtain one sided versions of the other known two sided conditions on f(x,y).

112

b. NAGUMO CONDITION.

THEOREM 2. Suppose that

lim f(x,y) x+O+ Y+Y O

exists. Furthermore, let f satisfy a one sided Nagumo aondition

on the strip S, that is

1 f(x,y)-f(x,z) ~ x (y-z)

for a < x < a, -00 < z < y < +00 Then the problem (1) possesses

at most one solution on the intel'val [a,a).

Proof. Suppose not; then there exist two solutions, y(x) and

z(x), of (1), on [a,a). Define th~ aLt.ciliary £vnction g, on [a,a), by

I y(x)~z(x) g (x) =

a

for x > a

for x = a

We first show that g is continuous at x = a (notice that g has a finite derivative, and hence is continuous, on a < x < a). For

x > a , we have, by the mean value theorem, applied to the dif­ference function y-z on the interval [a,x]

g(x) y (x) - z (x) - [y (a) - z (a) I x

y' (x)-z' (x)

f(x,y(x))-f(x,z(x))

where a < x < x. Hence

since the limit

exists.

lim g(x) a x+O+

lim f(x,y) x+o+ Y+YO

113

Since y of z on [0, a), there exist two numbers xo' xl' with the property that 0 ..; Xo <xl < a , that y(xo) = z (xO) , while y(x) of z(x) for Xo < x ..; xl ; and, without loss, as in the proof of Theorem 1, we can assume that y(x) > z(x) for Xo < x..; xl .

Hence, by the mean value theorem, applied to the auxiliary func­tion g on the interval [xc ,xl]' there exists a number x, with

Xo < x < xl ' such that

- [f (x, y (x) ) - f (x , z (x) ) ] _ [y (x) - z (x) ] x x

..; 0

where the very last inequality follows from the one sided Nagumo condition. This contradiction completes the proof.

REMARK 2. The results of Theorems 1 and 2 are similar to those of Diaz and Walter [6], except for the one sidedness of the condi tions on f.

c. KRASNOSEL'SKII-KREIN CONDITION.

THEOREM 3. Let f satisfy. simultaneously. a one sided Holder con

dition. and a one sided Nagumo type condition. on the strip S ;

that is. suppose that there are constants Cl. H. N. with

o < Cl < 1 o < H o < N

such that

f(x,y)-f(x~z) ..; H(y-z)Cl

and

f(x,y) -f (x, z) N ..; x (y-z)

for 0 < x < a • -00 < z < y < +00 ; and suppose. further. t'hat

114

N < 1

1 - 01

(this is no l'utl'iotion at aH. bJhen N <; 1). Then the pl'ob'lem (1)

has at most one 8o'lution on the intel''lJa'l [0 ,a).

Pl'oof. Suppose notj then there exist tbJO solutions, y(x) and z(x), of (1), on [O,a). Without loss, suppose that z(x) <; y(x) for 0 <; x < a,since the functions min (t(x),z(x)) and

x

max (y(x),z(x)) are also solutions of (1). For each number S, x

1 with 0 < a < T7a ' define the auxiliary function ga' on the inter

val [0 ,a), by I I(xH(x) for x > 0 xl!

g a (x) ..

0 for x = 0

We first show, by mathematical induction, that the function ga(x) is continuous at x = 0 • for the particular sequence of values

"m i am = Li_O 01 , where m = 0,1,2, ..• j this implies the desired con-

tinuity for 0 < a < 1~0I As in the proof of Theorem 2, we have,

for x > 0 , by the mean value theorem:

y(x)-z(x) = y' (x)-z' (x) x f(x,y(x))-f(x,z(x))

for some x, with G < x < x. Now, by the one sided Holder condi­tion, we have that

f(x,y(x))-f(x,z(x)) <; H(y(x)-Z(X))OI

Since yeO) = z(O) , this completes the function ga ' for a = 13 0 = 1.

continuity has been proved for ga

the proof of the continuity of Suppose now that the desired

h s o "m i and , were .. ~m = Li-O 01 ,

m is a non-negative integer. Consider the function ga' where

a • L~!~ OI i . For x> 0, we have, by the mean value theorem, that

there is a number x, with 0 < x < x , such that

g 2 m+I(X) l+a+a + ... +a

-x

115

Y(X)-z(X) m+l xa+ ... +a

y'(x)-z'(x) m+l xa+ ... +a

f(x,y(x»-f(x,z(x» m+l xa+ ... +a

E;;; H(y(x)-z(x»a m+l xa+ ... +a

- m+l H (.2S...r+·· .+a

x

- m+l H (...!...r+·· .+a

x

y(x)-z(x) r m xl+a+ ... +a

g (X) l+a+ ... +am

The desired continuity ef gS(x), at x = 0 , for S = E~!~ a i , now

follows from the induction hypothesis on m. This implies that

gS(x) is continuous at x = 0 , for all 0 < B < l~a

Since y '# z, and y ~ z , on [O,a), there exist two numbers Xo ' xl ' with the property that 0 E;;; Xo < xl < a , and y(x o) = z(x o) , while y(x) > z(x) for Xo < x E;;; xl' For any number S such that

1 y(xO)-z(x o) o < s < 1-a ' one also has that gS(x O) = S = 0 while

Xo

o , by

gs(x o) is meant simply zero).

Recall that, by hypothesis, the constant N, of the one sided Nag~

mo condition, is required to satisfy 0 < N < 1~a ' and consider

the auxiliary function gN' By the mean value theorem, applied to -the function gN on the interval [xc ,Xl]' there exists a number x,

with Xo < x < xl ' such that

116

- -N - - - - N - -(X1-X O) (x) ([ftx,y(x))-f(x,z(x)))-:-[y(x)-z(x)

x

..;; 0

where the very last inequality follows from the one sided Nagumo

condition. This contradiction completes the proof.

REMARK 3. As can be seen from the proof, the Holder condition on

y(x)-z(x) x N

f was used only to assure that the function tends to

zero as x tends to zero; this is the ess.Jltial fact used in the

proof. This same fact could be obtained if, instead of th~hy­

pothesis of the Halder condition for f, we substituted the hy-

pothesis that the limit of f(x,y) N-l x

exists as (x,y) tends to

(O+,yo)' For in that case, we have, by the mean value theorem,

for x > 0 ,

y(x)-z(x)_ N

x

y' (x) - z ' (x) N-l x

f(x,y(x))-f(x,z(x)) N-l x

Indeed, this alternative hypothesis was used, in conjuntion with

the generalized Nagumo condition, in the results of Bownds and

Metcalf (9).

d. OSGOOD CONDITION.

THEOREM 4. Let w be continuous on [0,00) , w(O) = 0, w(u) > 0

for u > 0 , and f· _(1 ) du = +00 Suppose that f satisfies the O+w u

one sided Osgood condition on the strip S, that is

11 7

f (x,y) -f (x, z) ,,;;;; w(y- z)

for 0 < x < a , - 00 < z < y < +00 Then the probZem (1) possesses

at most one soZution on [O,a).

Proof. Let W be a function such that W has a continuous first

derivative on (0,00) W(O+) , and W' (x) = w&r for x > 0

(hence W(x) > -00 for x > 0). For example, one may simply choose

I x 1 W(x) = ~ du

I wlU; for x > 0 .

Suppose that the conclusion of the theorem is false. Then there

exist two solutions, y(x) and z(x), of (1), on [O,a). Without loss, as was done in the proof of Theorem 3, it may be assumed

that z(x) ";;;;y(x) for O";;;;x <a. Since Y" z, and y ~z, on [O,a), there exist two numbers x o ' xl with the property that

o ,,;;;; Xo < xl < a , and y(x a ) = z(x o) , while y(x) > z(x) for

Xo < x ,,;;;; xl' This means that one also has that

O h "l eW(y(x)-z(x» > 0 , w 1 e

for Xo < x ,,;;;; Xl . Then, by the mean value theorem, applied to

the (product) function e-xeW(y(x}-z(x» on the interval [xo,xll ,

there exists a number x, with Xo < x < Xl ' such that

-Xl W(y(xl)-z(x l » -Xl W(y(xl)-z(x l » -x O W(y(xo)-z(x o» 0< e e = e e -e e

( ) -"x W(y(x)-z(x»F' (x)-z' (x) -1} xl-xO e e w(y(x)-z(x))

( ) -x W(y(x)-z(x»{f(x,Y(X))-f(X,z(x)) -1} xl-xO e e w(y(x)-z(x))

,,;;;; 0

where the very last inequality follows from the one sided Osgood

condition. This contradiction proves the theorem.

11 8

e. MOYER CONDITION.

In each of the preceding four proofs we examined an expression, call it E(x,y(x)-z(x)), involving the independent variable, and the difference of two solutions.

-Lx (y(x)-z(x))e y(x) -z (x) x

(They were, respectively, y(x)-z(x) and -x W(y(x)~z(x})

Nee ) . x

In each case, E is positive if y-z is positive, and E is equal to zero if y = z. Then, by the mean value theorem, we showed that E(x,y(x)-z(x)), under the preceding assumptions, must, indeea, always be zero.

Pursuing this simple approach further, we are now able to place the preceding results in a more unified setting, and in the process, obtain theorems which would include these results, as well as others. We will list the assumptions required, as their need arises.

We begin by considering a function E(x,r), defined on (O,a)x(O,-), such that

(i) E is continuously differentiable on (O,a) x (0,00) ,

(ii) E(x,r) > 0 on (O,a) x (0,-) , and E(x,O+) = 0 on (0 ,a).

Suppose that problem (1) possesses two solutions y,z; then there

are numbers xo' xl such that 0 < Xo < xl < a and (without loss) y(x)-z(x) > 0 for all xo < x < xl and y(xo)-z(xo) = O. Therefore, E(xI,y(xI)~z(xI)) > 0, and E(xo'y(xo)-z(xo)) = 0, provided that Xo f o. If Xo = 0, then E(xo'y(xo)-z(xo)) is so far, not defined. Thus, we assume, further, that

(iii) E is such that E{x,y(x)-z(x)) -+ 0 as x -+ 0+. (We will understand E(O,y(O)-z(O)) to mean lim E(x,y(x)-z(x)) ). x+o+

-Then, by the mean value theorem, there exists a number x, with

Xo < x < xl ' such that

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+ E (x,y(x)-z(x)) (y' (x)-z' (x))} r

+ Er(x,y(x)-z(x))(f(x,y(x))-f(x,z(x))} .

If f is such that there exists a function E satisfying (i), (ii), and (iii), and such that the expression inside the bracket~ above is non-positive, then we would have the contradiction

that is, problem (1) can have at most one solution. We therefore assume that f satisfies the one sided condition:

(iv) f is such that, for all 0 < x < a

Ex(x,y-z) + Er(x,y-z)[f(x,y)-f(x,z)) .;;; 0

This is the "one sided" condition of Moyer [4).

It now remains only to reformulate (iii) in a more suitable form, which does not involve, explicitly, a knowledge of the "two" solu­tions y(x) and z(x), so to speak. One condition sufficient for (iii) to hold, independently of the knowledge of the two solu -tions y(x) and z(x), is:

(iiia) lim E(x,r) 0 x-+-O+ r-+-O+

This is actually the case, in particular, in the Lipschitz and

Osgood conditions, where E(x,r) = re-Lx and E(x,r) = eW(r)e-x , respectively. And we have, already, in the preceding discussion, proved:

THEOREM S. Suppose f is suah that thepe exists a peaZ vaZue.d

funat:ion E(x,r} ., defined on (O,a). x (0:.",,) •. satisfying t/iig roZZ0ld.

t. ng

(i) B is aontinuousZy diffeNz.n'tiabZe on (O,a) x (0,00) •

(ii) E(x,r) > 0 on (O,a) x (0,00) and lim E(x,r)=E(x,O+)=O r-+-O+

on (O,a) ,

120

(iiia) lim E(x,r) 0, x .... 0+ r .... O+

(iv) Ex(x,y-z) + Er(x,y-z) ~(x,y)-f(x,z)l < 0 for all

o < x < a -",<z<y<oo.

Then the initial value problem (1) possesses at most one solution

on [O,a).

If Y and z are solutions of (1) sucht that y(x) > z(x) for

o < x < xl ' for some xl ' then

E (x,y (X) -z (x)) E(x,y(x)-z(x)-y(O)+z(O))

E(x,x(y' (x)-z' (x))) o < x < x

E (x,x [f (x,y(x)) -f (x, z (x)) 1)

Hence, another condition sufficient for (iii) to hold, indepen­dently of the knowledge of the two solutions y(x) and z(x), is:

(iiib) lim f(x,y) exists; and lim E(x,xh(x)) o , where X"" 0+ x .... o+ y .... yo

hex) is any non-negative function, defined on (O,a), such that lim hex) = O.

X"" 0+

This is actually the case in the Nagumo condition, where

E(x,r) = ~

proved:

And we have already, in the preceding discussion,

THEOREM 6. Suppoee f is such that there exists a real valued func­

tion E(x,r). defined on (O,a) x (0,00), satisfying hypotheses (i), (ii) and (iv) of Theorem 5, and

(iiib) lim f(x,y) exists; and x .... o+ y .... yo

lim E(x,xh(x)) = 0, where hex) is any non-negative x .... o+

function, defined on (O,a), suc h t hat lim hex) = 0 x .... o+

Then problem (1) possesses at most one solution on [0 ,a) .

121

To obtain "one sided" generalizations of most other "two sided" uniqueness theorems (such as that of Krasnosel'skii and Krein [3]), one need merely retain hypotheses (i), (ii), (iv) of Theorem 5, and derive other conditions sufficient for (iii) to hold. For the sake of brevity, they will not be listed here; the reader is referred to Moyer [4] for a compilation of some of these "two sided" uniqueness conditions just mentioned.

REFERENCES

[1] M. NAGUMO, E~ne H~nne~ehende Bed~ndung 6an d~e Un~tat den Lo~ung von V~66enent~atgte~ehungen, Japan J. Math., 1 (1926), 107-112.

[2] W. OSGOOD, Bewe~~e den Ex~~tenz e~nen LO~i..ng den V~66enen,Ua:!,:

gte~ehungen ~ = f(x,y) ohne H~nzunahme den Cauehy-L~p~eh~tz Be~indu~g, Monatsh. Math. Physik, ! (1898), 331-345.

[3] M.A. 'KRASNOSEL'SKII and ~.G. KREIN, On a Cta~~ 06 Un~quene~~ Theonem~ 60n the Equation y'=f(x,y), Uspehi Mat. Nauk, (67), !l (1956), 209-213. (Russian).

[4] R.D. MOYER, A Genenat Un~quene~~ Theonem, Proc. Amer. Math. Soc. II (1966) , 602-607.

[5] W. WALTER,V~~6enent~at-und Integnat-Ungte~ehungen, Springer­Verlag, 1964.

[6] J.B. DIAZ and W.L. WALTER, On Un~quene~~ Theonem~ 60n Ond~­nany V~66enent~at Equat~on~ and 60n Pant~at V~66enent~at Equat~on~ 06 Hypenbot~e Type, Trans. Amer. Math. Soc., ~ (196{)) , 90-100.

[7] John H. GEORGE, On O~amuna'~ Un~quene~~ Theonem, Proc. Amer. Math. Soc., ~ (1967), 764-165.

[8] H. OKAMURA, Cond~t~on n~ee~~a~ne et ~u66~~ante nempt~e pan te~~quat~on~ 'd~66~nent~ette~ ond~na~ne~ ~an~ po~nt~ de Pea­no, Mem. ColI. Sci. Kyoto Univ. A24, (1942) 21-28.

[9] J.M. BOWNDS and F.T. METCALF, An Exten~~on 06 the Nagumo Un~quene~~ Theonem, to appear in Proc. Amer. Math. Soc.

Washington, D.C. Rensselaer Polytechnic Institute, Troy, N.Y.

~ecibido en setiembre de 1970.

Revista de la Union Matemitica Argentina Volumen 25, 1970.

ON THE BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS Stephen Vagi

fo P406e~~o4 Albe4to Gonz~lez Vom~nguez with a66eetion and g4atitude

I. INTRODUCTION.

The purpose of this note is to present a characterization of the Hardy space H2 for Siegel domains of type II. For domains of type I, i.e., for tube domains over regular convex cones Bochner's theorem [1] characterizes H2: a function belongs to H2 if and only if the Fourier transform of its boundary'function vanishes outside a certain cone. For domains of type II only part of this result' carries over: it follows from Gindikin's representation theorem [2] , (5) that a certain parotiaZ Fourier transform (Le., one not involving all variables) of an H2 function must be ze,ro outside a cone attached to the domain, however, this condition is

, not sufficient. Therefore, one has to search for additional con­ditions which together with the one just stated will yield a suf­ficient one.

Among the Siegel domains of type II those which are hermitian sy~ metric spaces form an important subclass. Every such domain is holomorphically equival~nt to a bounded domain in some Cn to which it bears the same relation as the classical upper half-plane bears to the, unit disc. In a recent paper [8] W. Schmid proved that for the bounded realization of a hermitian symmetric space, provided it contains no irreducible factor of tube type, the boundary values of H2 functions are characterized by the fact that they sa! isfy the tangential Cauchy-Riemann equations on the distinguished boundary of the domain. This circumstance and the unit disc--half­plane analogy naturally suggest an additional condition for the membership in H2, viz., that of satisfying the tangential Cauchy­Riemann equations on the distinguished boundary.

It will be shown here that the two conditions outlined above do suffice to characterize H2 of an arbitrary Siegel domain of type II. Moreover, it turns out that if the d-omain is hermitian sym­metric and contains no irreducible factor of type I, then, in a­nalogy with Schmid's result, the second condition alone is suf­ficient.

124

The author thanks Adam Koranyi for several helpful conversations,

and both him and E.M. Stein for access to their unpublished manu­script (5).

2. STATEMENT OF RESULTS.

For all definitions and basic facts about Siegel domains of type II and their Hardy spaces, we refer to (2), (3), and (4). We shall

follow the notation of (3), and make one additional definition; if

D is a Siegel domain and B its distinguished boundary, we shall de note by H2(B) the closed subspace of L2 (B) which consists of the -

boundary functions of elements of H2 (D). H2(B) is isomorphic, as a Hilbert space, to H2 (D).

We now state our results in detail. D is a Siegel domain of type

II, Bits Bergman-Silov boundary, VI' V2 , Q and 4 have their usual meaning (3).

THEOREM. If f is a function in L2 (B), then it belongs to H2(B) if

and only if the following two conditions are satisfied:

(a) For almost every ~ E V2 the Fourier transform of

f(·,~) vanishes almost everywhere outside Q', the

dual cone of Q.

(a) f satisfies the tangential Cauchy-Riemann equations

on B in the sense of distributions.

LEMMA. Hypothesis (a) of the theorem is implied by (a·) if and

only if:(y) For every A ~ n' there exists a ~ E V2 for which

(A,4(~,~) < O.

The condition (y) can be analyzed in terms of the geometry of the

domain;we shall say that a subset of n generates Q if the set of

non-negative linear combinations formed with elements of that sub­set contains Q. We have the following

PROPOSITION. (i) (y) holds if and only if

1: = {XEReVIlx = 4(~,d, ~ E V2 } generates Q.

(ii) If D is homogeneous, 1: generates Q if and only

125

if it spans ReVI .

(iii) If D is hermitian symmetria. then E spans ReVI if and only if D aontains no irreduaible faator

of type I.

Assertion (iii) of the proposition is essentially an unpublished result of Koranyi, the proof given here is ours. The above state­ments clearly imply the following

COROLLARY. If D is hermitian symmetria. then the following two

faats are equivalent:

(i) D has no irreduaible faator of type I.

(ii) H2 (B) aonsists of exaatly those funations of

L2 (B) whiah. in the sense of distributions.

satisfy the tangential Cauahy-Riemann equations , on B.

3. PROOF OF THE THEOREM AND THE LEMMA.

We begin by determining explicitly the tangential Cauchy-Riemann equations on B. To do this we first introduce suitable coordi­

nates in VI x V 2 • The usual way of putting coordinates on VI x V2 is to choose a euclidean coordinate system in VI which is adapted to ReVI , and any euclidean coordinate system in V 2 • Thus, if

and

B O} ,

then

~(Z2'Z~) in this system has coordinates ~k(z2,z2)' k=1,2, ... ,n , where each ~k is a (numerical) hermitian form. We introduce a

126

slight modification as follows: Set

x=(xl'···.x ) • XJ.=X1 j n 1

t=(tl· .. ··t ) n l

In this coordinate system we have D = {(x,t.~)lt E n} and B = {(x, t, ~ ) I t = O} •

We shall denote points of B by (x,~), x E ReV l , ~ E V2 . All func­tions defined on B will be given, unless specifically stated to the contrary, in terms of the (x,t,~) coordinate system. The measure de on B [3] is just euclidean measure on ReV l x V2 and will be written whenever convenient as dxdV~. Clearly we have

_a_ ax j

1

a ax.

J

a a ax j a <

2 ~j

_ a_ a z j

1

a az j

2

~ ( 2

+ i

a at.

J

_a _ at.

J

1,2, ... ,n l

1 ,2, ••. , n 2

1,2, ... ,n l

1,2, .•. ,n2

b oo If we denote the basis elements of V2 y ~l ... ~n ' it is easy to see that 2

Now let x x j _a_ + 1 a z j

1

x j a 2 a z j

2

127

be a COO vecto~ field of type (0,1) defined in a neighborhood of B. The condition that it be tangent to B is that expressed in (x,t,~)

coordinates the coefficients of the a/atj's vanish on B, i.e.,

A function f E C1 (B) is said to satisfy the tangential Cauchy-Rie­mann equations on B if it is annihilated by the restriction to B

of all such vector fields, i.e., if

Since we can find for every point of B COO functions"x z j defined

in,a neighborhood of B,and such that at the point x2Jo = 1 and XzJ = 0 for j # jo' it follows .that f satisfies the tangential

Cauchy-Riemann equations on B if and only if

(3. 1 ) o j=1,2, ••• ,n2

We denote the operator on the lefthand side of (3.1) by Zj' If

f E C1 (B) and ~ E C; (B), one verifies that

IB f Zj~dS = - I ~Zj f dS

and, therefore, a function f E L11 (B) is said to satisfy the oc tangential Cauchy-Riemann equations, in the sense of distributions, on B if for every ~ E C 00 (B)

o

(3,2) o

We now proceed to the proof of the theorem; in what follows We carefully distinguish between "functions in L2", i.e., measurable

and square integrable functions, and "elements of L2", i.e., equi­

valence classes of such functions. Let f be a function in L2(B) = L2 (ReV 1 x Vz). Let S be a subset of full measure in Vz such

that for every ~ E S the function f("~) is in LZ(ReV 1).

128

For fixed ~ E Sand E > 0 consider the function

Fr;E(X) = f e- 2wi <1 , x>-2w!x!E f(x,r;)dx

ReV 1

There exist.s a function Fr in L2 ((ReV1)') such that F E + F in .. .. I; I;

quadratic mean and pointwise for 1 E (ReV1)' I except when 1 is in

a set Er; C (ReV1), of measuJe zero. Fr; is (a representative of)

the L2 Fourier transform of f(·.t). Observe. that the function

is measurable on (ReV1), x S. By Plancharel's theorem and known

properties of the Poisson integral,

The righthand side of this inequality is an integrable function of

r; by Fubini's theorem. Therefore I by the Lebesgue dominated con­

lergence theorem,

J dV J ! F E (1) - F r n (x) 12 d1 + 0 if E I n + 0 • S I; (ReV) ,I; ..

1

Consequently, there exists a function f in L2 ((ReV 1)' x S) to

which the function (1,r;)~ F E(l) converges in quadratic mean r;

The set E = UI;ES Er; x {I;} is of measure zero, therefore, since

Fr;E(l) -+ FI;(l) on (ReV1), x S - E we have that almost. everywhere

on this set

f(l,r;) = FI; (1)

We can, of course, extend the definition of f to all of V2 by set­

ting it equal to 0 outside (~eVl)' x S. To sum up we have shown

that for 1; E S the function f(',I;) is almost everywhere in (ReV!)'

equal to (a representative of) the L2 Fourier transform of f(·,I;).

Assume now f satisfies the hypotheses raJ and (6) of the theorem.

We take S to be the set of r;'s for which f(',I;) E L2 and Fr;(l) =0

for almost every 1 ~ n'. Consequently,f(l , r;) = 0 for almost every

(l,r;) E (n')£ x S. Let now $1 E CoOO(ReV 1) I $2 E Cooo (V 2), and set

129

By assumption (3.2) holds for j = 1,2, ... ,n2

(3.3) o = f fZ.~dS = f dV f Z.~fdx B J V ~ ReV J 2 I

SinceZ.~ E C - there is no difficulty in taking its Fourier J 0 transform in x, the resulting function will be in C'"((ReVI )'xV2), and no problems of measurability arise. Denoting by F the Fou -rier transform in x, the inner integral by Plancherel's theorem is equal, for s E S, to

(3.4) f Fn F lA) dA = f Fn f O. , d dA (Re V )' J S (Re V )' J I I

The integrand of·the righthand side of (3.4) is in LI ((ReVI )'xV2), so after substituting back into (3.3) we can use Fubini's theorem to obtain

(3.5) J d' f F~ f(A,~)dV - 0 (Re V )' A V J ~ I 2

Now

d ~2 (d] (F$I) CA) d ~ j

_d_ [e-211<A,<I>(S'~»~ (s)le211<A,<I>(s'~»(F~ )CA) d Z. 2 I

J

Substituting this into (3.5) we have: (3.6)

J dA(F$l)(A)f ~[e-21f<A,<I>(s,s»~2(S)le211<A,<I>(S,O>f(A,ddVtO (ReV I )' V2 d Sj .

Since the functions F$I are dense in L2 ((ReVI )'), the inner inte~ ral vanishes for almost all A E (ReVI )'. Note that any Co- func­tion lJ! on V2 can be written as exp(-<A,<I>(~,s»H2(d with suitable ~2' therefore, we obtain from (3.6), that

(3.7) o j = 1 ,2, .•• , n 2

for all lJ! E C -(V) and almost all A E (ReVI )'. Note, however, o 2 that the set of A'S where (3.7) holds may depend on lJ!. Let us set

130

21T<A <p(r r» ~ gCA,r,;) = e ' .. , .. f(A,r,;)

Let K be a compact subset of 0' ,and h E L2 (K). Then by (3.7) for

every ~ E Co~(V2) and j = 1,2, ... ,n 2

r heAl [I a~ g(A ,r,;)dV )dA = J ~[J h(A)g(A ,r,;)dA)dV = o . . K V2 ar,;j r,; V2 a~j K r,;

Therefore, the inner integral is almost everywhere equal to a hol omorphic function of r,; on V2 , and, consequently, r,; ..... g Co ,r,;) isa weakly, and hence also strongly, holomorphic map of V 2 iI),to L2 (K) •

Since g(A,r,;) = 0 almost everywhere outside n' x V2 we have

and thus can conclude that g is contained in the space £2 of [5). Therefore, we now can use the theorem of Gindikin [5, Theorem 4.1),

and swi tching to the (zl' z2) coordinates, obtain finally that the function F is defined on D by

belongs to H2 (D).

We still have to show that the boundary function of F coincides

with f almost everywhere on B. If zl = x + it + i<P(Z2'Z2)' tEO, Z2 = r,;, define Ft on B by

Ft E L2 (B) and we know [2) , [5) that Ft tends in L2 (B) to a func -tion FE H2 (B). Now

21T<A t>~ A For r,; E S, e- , f(A,r,;) + f(A,r,;) = Fr,;(A) as t + O. The con-vergence is dominated so we have also convergence in L2 ((ReV1 )'),

and, therefore, by Plancherel's theorem

Ft (. ,r,;) + fl, ,r,;)

131

in L2 (ReV 1)' T~is together with the fact that F t .... F in L 2 (B) imm~ diately yields F = f almost everywhere on B. This finishes ,the

proof of the sufficiency of the conditions (a) and (s).

The necessity of these conditions is a straightforward conse -quence of the existence of boundary values of H2 functions, and

(for (a)) of a lemma of Stein [9].

We now come to the proof of the lemma. Let S consist of all ~ E V2 for which f(.,~) E L2 (ReV1 ). Construct g as before. Note that g has aU the properties it had before, except that possibly it is not zero outside n' AX V2 . We want to conclude that if (y) holds g, and, therefore, f is zero almost everywhere outside n' X V2 . By Lemma 3.2 of [5] one can modify g on a set of measure zero so that for almost all A E (ReV1 ), , g(A,') is a hol­omorphic function on V2 . Now it follows from

f e - 41r < A , 'II( ~ , ~ ) > I g (~ , ~) I 2 dA dV ~ (ReV 1)'xV2

J If(A,~)12dAdV~<" (ReV 1)'xV2

that for almost all A E (ReV1 ),

(3.8) J e-4n<A"(~'~)'g(A,~)IZdV~ < .,

V2

Using the fact that V2 - {O} can be considered as a complex line n -1

bundle ove'r the complex projective space of dimension nz -l,P Z

we can parametrize Vz- {O} by a non-zero complex number, and a n -1

point in P Z Introducing these new coordinates in (3.8) we

check that for almost all ~o E Vz

(3.9)

where C(~o) is the complex line determined by ~o' ZE C, and dV~ the euclidean area element on C(~ ). We, omit th'e routine but

o somewhat cumbersome details. Now let A in' such that g(A,') is holomorphic and (3.8) holds. Then there exists ~ E V2 such that <A"(~'~» < O. By continuity this inequality holds in a whole neighborhood N of~. Let ~o E N such that (3.9) holds; then since

132

we have

J Ig(A,Z~ )1 2dV < "" C ( ) 0 Z

~o

This implies that the holomorphic .function

is identically zero, in particular, g(A,~o) = O. We can repeat

this argument for almost all ~o E N, and conclude that g(A,') va~

ishes on a set of positive measure, and hence identically in V 2 .

To prove the converse statement we use the.fact, proved in Sec -

tion 4, that if (y) does not hold, then there exists an open set

of A's in (ReVl )' -"il' such that for A in this set, <A,<I>(~,~»>O, for all ~ # O. Let N be a closed ball contained in this set. Let

g be an entire function on V2 such that

A EN,

e.g., g = gl ® g2 @ ... ® g with each g. entire of exponential " n 2 J

type will do) because I<A,<I>(~,~»I > cl~12 with c > 0 since N is compact .and bounded away from the origin (it cannot intersect "il'). Let ~ E C""((ReVl )') with support contained in N. Set

(3.10) o

Let

f(x,s) J e2ni<A.x>h(A,~)dA (ReV l )'

133

f E L2(B) n C~, and by (3.10) satisfies the tangential Cauchy-Rie mann equations on B; moreover, for every ~ E V2,f(. ,~)ELl(ReVl)­and, therefore, for every (A,~),F (A) = h(A,~), but the support

~ of h is contained in N C n'£ x V2 . This concludes the proof of the lemma.

4. PROOF OF THE PROPOSITION.

Let w denote the set of linear combinations formed with elements of E and non-negative coefficients, i.e., the convex cone spanned by E. If E generates 0, then 0 ewe 0, and, therefvre, w', the dual.cone of w, is equal to 0'. If <A,~(~,d>;;;. 0 for all ~ in V2 , then <A,X> ;;;. 0 for all x in 0 and, therefore, by continuity, also for all x in' n, i.e., A EO'. This proves statement (i) in one direction. To prove the converse, let us note ·first ~hat a convex dense subset of a convex open set U must equal U, and sec­ond that the boundary of a convex set is nowhere dense. Suppose now that 0 is not contained 'in w, then by the above remark the interior of 0 - w is nonempty, and, consequently, w' contains 0' properly and the interior of w' - 0' is nonempty. It follows that the interior of w' - O'is nonempty as well. Let A be a point in the interior of w' - n'. Then <A,X> > 0 for all x in w,

in particular for all ~(~,~).

To prove assertion (ii) let us recall that if D is homogeneous, then there exists a group of linear transformations of VI x V2 which carries D into itself, which acts transitiveLy on 0 and whose elements g have the property that

(4.1)

If 1: spans ReVl , then pick a basis in E, and 'consider the convex :one generated by this basis. This cone is contained in wand its interior is nonempty, and, hence, intersects the interior of o because ao is nowhere dense. Now by (4.1) and the transi -tivity on 0 of the group whose elements are g, non-negative line­arcombinations of the ~(~,~)'s fill up 0 if w n 0 ~ , and, there­fore, 0 C w. The converse is clear from the last remark.

We finally come to (iii). One half of the statement is true with

134

out assuming D to be hermitian symmetric or homogeneous. Let D be the product of two domains, Dl and D2 ' Dl being of type I. Let, with obvious notation, Al ~. n'l and A2 E 0'2' Then

Now ~ • (O'~21 , therefore, .. we have

In other words (y) does not hold and, hence, by (i) I: does not generate g.

Now let D be hermitian symmetric. We need some of the more detail ed knowledge about the structure of D, as given in [6], and begin by stating the relevant information. There is a hermitian1nner product ( , ) on Vl x V2 which, restricted to ReVl , is a real in­ner product. 0 is se1fdua1 with respect to ( ,). A Lie group of linear transfe'rmations r acts on Vl xV2 • r leaves Vl , ReVl , and V2 invariant, and is transitive on 0. 1 The elements .of rare ho10morphic automorphisms of Vl x V2 , and their restrictions to D carry D into itself. Moreover, we again bave

(4.1) g~(~,~) = ~(g~,g~) g E r .

If A" denotes the adjoint relative to (, of the linear trans -formation A of Vl x V2 , then for g E r we also have g" E r.2

Suppose now that the subspace of ReVl spanned by I: is proper; we have to show that D as a hermitian symmetric space is ·a product one of whose factors is a Siegel domain of type I. By (4.1) M is invariant under r. Since g E r implies that g" E r, it follows that Mi, the orthocomp1ement of M in ReVl , is also invariant. Therefore, denoting by~ the orthogonal projection of ReVl onto M, we have

(4.2) g E r

Let us note first that since ~ is linear and open, ~O and (I-~")o

are open convex cones in M and Mt"respective1¥. (4,2) implies

1. In the notation of [6] :vl=~~,Revl=~~,v2=~2,{l-.£,(, )~<'>vJr=ad(K;),

2. This follows easily from the fact ad(V)"=-ad(vV),[6,p.282] ,and that

" Kl is invaria.nt under v [6,p.284].

that for every V E gC one has " -

~l = 1!t +i:!l ' the Lie algebra 0 f

135

that r acts transitively on both. Let III be as above. Clearly, III C M, but III is contained in ao too; if it were not, III no would be nonempty, and then by the final remark in the proof of (ii) we

should have 0 C Ill, which contradicts the fact that M is proper. Note further that by (4.1) rill C Ill, and that

(4.3) III = 1T1ll C 1TO C 1T0 where the la5t inclusion follows from the continuity of!T. Final ly, note that the interior of III is nonempty. Therefore, by (4.3) III n 1T0 '" <p. Let x E III n 1T0. Then 1T0 = rx C III C ao, hence,nncw, and so by (4.3) !TO = W. Now let x E 0, yEO, then the three n~ bers (X,1TX), (x,y), (y,1TX) are positive. In the three dimension­al subspace of M spanned by 1TX, x, y, orthonormalize these three vectors in this order. In this coordinate system x = (x l ,x 2 ,O) ,

y = (Yl'Y2'Y3)' and 1TX = (xl,O,O) where all the nonzero coordi -' nates are positiv~. Consequently, we have

(X-1TX,y) (X,Y)-(1TX,y) x 2Y2 > ° Since for fixed x this holds for arbitrary y in 0, by the self­duality of 0 it follows that (X-1TX,<P(~,~)) > ° for all ~ in V2 • Consequently, we have (I-1T)O C ao.

Now let ~ = 1T0 + (I-1T)O. This is an open convex cone in ReV l . Clearly, 0 C ~, but on the other hand, since 1TO and (I-1T)O are contained in 0, we also have ~ C O. Since ~ is open, and (an o­pen convex set being the interior of its closure) 0 is the inte -rior of 0, we have 0 =~. It now follows that 1TO and (I-1T)O are regular: let x E 1TO, then x E O. Therefore, if -x were contained in 1TO, then it also would be in 0, which contradicts the regular­ity of o. S'ame argument for (I-1T)O. The cone 0 is, therefore, the product of two regular convex cones. Note next that the splitting of ReV l into M + Ml also induces a splitting of Vl x V2 (we identify Vl + V2 with Vl x V2 to unify the notation):

V 1 + V 2 = (M + iM .. V 2) + (M 1 + iM'

By expanding ~(~+~' ,~+~') and ~(~+i~',~+i~') and using the fact that H~,~) E 1TO, we find that ~(~,~') EM + iM. So Ml + iMl and (I-1T)O determine a Siegel domain of type I, D'. Similarly, M + iM, 1TO, V2 , ~ determine a domain of type II, D". Clearly, D is the product of D' and D".

136

To conclude the proof we have to check that D as a hermitian sym­metric space is also the product of D' and D". To do this let g E r, and define a map f: V1 +V2 + V1 +V2 as follows:

on

on

f clearly is a linear automorphism of V1 + V2 and, hence, holomor phic; its restriction to D is a holomorphic automorphism of D which leaves D' pointwise fixed. Therefore, using Lemma 3 on page 134 of [7], we infer that D' is a hermitian symmetric space. A similar argument works for D". Since. D can be a hermitian symmetric space in only one way, the truth of the assertion (iii) and of the propositiDn now follow.

REFERENCES

[11 S. B.OCHNER, GJtoup .i.nvaJt.i.a.nc.e 06 Ca.uc.hy'.6 60Jtmuia. 60Jt .6evelLa.i va.IL.i.a.bie.6, Ann. of.Math., 45 (1944), 686-707.

[~] S. G. GINDIKIN,Ana.iy.6.i..6 .i.n homogeneou.6 doma..i.n.6, Uspekhi Mat. Nauk, 19 (1964), 3-92 (in Russian).

[3] A. KORANYI, The Po.i..6.6on .i.n~egJta.i 6011. genelLa.i.i.zed ha.i6-pia.ne.6 a.nd bounded .6 ymme~IL.i.c. doma..i.n.6 ,Ann. of Math. ,82 (1965), 332-:350

[4] A. KORANYI, HoiomolLph.i.c. a.nd ha.lLmon.i.c. 6unc.~.i.on.6 on bounded .6ymme~IL.i.c. doma..i.n.6, C.I.M.E. Summer course on bounded sym -metric domains, 125-197, Cremonese, Roma, 1968.

[5] A. KORANYI and E. M. STEIN, In~eglLa.i lLeplLe.6en~a.~.i.on.6 06 H2 6unc.t.i.on.6 on genelLa.i.i.zed ha.i6-pia.ne.6, to appear.

[6] A. KORANYI and J. A •. WOLF, Rea.i.i.za.~.i.on 0 6 helLm.i.~.i.a.n .6 ymme~Jt.i.c. .6pa.c.e.6 a..6 genelLa.i.i.zed ha.i6-pia.ne.6 ,Ann. of Math., 81 (1965), 265-288.

[7] I. I. PIATETSKY-CHAPIRO, G~oml~lL.i.e de.6 doma..i.nu c.ia.u.i.que.6 e~ ~hlolL.i.e de.6 6onc.~.i.on.6 a.u~omolLphe.6, Dunod, Paris, 1966.

[8] W. SCHMID, V.i.e Ra.ndwelL~e hoiomolLphelL Funkt.i.onen a.u6 helLm.i.~e.6c.h .6ymme~IL.i..6c.hen Ra.umen, Inventiones. Math., 9 (1969), 61-80.

~] E. M. STEIN, No~e on ~he bounda.ILY va.iue.6 06 hoiomolLph.i.c. 6unc.­~.i.on.6, Ann. of Math., 82 (1965), 351-353.

De Paul University. Chicago.

Recibido en agosto de 1970.

Revista de la Uni6n Matematica Argentina V~lumen 25, 1970.

SINGULAR INTEGRALS AND MULTIPLIERS ON HOMOGENEOUS SPACES

Ronald R. Coifman and Miguel de Guzman*

Ved~eado a! p~o'e~o4 A!be4to Gonz4!ez Vomtnguez

The purpose of this note is to .show that, with very little modifi cation, it is possible to extend the estimates of Calder6n and

Zygmund [1] to homogeneous spaces.**

As an application we obtain a criterium for the boundedness in LP for operators commuting with a group action. This condition in

the Euclidean case yields the Marcinkiewlcz multiplier theorem. (As proved by L .. Hormander [2]). In the case of the sphere we olz. tain a multiplier theQrem for expansions in spherical harmonics •

iince the natural setting for our theorem is locally compact Abe­lian groups or compact Riemannian Symmetric spaces, we chose to state our results in such a general context.

§ 1 •

Let X be a topological space and p(x,y): X-+- R+ a "distance" func

tion on X satisfying:

*

**

1° p(x,y) p(y,x) > 0 for all x # y

2° There is a constant C > 0 such that for all x,y,z p(x,y) ~ C(p(x,z) + p(z,y)

3° The sets

Sr(x) = {y EX: p(y,x) < r}

form a base of open neighborhoods of x

The research of the first named authQr was supported in part by the U.S.Army Research Office (Durham), Contract No. DA-31--124-ARO(D)-58.

Similar extensions have been obtained independently by N .. Ri­viere and A.Koranyi and S.Vagi by somewhat different methods.

138

4° There exists a number N such that for every

x E X and r > 0 there are no more than N

points xl.' E Sr(x) such that p(x. ,x.) > ~ . l. J 2

We should observe that condition 4° is a "homogeneity" condition

on the space X; it is satisfied when~ver there exists a Borel

measure ~ on X such that for some C'· > 0 and for all x,x' E X

and r > 0 we have:

(1 . 1 )

The following lemma is an analogue of a lemma of Whitney.

LEMMA 1, Every open set 0 eX, 0 # X aan be represented as

o = U S (x.) i=l r i l.

where eaah point in 0 belongs to no more than M "baZls" Sr. (Xi)' l.

(M,,;;;; 2NR.g 2 (SC 2 )). Moreover, for eaah i, Skr. (Xi) Cf 0 for k>SC 2

l.

The next lemma is well known in a different setting and a gener­

alization of Wiener's covering lemma.

LEMMA 2. Let E C X be a bounded set (i.e. E is aontained in some

"balZ") and Sr(x)(x) a aover of E. The: there exist xi E E suah

that Sr(xi) (Xi) are disjoint and E ~ Ui=lSkr(x.)(xi), where k is

a aonstant depending only on C and N. .l.

We now let ~ be a measure satisfying condition (1.1). By a stan­

dard argument, one obtains from lemma 2 a Hardy-Littlewood maximal

estimate, which, together with lemma 1, yields the following anal­

ogue to the estimate of Calder6n and Zygmund. (See also E.M.Stein

[3] ) .

LEMMA 3. ret f(x) > 0, f E L'(X,d~). Then, for every A > 0 ,

there exisi; Sr. (Xi) suah that: l.

a) f(x)";;;;A for a.e. xtf,USr.(x i ) l.

b)

c)

d)

1;1(8 ex.)) Ir _ J;. ];

139

I S (x.) ri 1

/xi (y) dll (y) Lill(8r. (xi» .;;; K -----

1 A

No point in X belongs to more than M "balls"

S (x.) ri 1

where the constants K, M depend only on C' and C. (*)

From lemma 3 we obtain the following genera1i'zation of the theorem of Calderon and Zygmund.

THEOREM 1. Let M bea bounded operator on L2 (X,dll) of the form

M(f)(x) = fxk(X,Y)f(Y}dll(Y)

If there exist k and C such that. for aLL x,xo,r > 0 • we have.

(1 .2) J X-Skr(xo )

then M is a bounded operator on LP(X,dll) for 1 < p .;;; 2 and

(1 .3) Il ({x E X

§ 2.

II fill M(f) (x) > A}) .;;; C'

A

Let G be a locally compact a-compact group, dx a left invariant HaaF measure on G. We assume the existence of a base Ui , i E Z, of open neighborhoods of the identity e of G, satisfying the fol­lowing conditions:

(*) Lemma 3 is of independent interest since it implies also that the estimates of John and Nirenberg concerning func tions of bounded mean oscillation are valid in this gene£ a1 setting.

ZO

4°

u'" U i i __

aD G

140

< ... , where lu. I is the left Haar measure 1

of Ui' Examples of groups for which such a base exists can be found in Edwards and Hewitt [4].

We now define

p (x) inf {I u·1 1

and

( -1 px,y)=p(xy)

It is immediate that p(x,y) satisfies the conditions 1°,2°,3°,4° of § 1.

We should observe that for convolution operators the condition (l.Z) of theorem 1 reads ~s follows:

(Z.1) f IK(x y-1) - K(x)ldx <; C for all y E Ui X-ui+1

Our purpose now is to give conditions under which a bounded linear operator on L2(G), commuting with left or right translations, can be represented as a "convolution" with a kernel satisfying condi­tion (Z.l).

Let 'r ~ 0 be a family of functions in L'(G) n L2 (G), 0 < r < ...

such that

a) I ' (x)dx = 1 I ,2 (x)dx <; c

G r G r r

b) 'r*'s ~s* 'r

c) J 'r(x)dx <; C(f)£ and p (x) > t

141

I 14> (xy-l) -4> (x)ldxo;;;;C·(eJli)e:' r r r

for some C,C' ,e:,e:' > 0

If we now define $r

ing theorem:

4>;r - 4> r /2 ' for r > 0 , we have the follow-

THEOREM 2. Let M be a bounded linear operator 'on L2(G) oommuting

with left (or right) translations of G. If. for some e:, C > 0 we

have

(2.2)

then M is a.bounded operator on LP(G) for 1 < P < 00 and i8 of weak

type (1.1).

A basic example of an approximate identity 4>r satisfying the con­ditions a), b), c), is obtained in the following way.

Let

(2.3)

and

(2.4)

S r

{x E G p (x) < r}

1

{~ if

if xES r

x $. S r

Condition b) is satisfied if G is Abelian or G is compact, and

the sets Sr are invariant under conjugation. As for condition c) we ~ust add the assumption

(2.5) for some C,e: > 0

which is satis.fied for the examples cited below. The other con­ditions are then obvious.

14Z

Most approximate identities useful in Analysis, like the 'Poisson kernel or Gauss-Weirstrass kernels, satisfy the conditions a),b), c). We should observe also that. instead of defining wr as above, we can. with suitable modifications. let

and theorem Zo remains valid.

Let now G/K be a compact Riemannian symmetric space. Fix a G-in­variant Riemannian metric on G/K and define p(x) to be the volume of the smallest closed ball centered at 0 containing x. (0 is the base point of G/K). Define Sr'~r by (Z.3) and (Z.4); condi­tion (Z.5) is then satisfied for e: = k ' nbeing the dimension of G/K. Theorem Z remains true for dperators on L2(G/K) which com­mute with the action of G. As a conse~uence of theorem Z we ob­tain in the case of the sphere l:n = SO(n+l)/SO(n). a multiplier theorem for expansions in spherical harmon..iG:.s~

Let Y~(x) • i = l"Z.; ..• dk base of sphe:f'ilCaI harmonics. An mutes with the action of SO(n+1)

(Z.6)

where ~ is a bounded sequence.

k = 0.1,Z •... , be an orthonormal operator M on L2(E ) which com -

n is of the form:

THEOREM 3. An operator M defined by (2.6) is bounded on LP(E ). . n 1 < P < 00 • and is Of IJeak type (1-1) if the sequenae {mk } is

bounded and satisfies

(Z.7) 2N+1

\ 1 J' 12 Z>-(2J'-1)N I.. t. ~ "C

K=2 N for j=[!]+1

Remark: condition (2.7) is satisfied if 1 fiimk 1 < n

[2'] + 1.

143

In conclusion, we observe that, in the case of Rn , Tn, and the p­adic number fields (see Taibleson [5)), condition (2.2) is easily seen" to be equivalent to a condition involving estimates on deri~ atives or differences of the Fourier transform of M. This equiv~ lence is still valid for compact symmetric spaces as will be shown explicitely in a forthcoming paper.

REFERENCES

[1) CALDERON A.P. and ZYGMUND A., "On tb.e EX.illtenc.e 06 CeJtta.in S.inguta.Jt IntegJtatll", Acta Math., Vol. 88 (1952), pp. 85-139.

(2) HORMAN])ER L., "Ellt.imatu 60Jt TJtanlltat.ion InvaJt.iant OpeJtatoJtll .in LP Spac.ell", Acta Math., Vol. 104 (1960), pp. 93-140.

(3) STEIN F .M., "A CouJtlle .in flaJtmon.ic. AnatYll.ill on Euc.t.idean Spac.e6 N , Presented to the University of Paris at Orsay in 1961-8. Notes in French.

(4] EDWARDS R.E. and HEWITT E., "Poi.ntw.ille Um.itll 60Jt Sequenc.ell 06 Convotut.ion OpeJtatoJtll", Acta Math., Vol. 113 (1965) , pp. 181-218.

(s) TAIBLESON M.H., "flaJtmon.ic. AnatYll.ill on n-d.imenll.ionat Vec.toJt Spac.u oveJt Loc.at F.ietdll 111. MuLt.ipt.ieJtll", Math. Ann., Vo). 187 (1970), pp. 259-71.

Washington University.

St. LQuis, Missouri.

Recibido en julio de 1970.

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Re17ista de 1a Uni6n Matematica Argentina Vo1umen 25, 1970.

MULTIPLIER TRANSFORMATIONS OF FUNCTIONS ON

SU(2) and l2

by Ronald R. Coifman and Guido Weiss *

Dedi~ado a! p~o6e~o~ A!be~to Gonz4!ez Dominguez

1. INTRODUCTION. Suppose f is a function belonging to LP(-1I,1I) ,

1 ~ P ~ 00. We write

(1 .1) lOO f(k)e ik6

to indicate that the sum on the right is the Fourier series of f.

An important class of linear operators mapping a space LP(-1I,1I)

into a space LQ(-1I,1I), 1 ~ q ~ 00, is the class of muZtipZier tran~

formatio~s. Each such transformation M is characterized by a se­

quence (M(k)} having the property that if f has the Fourier series

(1.1) then Mf has the Fourier series

(1 .2) lOO M(k)f(k)e ik6 • k=-co

Considerable work has been done on the problem of

tions on the sequence {M(k)} which guarantee that

linear transformation from LP (-11 ,11) to LQ (-11 ,11).

finding condi­

M is a bounded

Perhaps the

best known result of this type is a theorem of Marcinkiewicz [3)

which can be stated in the following way:

A 2k 1 A A

THEOREM. If {M(k)} and {l. -k_1IM(j+l)-M(nn are bounded then

J=2

M is a bounded Unear transformation from LP(-1I,1I) into itseZf fof'

1 < P < 00. More preciseZy, if f E LP(-1I,1I) has the Fourier ser­

ies (1.1) then, under these conditione, (1.2) is.the Fourier ser­

ies of a function Mf E LP(-1I,1I) and there exists a constant Ap ,

independent of f, such that

* Research supported in part by the U.S.Army research office

(Durham) under contract No. DA-31-124-ARO(D)-58.

146

w 1 w

J t (Mf) (e) IPde)P OMfD < A DfO • A ( J P P P P

-w -w

The purpose of this paper is to establish an analogous result for functions on G = SU(Z.) , the special unitary group of 2 x 2 complex matrices. In order to dp this we will have to set up some nota­tion and announce ·certain classical results. We refer the reader to Vilenkin (5) and to Coifman and Weiss (1) for. their proofs.

An element u E SU(2) is a 2 x 2 matrix having the form

where zl and .z2 are complex numbers satisfying IZll2 + IZ212

= zl zl + zi z2 =

In general, if a = [a .. ] is an n x n matrix with complex entries, 1J

its Hilbert-Sahmidt norm Warn is defined by

10 a 01 2 n

Li,j=1IaijI2

The operator norm of a will be denoted by U an. That is, n all is the least constant A such that

for all n-tuples x = (x 1 ,x 2 ' ... ,xn) of complex numbers. We shall have occasion to use both of these norms. For technical reasons, however, we. use the Hilbert-Schmidt norm to introduce a metric d on G by letting d(u,v) = IOu - v HI. In particular, the function whose values are p (u) = !III u - e 01 , where e is the identity element of G, will play an important role in our development.

A function f on G is called aentral if its values depend only on the classes of conjugate elements of G. That is, f(v-1uv) = feu) for all u,v E G. It is easy to see that the function p we have just introduced is central since multiplying a column (row) on the left (right) by a unitary matrix does not change the Euclidean

147

norm o~ the column (row). Thus, the Hilbert-Schmidt norms of v-Iuv - e = v-I (u-e)v and u-e are the same. The fact tha-t p is c_entral enables us to obtain a particularly simple expression of p(u) in terms of the proper values of u. Since u is unitary and its determinant is 1,the proper

e- iA / 2 and e iA / 2 for 0 ~ A ~ 2~ matrix v such that v-Iuv is the

Consequently, since p (u)

we must have

values of u must have the form

Moreover, we can find a unitary diagonal matrix

(1.3) p (u) fl sin t

We shall also use the fact that integration of a central function with respect to Haar measure has a particulary simple expression in terms of the parameter A. Suppose f E Ll(G) is such that_

vA feu) - F(A), where e-l."2 are the proper values of u, then

(1.4)

where du is the element of Haar measure on G SU(2).

In order to discuss the analog of th.e Fourier series expansions (1.1) for functions defined on G we have to introduce some of the basic facts concerning the irreducible unitary representations of SU(2). First, we recall that a unita~y repr~8entation of G is a continuous map, u + T(u), of G in~o the class of unitary operators on a Hilbert space H that satisfies the relation T(uv) = T(u)T(v) for all. u,v E G. A subspace M cHis said to be invariant under the action of T if T(u) maps M into Itself for all u E G. If {a}

and H are the only invariant subspaces, then the representation T is said to be irreducible. A basic result in the theory of repr~ sentations of compact groups is

148

(l.S) If the ~ep~esentation T, aating on the HiZbe~t spaae H, is

i~~eduaibZe then H is finite dimensionaZ.

If T is an irreducible representation acting on H, we can choose an orthonormal basis of H, which must be finite by (1.5), and ex­press T as a unitary matrix [t .. ] with respect to this basis. We

1J let the symbol T represent the matrix [tij ] as well. In fact , for the remainder of this paper we will assume that our irreduci­ble representations are unitary matrix valued maps v + T(u) =

. I tij(u)) and the fact that multiplication is preserved under such ma,ppings is expressed by the formula

for all u,v E G and 1 < i,j < d, where d is the dimension of the space H on whi~h T acts.

Two representations Sand T acting on the Hilbert spaces Hand K are $aid to be equivaZent when there exists an invertible linear transformation L: H + K such that T(u)L = LS(u) for all u E G. A system {Ti } , i belonging to some indexing set t. of irreducible representations of G is said to be compZete if, given any irredu­cible representation T, there exists a unique index i such that T and Ti are equivalent.

Proposition (1~5), together with the following one, constitute a formulation of the Peter-Weyl theorem in the special case when G • SU(2):

(1.6) n Let l ~ {i : i = 2 ' n • 0,1,2, ..• } be the coZZection of

haZf-intege~s. A aompZete system of irreduci1:iZe matrix

vaZ.ued rep~88entations of SU(2) can be indexed by the set

t in such a ~ay that, if {T~} , i E t, is the indexed

system then:

(i)

(ii)

Ti • [ti ].is a e2i+l) x (2i+l) matrix. ~here m,n .

-i <; m , n <; i

the coZZection of funations 12I+T t i is an o~tho­m,n no~maZ basis of L2(SU(2)).

149

Thus, if f E L2(G) , G = SU(2) , we can define its (matrix value~ .. Fourier t;roansform f by letting

A I Q-1 f(Q) = G f(u)T (u )du

forQ E.t. It follows from (1.6) (ii) that

(1.7) f - f' (ZQ+1 )tdf(Q)TQ} , 2£=0

where the series on the right converges to f in the L 2- norm • This e! pansion is the desired analog of the Fourier series expansion (1.1).

If f,g E L1(G) we define their convolution f * g by letting

(f * g)(v) = IG f(u)g(vu- 1)du

for all v E G. It follows readily from this definition that if (f * g)A is the Fourier transform of f * g then

(1 .8)

for all Q E.t. It is well known that, in the classical case, mul tiplier transforms arise from convolution operators (generally , from convolution with a distribution). Motivated by the defini­tions we have made(1), therefore, the multiplier transformations M that we consider are those that transform a function with devel opment (1.7) into one, Mf, whose development is

(1. 9) f" (2h 1 ) t r (M (Q ) f (Q ) rll ) 21l =0

As in the classical case, those multiplier transformations that map L2(G) boundedly into L2(G) are the easiest to characterize:

(1) Since SU(2) is not commutative, the operation of convolution we introduced is not commutative. The reader should observe, therefore, that the multiplier theorem we are developing is, really, a statement about "left" multipliers. We leave it to the reader to formulate the related results that arise because of this lack .of commutativity.

150

THEOREM (1.10). The mu~tip~iep tpansfopm M ma~s L2(G) boundedZy

into itse~f if and onZy if the opepatop nopms M(2) ape bounded in

dependentZy of 2 GO.£ .

We shall sketch a proof of this theorem. We first observe that if

ro (22+1) l2 a 2

22=0 m,n=-2 m,n

is the div~Zopment

= I f(u)t2 (u)du

of a function f E L2 (G) (thus, a2 m,n

G n,m , since r (u- 1 ) is the conjugate transpose

matrix to r), then~ by (1.6) part (ii)

(1.11)

If ~!,n are the coefficients of the multiplier matrix M(2) then

the L2 norm of a function having expansion (1.9) is

ro 22=0

(22 + 1) l m,n=-2

1,2 2 t. ~m,j j=-2

If the operator norms of the matricesM(2) are bounded, say

HM~)H < A for all 2 E.£, then

1,2 2 t. ~m,j j=-2

Summing over nand 2, we then obtain UMfl1 ~ < A2UfH ~. Conversely,

if the last inequality is satisfied by all f E L2(G) it follows

that 8M (2) U < A for all 2 E .£. We see this by applying M to func

tions whose development is, say,

(22+1) l2 CI~,l t~,l j=-2

Having set down this background material we can now turn to the

development of the multiplier theorem mentioned above.

We would also like to 'thank our friend and colleague I. I. Hirsh­

man Jr. for having read the manuscript carefully and having con -

151

tributed several valuable suggestions.

2. MULTIPLIER TRANSFORMATIONS THAT PRESERVE LP(G), 1 < p <-.

Our treatment of the multiplier theorem has certain features that

are similar to that of Hormander's version of the multiplier theo rem associated with n-dimensional Euclidean spaces. The similar­ity lies in the fact that the multiplier theorem is reduced to re

suIts concerning certain Calder6n-Zygmund singular integrals (se~

[2] ). These results have the following analog for SU(2): SUppOSt:1 M is defined on C-(G) by

(Mf) (u) lim s"'O e:>O

J -1 m(v)f(uv )dv p (v»e:

where m is locally integrable on SU(2) - {e} and satisfies

(2.1) J im(uv-1) - m(u)i du ~ C <-p(u»2p(v)

where C is independent of v. Then, if M is bounded as an operator

on L2 (i.e. there exists A < - such that UMfU 2 ~ AUfU 2 for all - . . fEe (G)), M is also bounded as an operator on LP(G), 1 < p <-; that is, there exists a constant A such that UMfU ~ A UfU for

P P P P all f E C-(G).

We state this result without proof since it is not substantially

different from that found in Hormander [2] (1). Our development

will make use of a result, obtained by de Guzman and the first author of this paper, which permits us to replace condition (2.1)

(1) If zj=xj+iY j , j=1,2, the correspondenc~ U = "-_ Zz 12 - _ZZ 21

-_" +--+-

+--+- (x 1 ,y 1 ,x 2 ,Y 2) can be used to obtain a natural identifica­tion of SU(2) with the surface of the unit sphere in Eucli­dean 4-dimensional space. The operators described above, in terms of this identification, are Calderon-Zygmund type Sin­gular integrals associated with the surface of the unit sphere in R4. Such operators have been studied by many au­thors. In particular, the results we have just announced can be obtained by applying the method of H~rmander to the singular integrals developed by Morley [4]. .

152

by a condition on the behaviour of M when applied to a specific approximation to the identity. This result, in its full general­ity , is contained in the preceeding article of this volume; at present, we limit ourselves to stating the special case associat­ed with SU(2):

= {UE G For 0 < r ~ 2~ let S r

p (u) < 18 sin f }. By virtue of +.>.

(1.3), if the proper values of u are written in the form e-~2 for

A = Au E [O,2~], this is equivalent to Sr = {u E G : Au < r}. Let

X be the characteristic function of S and ~r = Xs liS I, where, Sr r r r

in general, lEI denotes the Haar measure of E C SU(2). If ~r

= ~r - ~r and M is a linear operator which is defined on Loo(G) , 2"

bounded on L2(G), commutes with left translations and satisfies

(2.2) f I(M~ ) (u) 12 [p (u)] 4du ~ Cr SU(2) r

where C < 00 is independent of r, then M is an operator of the Ca!

der6n-Zygmund type described above. In particular, M is bounded

on LP , 1 < p < 00 (1)

Suppose, then, that we do have a multiplier operator of the type

described in (1.9). We shall suppose that the operator norms of the matrices M(n are bounded; thus, by (1.10) M maps L2(G) bound edly int.o itself. It is readily verified that M does commute wi th

left translations. Thus, if we can find suitable conditions which imply (2.2) we would have a theorem concerning multiplier trans -

forms. Our task, therefore, is to study the effect, on the L2

norm, obtained by applying M to the function ~r and then multiplr ing the resulting function by p2.

(1) The reader should observe that the analog of condition (2.2) for R3 (the number 3 being the dimension of SU(2»is equiva­lent to estimates of the L2-norm of the 1st and 2nd order de rivatives of the Fourier transform of M (applied to an appr~ priate analog of ~). A si~ilar situation arises in H~rman-de r [2]. r

153

We first suppose that MWr has the development

Moreover, let us agree that, for the remainder of this paper, we have chosen the particular complete system of irreducible repre­sentations of SU(2) that is introduced in the third chapter of Vi lenkin's book [5]. This will enable us to. refer directly to this source for many formulae and, in particular, for the calculations of certain Clebsch-Gordan coefficients. These coefficients arise when the tensor product of two irreducible representations are d,! compos~d into irreducible parts. In particular, products of the form t2: ,t2 can be expre·ssed as linear combinations of the m,n m,n functions t~,q with 12-2' I 0;;; k 0;;; 2+2' , -k 0;;; p,q 0;;; k. We shall be interested in the precise values of the coefficients involved in these linear combinations when J!' = .t and (m' ,n') is either

( t ' t lor (- t ' -t) . The reason for this is the following:

Because of (1.3), [p (u)) 2 = .8 sin2t .. 4 (1-cos .A

-1-

the proper values of u. But 2 cos t .. e 2 + I 1

til I) , where e 2 i1.

e 2 ~ trace(u) ..

are

.. t21 1 + tI 1 -2'-2 2'2

Thus, the effect of multiplication by ~(u)]2 1

is eas ily deri ve"d from the formulae expressing ~21 1 t~,n as li~ ±2"'±2"

These can be found in §8 of chap-

ter III of Vilenkin and, from there, we obtain

(2.3)

+\1'(2 -m+1) (2 -n+1)

_1_\ V(2 -m) (2 -n) 22 + 1

1 3 2=0'2,1'2,···,-20;;;m,n0;;;2

154

Let us now define A2 (£ ,6) =..;1 + £ (1+46m t for -2 " m ,,£ and m 2 22 +1

E .'

1 1 (1)

2'2 Then. from (2.3) we have (formally)

A DO 2 2 2 4sin2 '4 L (22+1) L a t

22=0 m,n=-2 m,n m,n

.. 2 {2 .~ 2 2 2+£ } 2 L (22 + 1) L 2a - t. 1 1 A (£,6) A (£, <5) a +cS +cS t . 22'-.0 m,n=-2 m,n £,6=-'2'2 m n m,n m,n

We see therefore, that the effect of multiplication by p2 is to produce cer~ain differences of the coeffi~ients involved in the

series deve1ope~nt of M\. In view of (1.9), these coefficients ~fnvolve both those of the multiplier transform and of the

mIn function Wr . We shall now examine this relationship carefully and; by obtaining estimates on the diffel'"ences of the coefficients

of !/Ir' we will t~en be able to determine conditions on the coef­ficients of ~he M (2) 's, and their differences, assuring us that condition (2.2) is satisfied. This will then give us a multipli­er theorem. whose basic features are not unlike those the the clas s~cal result of ·Marcinkiewicz.

By the nature of the definition of $r(and Wr ) in terms of the ce~ tral function p, it f~llows that $r (and Wr ) is also central. It is well known (see §4 of chapter I of Vilenkin) that the charac­ters

2 x (u)

of the irreducible representations of G form a complete orthonor-

(1) The reader should observe tha~ these coefficients will mul-

. 1 2+£ t1P y am+ 6 ,n+6' These last expressions maY have no meaning

1 when £ - - 2 and m (or n) equals 2 or -2; when this is the

2 case, Am(£,6) z 0 and, thus, these terms will not appear in

the summation occ~rin~ in (2.4).

155

mal system in the space of central fun~tions. In particular,

~ r (11 )

is a scalar (211+1) x (211+1) matrix. That is, if III denotes the

(211+1) x (211+1) identity matrix and

A

~ (11 ) . The development (1.7) becomes in this case

Similarly, if we let

and

we have

1jJ -r

A

'I'r (11 )

~ (211+1)tr{'(II) Til} 211=0

By making use of formula (1.4) and the fact that l has the values

11 x (u)

sin(lI+ t) A

• A Sl.n "2

(see §7 of chapter III of [5]), we can calculate the coefficients

;~ and ~~ explicitly. We f'irst observe that

Jr

o sin2 ~ dA

2 r-sin r

2

156

Thus,

----- Is l (u)du (ll!+1)ISr l r

Z fSin ~ sin(Z2+1) IdA

o (ll! +1) r-sin r

------- Jr [cos £A - cos(£+1)AJ dA . (ll!+1) (r-sin r) 0

Hence,

(2.5) Z2 +1

sin £ r £

sin (£ +1) r £ +1

r - sin r 1 3

£ =0 '2' 1 '2" ..

A£ A£ Aj2 Since <Pr - <p r / 2 =1jJr(2.5) will also give us an explicit formula

f h ff " A£ or t e coe lClents 1jJ .

. r

j2 1 3 we let D~£ For any sequence {et } ,£ =0 '2" 1 'Z,.. . ~ 1 1

2£ £ £+'2 £-'2 D et = Zet - et - et We then have:

LEMMA (2.6). Each of the series

is bounded by a constant times r; that is, each series is OCr) .

We shall indicate, briefly, how these estimates can be obtained. In each case it is convenient to split the sum into one involving

those terms for which Qr ..;; 1 plus one involving those terms for A£ A£ A£

which £ r > 1. In order to estimate ljJr = "'r - <Pr/2 we can make use of (2.5) and the power series expansion for the sine function

AQ to obtain ljJ = O(£2r2). Thus,

r

OCr) .

157

For the rest of the estimates it suffices to examine the coeffi-. 'Q

clents <P r •

sin rx/x we For example, applying the mean value theorem to

AQ AQ obtain <Pi = 0(1) for Q > 1/r. Thus, $r = 0(1) and

OCr) .

We can conclude, therefore, that

and this is the first estimate of lemma(2.6). Similarly, it can be shown that, for

OCr).

r 40{ L Q2} OCr). The mean value theorem can then be used Q ~l/r

to obtain estimates for D:Qr and D2: Q when Q r > 1 that allow

'I' 'l'r ' ,

us to complete the proof of the lemma. We leave the details to

the reader.

Suppose we denote the coefficients of the matrix M(Q) by Q

ll m,n' , . ,Q

-Q';;;m ,n.;;;Q ; that is, M(£) =[(\1 )D.. Then the coefficients m,n

involved in the series (l.9), when f = 1Ji r ' are those of the ma-

trix

Thus, the coefficients Q (l

m,n

(2.7) Q

(l m,n

involved in the expansion of M$r are

158

By straight-forward computations we obtain

LEMMA (2.8).

\' 2 II 2+<: l A (£,o)A (£,o)~ +' +,} 11m n m u,n u

£,0=2'-2

+

It follows from (2.4) and the remarks following (1.11) that

116 IG 1 (Mlji) (u) 12 [p (u)]4du L (2£+1) 2ll =0

where ~ll is the expression within the curly brackets in (2.4). m,n

But is clear from this expression, (2.7) and (2.8) that

I IMlji 12 p 4du is majorized by a constant times G r

where we recall that, for any (2£+1) x (22+1) matrix P = [Pm nD , , II

-ll .;;m,n.;;£ ,1IIP1Ii= (L Ip 12)1/2 denotes its Hilbert-m,n=-R. m,n

Schmidt norm and r:" r:,2 are the "difference" operators which, when

applied to the family of matrices M(£), have the coefficients

2

159

and

2 II \Im,n

It now follows immedJately from theorem (1.10) , (2.2) and lemma (2.6) that we obtain the following multiplier theorem:

THEOREM I. Suppose M is a multiplier transform with multiplier

matrioes M(ll) whose operator norms are bounded and. moreover.

satisfy

1

ti) 11M ell ) m= o (1l2) , (ii) IIIlIM(ll) 111= 0(£"2) , (iii) 3

IIllI 2M (ll) III = 0 (£2)

then M is a bounded operatbr on LP(G) , 1 c pc •. (1)

Let Us now investigate the operators II and lI 2 in more detail. We shall do this in order to obtain a better understanding of the lIeaning of conditions (ii) and (iii). A simple calculation shows that for -ll+l<m,n<ll-l

(1) At the beginning of this paper we ~tated the classical multi plier theorem of Marcinkiewicz. There we imposed conditions on certain sums of absolute values of differences of coeffi­cients. The reader can easily check that by making appro­priate changes in the statement of Lemma (2.6) we could ob­tain a result which, instead of conditions (i), (ii) and (iiU (iii) , we would have

Ihi(ll) 112)1/2 = 0(2k),( L IIllIM(ll) 1112)1/2 = 0(1;

2k- 1cllc2k

and ( L IDlI 2M(ll) 1112) 1/2 = 0(2-k )

2k- 1cllc2k

160

1 + 2" I

e: 6=-'!''!' , 2' 2

(2.9)

(iM(2) ) m,n

1 2-.!. 2+2' + 2

1.1 _ 1 1 1.1 1 1 22 2 mf2,n+t m-2',n-2' 2 2 2

A+l.I m,n -l.I m,n- ,A_l.I m,n-l.Im,n-2

2 222 and, finally, R (e:,6) = (Am(e:,6) - An (e:,6)) . m,n

2

2+.!. 1 2 +

2-2' 1.1 1 1 1.1 1 1

m-2',n-2' m+t,n+t

2

When m = -2 or m = 2 we have already observed that undefined terms 2-.!.

(like 1.1 i 1) appear with coefficients that are O. We can,ther~ 2+t,n+2'

fore, interpret the above formulae in this case as well by making

the convention that such undefined terms are absent. It is of in~

terest, however, to note that we can also rewrite these differ -

ences in these extreme cases in the following way when m = 2:

(2.9')

where

case £

(and n

A

(t.M(~ )) m,n

(2M(~ )) m,n

161

1 + 2~ +1 1 +

n-Z

+ , n - ~ E (02~+1 - £ £,0

.. 1 + 2~ + 1

+ \" (20£ (~-n) L 2£ + 1 £,0

) +

) +

E' denotes the summation over all £,0 = 2'-2 except the £,0 1 1 -2 and 6 = 2 Similar formulae are valid for m = -~

Q , -Q ) •

Part (i) of the following lemma follows immediately from the ine­quality (a2+b 2 ) (a-b)2 ~ la2_b 2)2 applied to the non-negative num-

~ ~ bers a = Am(£,o) andb = An·(£,6). Part (ii) is an obvious iden-tity.

LEMMA (2.10). (i) (m-n) 2

R~ (£,6) ~ --------m,n Im.,...j 2(2~+1)(Q-~)

2

(ii) 2-n 2riT

162

-2 <: m ,n <: I! •

We can now obtain the following more detailed version of theorem I

THEOREM II. 2

SupposeM(I!) = l(J.1m,n)1, -I! ~ m,n <: I! 1 3 2 =0 '2' 1 '2' ...

is a sequenae of mat~iaes having bounded ope~ato~ no~ms and satisf~

ing the aonditions

I! Ca) ~

m,n=-2 (21! -lm+nl)2

-3 1 1 0(2 ) ,6=-2'2 '

then the mu"Ltip"Lie~ ope~ato~ defined by the sequenae M(I!) is a

bounded ope~ato~ on LP(G), 1 < p < ~.

I! Since ~

m,n=-I! m"n

I / I 2 , condi -2 m,n

(22-lm+nl)

tion (a) assures us that the contribution to the Hilbert-Schmidt A

norm of M(2) by the terms off the main diagonal is less than or equal to a constant (indepen~ent of I!) times If". Since the ope!. ator norms of the sequence {M(£)} are bounded,each of the 22+1

A

diagonal elements of M(2) has absolute value not exceeding a bound for these norms. Thus, the diagonal elements also contribute, at most, a constant, independent of 2, times If to the Hilbert­Schi~dt norm of M(I!). Therefore, it suffices to show that condi­tions (ii) and (iii) of theorem I are satisfied. In order to this we first s"how that (b) implies

(Z. 11 J I! I! ~ n lJ.1m,n m,n--'"

1 2+z 2

J.1m+ 6 ,n+6 1

163

For 8 1 2 we first establish the identity

(2.12) 2 L 2j =1

£ +j /:, 2 II

+ m+j,n+j

This equality is a consequence of the fact that the series on the

right is absolutely convergent (condition (b) assures us that this

series is termwise majorized by a constant times L j-3/2) and 2j ~1

£+j+1/2 the fact that a j = llm+j+1/2,n+j+1/2

£ +. II +~ +. tends to 0 as j+oo

m J,n J

(from (2.12) applied to a. and condition (c) we see that {a.} is J J

a Cauchy sequence; thus, lim a. exists. J

The fact that this limit j+oo

is 0 is a consequence of the fact that the a.' s are differences of J

the bounded sequence

£+.!.

£ +j 1 II ) When 8 = --2 we have an analogous m+j ,n+j .

2' identity for II 1 1

m-2:,p-2: £

- llm,n in which there is an obvious change

in some of the signs preceeding the indices j and 1/2. We now

can obtain (2.11) by taking the Hilbert-Schmidt norms of the ma­

trices involved in (2.12).

By examining the expression for /:,2M(£.) in (2.9) we see, therefore,

that each of the first two differences, involving /:,~ and /:,~ , give

us matrices which, by virtue of (b), have Hilbert-Schmidt norms that are 0(£-3/2). Because of (2.11) the same is true for the

terms involving /:,~ and /:'~. Lemma (2.10) and condition (a) show that this also holds for the last term involving the coefficients

~,n' Hence, (iii) of theorem I is satisfied (care should be taken to account for the extreme cases arising when m or n are ±£

by making use of (2.9') and (2.10), part (ii)).

For the same reasons, the first two summands occuring in the ex­

pression for /:'M(£) in (2.9) have Hilbert-Schmidt norms that are 0~-1/2). The fact that mM~)m = 0~1/2), condition (a) and lem­

ma (2.10) assure us that the same is true for the remaining terms.

Thus, (ii) of theorem I is satisfied; this proves theorem II.

164

An immediate corollary of this theorem is the following result

for "diagonal" multipliers:

A

COROLLARY 1. Suppose the rrratrices M(2) are diagonal; that is.

2 0 0

I II -2

2 0 0 II -2+1' ..

M(2 ) ........ _ ........ 0 0

2 ..... 112 -

T~en the multiplier operator they define is bounded on LP(G) ,

1 < P < ... provided there e:x:ists a constant C > 0 such th'at

1

-2 I ~ C!/-2 for alL m.!/ (with llm_o

If we restrict ourselves still further to those "diagonal" mul­

tiplier operators of the /12 (that is, ll~ = 112 for -!/ ~ m ~ 2)

we obtain the special case (for SU(2)) of theorem 3 in the preceed

ing paper of Coifman and de Guzman:

2.J. 2_1 If / - 0(1) and 2112 - II 2 - II 2 = 0(!/-2) then the muLtiplier'

operator induced by the matrices M(!/) = 112I!/ is bounded on LP(G),

1 < p < ....

In view of the fact that the surface L of the unit sphere in 2

three dimensional Euclidean space can be realized as the homoge-

neous space SU(2)/SO(2), we can use theorem II to obtain a mul­

tiplier theorem for functions defined on L' We recall that a 2

basis for the spherical harmonics of degree !/(!/ an integer) can

be identified with the functions t;,o , -2 ~ m ~ 2 (see Vilenkin

(S), pages 167-8). In fact, using Vilenkin's notation, a function

165

in L2 (r ) can be represented by an expansion of the type 2

f(.p ,0) '" .I! l! r r ~ Yk ,l! (.p ,e)

l! =0 k--l!

Let M(l!) = (jl~ n) be a sequence of (211+1) x (211+1) matrices, , l! = 0,1,2, ...

In terms of these matrices we define (formally) the multiplier oE erator

(2.13) (Mf) (.p ,e)

We then have the following result as a consequence' of theorem II:

COROLLARY II. The operator M is a bounded transformation of

LP cr ) into itse~f. 1 < P < co • provided: (a) the operator norms 2

A

of M(l!) are 0(1) ,

l! (b) L and

m,n=-l!

(c) '\ I 2 l! l! -1 l! -1 I 2 L jl -jl -jl , Im-ol ,In-ol~Il-1 m,n m+o,n+o m-o,n-6

If the matrices M(l!) are diagonal we obtain the simpler operator

(2.14 )

In this case the last result becomes:

COROLLARY III. M maps LP(r ) into itseLf for 1 < P < co provided 2

166

2 ! 2 2+1 2-1 2 there e:x:ists C > 0 such that !Il k ! < C and 2Il k -llkHl-ll k_1l1 < C2-

for !k-Il! <2-1.2 = 0.1.2 •••• and t5 = -1.1.

By restricting our attention to other special classes of functions

on SU(2) we obtain other corollaries that gives us multiplier the9,.

rems for expansions in terms of Jacobi polynomials with integral

indices. These re.sul ts. together with "weak- type" theorems will

appear elsewhere. At this point we simply assert that there ex­

ists a weak type (1.1) result associated with each of the theorems

and corollaries we oliltained.

BIBLIOGRAPHY

[I] COIFMAN R.R. and WEISS G., "Rep4e6entatlon6 06 Compact G40Up6 and Sphe4lcal Ha4monlc6", L'Ens. Math., t. XIV, fasc.2(1968), pp. 121-173. .

(2] HORMANDER L. ,"E6tlmatu 604 T4aMlatlon Inva4lant Ope4at046 in LP Spacu", Acta Math., Vol. 104 (1960), pp. 93-140.

[3] MARCINKIEWICZ J., "SU4 .i.e6 Multlp.i.lcateu46 de6 S€.4le6 de Fou-4le4", Studia Math •• Vol. 8 (1939), 78-91.

[4] MORLEY V., "On Slngu.i.a4 Integ4a.i.6 on th-e. Sphe4e", Ph. D. The­sis, Univ, of Chicago (1958).

[5] VILENKIN N., "Speclal Functlon6 and the The04y 06 G40Up Rep4!. 6entatlon6", M~scow (1965).

Recibido en agosto de 1970.

Washington University, St Louis, Missouri.

Revista de la Union Matematica Argentina Volumen 25, 1970.

NOTE ON MEAN CONVERGENCE OF EIGENFUNCTION EXPANSIONS

by A. Benedek and R . Panzone

Ved~cado al p~o6e¢o~ Albe~to Gonzalez Vominguez

ABSTRACT. In this paper the mean convergence of series of eigen­

functions of the equation

(x 2a y I ) I + A Y = 0

is studied, completing results of Generozov, {Math. Review '527, (1970)).

I. INTRODUCTION.

1. The problem that we shall consider in this paper is a partic­

ular case of the following general question: let {~ (x)} be an 2 n

orthornormal (L -) complete system over an interval (xo'x l ) ,

-~ < Xo < xl <~, with respect to a measure p(x)dx. Determine

the greatest interval a < p < b or A ~ P ~ B, for which this sys­tem is a basis in LP( = the space of p-integrable functions over

(xo'x l ) with respect to the measure pdx).

We shall always suppose that p E (1 ,~) and that ~ E LP n LP*. n

Then it is easy to see that if {~n} is a basis in LP , the the ser

ies L~l a ~ that converges in norm to f E LP is its Fourier ser­n n

ies. That is,

a n

Jx 1 f(xH (x)pdx x n

o

It is known (cfr. [6], p. 268) that if for every f E LP , the Fou

Este trabajo fue presentado a la reunion anual de la U.M.A., el 7 de Agosto de 1970.

168

rier series of f converges weakly to f, then for every g E LP*, IIp + IIp. = 1, the Fourier series of g converges in LP* to g. Therefore, if {+n} is a basis in LP, it is also a basis in LP*. So a and b· are conjugate exponents. If we know that the system

is complete in L2 and rl pdx < ~, the to prove that it is a ba-Xo

sis in LP, it is enough to prove:

N , (1) I}; a (fH I <; C IfH

Inn P P P Y f E LP

with C independent of Nand f. P

In fact, by the L2-completeness,

(2) N

I}; a (g)t -gl + 0 for g E LP n L2 , P < 2 • Inn P

Then by (1), (2) holds for all g E LP. So {tn} is a basis in LP and also in LP* by the previous argument.

If IXI pdx ..... , then instead of the L2-completeness, it is suf­Xo

ficient to ask that (2) holds for g E EP = a dense subset of LP. Then again (1) implies that {tn} is a basis in LP.

The mo.st celebrated particular case is one due to M. Riesz: the system {cos nx,sin nx, n=O,l,2, •.. } is a basis in LP(O,2w) for 1 < P < • (cfr. [18], ch. 7, §3). This is false for p = 1.

2. Particular orthonormal systems appear as solutions of 5turm­Liouville problems. The simplest case is that of the equation

(3) y" + ).y = 0

with the boundary conditions

(4) y(O)cosa+ y'(O)sin a o , ,

y(2w)cos II + y'(.2w)sin II o -w/2 < a , II <; w/2.

169

For the following particular values of a and B the orthonormal systems are:

case a 0 !3 0 {sin(nx/2)} n = 1,2, ...

case a 0 13- 1112 {sine (2n+l )x/4} n = 0,1 ,2, ... ;

case a 1112 !3 11/2 {cos(nx/2)} n = 0,1 , ...

case a 11/2 13 0 {cos((2n+l)x/4} n = 0,1 , ...

It is easy to see that these four systems are bases in LP for

1 < P < -. For example, we shall prove this for the second sys~ tem. Let us consider the family of periodic functions of period

211, odd with respect to 11 and even with respect to 1112. The Fou­

rier series of such a function is a series in sin(2n+l)x

n = 0,1,... . By the theorem of M. Riesz, it converges to f in LP(0,211) if f E LP. On the other hand, such a function can be

defined arbitrarily on [0,11/2). So, the system just mentioned,

restricted to [0,11/2), is a basis in LP(O,1I/2). This means that

{sin((2n+l)x/4}; n = 0,1, ... is a basis in LP(O,211). Q.E.D.

A more general theorem holds:

THEOREM 1. The system of eigenfunations of the equation

(5) y" - q(x)y .,. AY o o .;; x .;; 211

with the boundary aonditions (4) is a basis in LP, 1 < p < - , if

q(x) is aontinuous in [0,211) ([19)).

In fact, this is a consequence of the preceding result and the

following theorem (cfr. [15), ch. I, th 1.9): let f E L1 (0,211)

and call Sn(f) the nth partial sum of its Fourier series with re­

spect to the system of eigenfunctions of the boundary problem (5),

(4). Call {Tn (f)} the partial sums of the Fourier series of f

with respect to the eigenfunctions of equation (3) and boundary

conditions (4) but with Isgn al1l/2 instead of a and Isgn 13111/2

instead of S. Then

(6) uniformly in x E [0,211) <

Now theorem 1 follows easily. Q.E.D.

170

3. Other cases of the problem have been treated by several au­thors. Pollard in [12], [13] , [14], considers the cases of series of Legendre,Jacobi,Laguerre and Hermite and his results are com­

plemented by others. In [2] series of Laguerre and Hermite are CO!!

sidered and in [10] an open question left by Pollard is settled.

[1] gives a review of the problem and Wing in [17] extends some

results of Pollard and investigates Jacobi series. Muckenhoupt in [9] completes results of Pollard and Wing. The intervals where the mean convergence is considered for the polynomials of Jacobi,

Laguerre and Hermite are (-1,1) ,(0,00) and (-00,00), respectively.

Essentially, these are the only orthogonal polynomials generated by a Sturm-Liouville problem: Merlo, [8] , proves that if an ortho­

normal system has the form {Pn(x)/p(x) ; n; 0,1, .. , witb Pn a polynomial of degree n and is a solution of a problem of Sturm­

Liouville, then except for a change of scale the Pn (x) are the polynomials of Jacobi, Laguerre or Hermi te. [7) contains some re­sults in this direction which are used by Merlo.

Eigenfunctions of the Bessel equation

give place to Fourier-Bessel expansions whose mean convergence has been treated in [17] for v;;;'-1/2 and in [3]when -1<v<-1/2.

Generozov in (4) studies this for expansions in eigenfunctions of the equation

(7) Oo;;;o.<l,Oo;;;xo;;;l, w(l)=O ,

w(O)=O if 00;;;0. < 1 /2 ; w bounded if 1/20;;; 0. <1

In this paper we shall continue the study of this equation.

II. THE EQUATION (x 2o.w')' + AW O.

4. Let us consider an equation of the form

(8) k (P ¥X) + (Ap-Q)X ° ,O<x<l p;;;,o,P>O.

171

If (p/p)1/2 is integrable on (0,1), then the change of variables

t t (x)

and the change of function

u(t) = (pp)1/4X (p (x(t)) p(x(t))f/ 4X(x(t)) ,

transforms equation (8) into

(9) u"(t) + (>.-q(t))u = 0 , 0< t< T = r (p/p)1/2 dx o

where q(t) 1 P 'p , + ---8 P p

+ ~ + P" 4P 4P

S. If instead of the given transformation we use T function v(d = u(Td , verifies

v"Ct) + T2 (>.-q)v

or, what is the same thing

(10) v"(d + (A-q)V

where A

(,=L) dt

tIT, the

6. Let X be the eigenfunction corresponding, to A in equation n n (8) and to certain boundary conditions, say

(11) X(l) = X(O) = 0 or boundedness of X near a singular endpoint.

Then u (t) = (pp)1/4X (x(t)) is an eigenfunction for equation (9;. n n and the boundary conditions corresponding to (11). Let f(t) and F(x) be related as u(t) and X: f = (pp)1/4p. Then

(11 ' )

172

Also the Fourier coefficients of f with respect to {un} are the

same than that of F with respect to {X n} andp. That is

T

f _ t"anun ' F _ t"anX with a 1 1 . n n

fa f(t)un (t)dt

JT 2 oUndt

It is easy to see then that

(1 2) JT N 2 (I N 2 III an un -fl dt ->- 0 iff II a X -FI p dx ->- 0

• 0 1 n n o N->-oo N->-oo

and in general; for 1 < q < 00 ,.

(13) J'TIIN a u -flqd~->- 0 iff JIlIN a X _Fl qpq/4+1/2 pq/4-1/2 dx->- 0 ol nn ol nn

7. Analogous results hold if we consider the equation (10) instead·

of (9): the eigenfunction associated to (10) and An = T2\ is now

wn(t) = IT (pp)1/4 Xn (X(t)). Here t = T- 1 I:(p/p)I/2 dx .

Let f and F be related in the same way as wand X: f = Tl/2(pp)1/4F.

Then

(14 )

and the Fourier coefficients of F with respect to {Xn } and p and

that of f with respect of {wj are the same. Consequently,

II N 2 II N (15) I ~ anwn -fl dt ->- 0 iff I II anXn -F 12 p dx ->- 0

o 0

Since r I~ anwn-flqdt = r II: anxn-Flq pq/4+1/2 pq/4-1/2 Tq / 2 -1 dx, o 0

(16) II N II N / II a w -flqdt ->- 0 iff II a X -Flq pq/4+1/2 pq/4-1 2dx ->- O. 1 n n 1 n n

o 0

The preceding considerations are implIcitly used in the proofs of

part II 1.

173

8. An interesting particular case we have if pP case (13) reads:

,1 N

1. In that

(17) JT N II a u -£Iqdt + 0 o l·n n iff ) I I a X - F I q p dx + 0'. o 1 n n

That is {Xn } is a basis in Lq with respect to the measure pdx iff {un} is a basis in Lq with respect to the Lebesgue measure. This last property can be often established by theorem 1.

The Tchebicheff polynomials constitute an important example of this case. They satisfy the equation

-1 < x < 1

which is of the form (8) with P = (1_x2)1/2 , Q = 0 , pP = The corresponding equation (9) is u"+>.u = 0 , -11 < t < O. So in this case Theorem 1 and the preceding observation assure that the system of Tchebicheff polynomials is a basis in L q with respect. to the measure (1_x2 )-1/2 dx for 1 < q < ""

9. Let us consider now the equation (cfr. (4) )

(18) O<x<l.,

with one boundary condition y(l) = o.

Let a < 1. By the transformation of section 4, which in this case 1 '2 is t = x -a/(l-a) , u(t) = x al y(x(t)), we get

(19) u"(t)-q(t)u(t)->.u=O, 0 < t < 1/(1-a) , u(l/(l-a))=O

with

(20) q (t) \I = I l-Za I !Z(l-a)

Two linearly independent solutions of this equation are

174

{ rt J, (tli"J , ItJ -v (tfi) if v 'i integer ,

(21) It J v (tfi) It Yv (tfi) if v integer

If we add the boundary condition

{ y (OJ - 0 in case ex < 1/2

(22) Y bounded in case 1/2 ..; ex < 1 ,

we obtain the solutions of (18)

(23)

where sn are the positive solutions of Jv(sn)

If ex ;;;. 1, then a so~ution of (19) that vanishes at x=1 does not

be~ong to L2, so no eigenfunation exists.

In fact, if ex=1, then for different values of A, the general solu­

tions of (18) are:

Cl A > 1/4 Y IX sin(h-1/4 19 x + C2)

A 1/4 y (C1lg x + C2)//X

1/4 C1 --

A < . , y IX sinh(/1/4-Alg x + C2)

None belongs to L2. If ex > 1 we can reduce equation (18) to (19)

by the same transformation, only now _00 < t < 1/(1-ex) < O. We con­

sider separately the cases A> 0 , A < O. In the first case the ge!!

eral solution of (19) is:

Therefore, the general solution of (18) is

(25 ) y

Using the asymptotic formulae for H(l) and H(2) , we obtain v v

175

(26)

This function does not belong to L2 unless bi = b~ = O.

If A < 0 the general solution of (18) is

(27) y

again using the asymptotic formulae for H~i) • we get.

(28) i-a ~ i-a ~/

y _ x- a / 2 (Bi e-x i-A/(a-i) + B~ eX i-A (a-i»)

Since H5 i )(ix) # 0 for real x > O. (efr. Watson [16]. pp. 78 and

511). the solution that vanishes at x = 1 must correspond to some

b Z # O. But then (Z8) does not belong to LZ. We have prove.d so

that:

If equation (18) admits a system of eigenfunations whiah is a ba­

sis in LZ' then a < 1.

Gemerozov proved in [4] for 0 ..; a < 1 the following theorem (cfr.

fig. 1).

THEOREM 2. Let -'" < a < 1. The system {Ynl Ilyn"Z

(see (23)) af eigenfunations of the equation

(Z9)

with boundary aonditions,

n = 1.Z •••• }

y(1) = 0 ; yCO) = 0 for -'" < a < lIZ, y bounded for 1 >a';>1/2.

is a basis in LP (O.1) for

(30) Z <p<_Z_ Z-(avO) avO

(aVO means sup(a,O)).

FIGURE 1.

~1 o

-V=-12~~:~) I ~,. __ - --1/2

-----~

-1

176

v

I 2CL-1 I V= 2(1-CL)

, \ I 1 , \ 1 1

, \ I 1

" 'I 1

~-: -----\: , 1

" " ________ .1

10. We consider now, for CL < 3/4, the system of functions

(31) y n ex) 11 - 2CL I

2 (1- CL )

which are solutions of equation (18) with A An' determined by

the condition Yn(1) = 0, (cfr. figure 1).

(For 1 > CL ~ 3/4. (23) gives the only solution of (18) wich be­

Zongs to L 2 ).

177

These functions verify at the origin

(32)

where c= min(1-2a.O). These conditions uniqueZy deter>mine, but

for a constant factor, the solutions (31).

(32) implies the orthogonality of these functions as we shall see

bdlow and might be considered as the boundary condition at the or

igin. So. the functions Yn defined by (31) are aZso soZutions of a

G Sturm-LiouviZle problem.

The orthogonality of {Yn} follows from,

p, - A ) fly Y dx m n n m e:

= by(3Z) o (1) ,

(recall a < 3/4).

The following theorem holds:

THEOREM 3. Let a < 3/4. The orthonormaZ system {Yn/IlYnI12} • Yn

defined by (31), is a basis in LP (O,1) if

i) _Z_<p<!. 2-a a

when 2/3 ~ a ~ 0

ii) 1 2(1-a) < p < 2a-l

178

when 3/4 > a > 2/3

iii) < p < 00 a < 0

We consider first the case p = 2.

LEMMA 1. The orthonormal syst.ems of theorems 2 and 3 are aomplete

in L 2 •

Proof. This. follows from .the equality

and the fact that when v > -1, {J (ts )tl/2} is a complete system 2 v n

in L , (cfr. [3) and the references there mentioned).

I I I. PROOF OF THE MAIN RESULTS.

11. Proof of theorem 2. We follow the same line of proof as Ge­nerozov, .[4). Improved estimations will permit to cover a greater range of a. Let _00 " a < 1, and v ~ -1/2. Actually we shall prove

that the system Yn(x) x(1-2a)/2Jv(xl-aSn)' sn = positive zero of

J (x), is a basis in LP for v

2 2 < p " --2-(avO) avO

Then, by taking v = I---Zl-2a I we have theorem 2 and by taking (l-a)

v = - I 1-2a I we have i) of theorem 3. 2(1-a)

Since by lemma 1, the system Yn(x) is complete in L2, it is enough to prove for our purpose that the Dirichlet kernel

verifies

179

( 33)

for 2/(2-(flVO)) ~ P ~ 2/(fl VO). Let us call

(34)

(35)

Making the change of va~ables x = tl/(l-fl), ~

and observi.ng that RYn"~ = (l-Il)IIYn"~ , we get (cf.!j4-7):

Finally calling f(t) = I'(t l / O - fl ) )t,./p(l-,') , we see that (33)

holds iff

(37)

But, (cfr. (17)),

(38) fuAN ·Jv(ANt)·Jv+l(ANT) +

2 ., - t

+ O(lt+ ~ T+t 2-t-T

c' fll f(t) IPdt p 0

Since It .Jv(t) is bounded for v ;;;. -1/2, the first two terms are of

the form h(T)k(t)/(l-t), where hand k are bounded independently

of N. Then, to prove that (37) holds it is enough to prove that

the kernels

180

with B = (-21_1)~ define continuous operators in LP(O,l). p I-a

The operator

(39) Il a If(T) = (1) ~ dt

T 2-t-T o

is continuous in LP (O,1) if (cf.[3),§6),

(40) avo < IIp < lA(l+(3)

In our case (40) reads

or equivalently

( 41) 2 2 avO> p > 2-(aVO)

Also the operators

(42)

are continuous in LP (0,1) (cf. (5) or (111) for -l/q < S < lip ,

1 < p < 00 , IIp+1lq = 1, which is just (40) and then in our case (41). Q.E.D.

12. ppoof of theopem J. In the proof of theorem 3 we need the

following auxiliary result.

LEMMA 2 •. The opepator

( 43)

i8 continuous in LP(-oo,oo) if -lIp < -8 , -a-a < llq

181

It is an easy consequence of the lemma in p. 308 of Muckenhoupt's paper, (9) •

(It can be proved using Theorem 3 of (3) with r(x,t) = I x/tI 1/ pq).

To prove the theorem we first observe that we have already proved i) in § 11 • For the rest of the theorem we follow the proof of th!., orem we follow the proof of theorem 2 but taking -1< \I < -1/2 • \I = -11-2al/12(1-a)l. Then again {Ynlx)} is a basis in LP(0.1) if (37) holds. Now the estimation (38) of the. Dirichlet kernel DN(t.d does not hold but instead we have (cf. (3) ,.14):

where K(A,T,t)

To prove (37) for some p. it is then enough to see that each term in the right hand side of (44) multiplied by It/Till • a = (1/2 - 1/p)a/(1'-a). defines a continuous operator in LP(O<T<l). whose norm is uniformly bounded in N.

We call these operators respectively K.K*,H,H*,l,J, and show that:

a) K and H are uniformly continuous in LP(0<T<1) for

b)

1 ---< 2(1-a) p < ~

a when 2 3 "3 < a < 4"

1 < P <.. when a < 0

K* and H* are uniformly continuous in LP for ".

2 1 2-a < p < 2a-1 when

1 < P < .. when a < 0

182

c) I is continuous in LP (0,1) for 2 2 1 2- (avO)

< p < av 0 ' a <

d) J is continuous in LP for

1 1 whem 2 3 2 (l-a ) < p <

2a-l 3" < a < 4

when a < 0

Then for a ~ 0, there is no restriction on p and iii) follows. If 2/3 < a < 3/4, the interval 1/2(1_a) < p < 1/ (2a-1) given by d), contained in the remairting intervals given by a), b) and c), and ii) follows.

c) we have already proved in §11, by considering operator (39).

d) The operator

1 1

[1 ~v+Z-Stv+t+S

(45) . f(t)dt o

That is, if

(46) v + 1. - S > 1 and v + ,t. + S >_1. 2 p ~ q

Replacing v by -!Za-1!/!2(1-a)! and S by (1/2 - l/p)a/(l-a) , (46) is equivalent to

I if 2 3

< P < 2a-l - < a < '4 Z(l-a) 3

(47) < p < '" if a < 0

That proves d) .

b) We consider the operators

(48) Pov.J'" o

183

An easy change of variables shows that their norms as operators in LP(O,~) do not depend on A. So we take A = 1 and determine the p's for which (48) is continuous in LP (0,1).

Let- us call

(49) 1 a = - (\1+2) (1/2 - 1/p)a/(1-a) •

Since 0 < a < 1/2 , we have:.

o < x < ~

(SO)

I 1/2 -a -1 x J\I+l (x) = 0(1) (1+x ) , o < x < ~

This implies that the operators defined by (48) are continuous in LP(O,~) whenever the operator (43) wtth a and a given by (49) is continuous. It is easily verified that it is true for any p E (1,~) if a < O. When a E (2/3,3/4), the mentioned operators are continuous if 1I(2a-1) > p > 2/(2-a). This is the condition to which reduce those of lemma 2.

a) follows from b) by duality if we observe that K" with a 1 1 a (,; - ph'-=-a as operator in LP is the adj oint operator of K with

a (1 1) a ';-qr-a as operator in Lq , (cf. (3) ,§6). Q.E. D.

ADDED IN PRODF. Theorem 1 still holds even when one only requires that q(x) E L':

M.M. Crum, "On the Sturm-Liouville expansion", The Quat. J. of Math., Vol. 6, N°24, (1955), 288-292.

[ 1]

[2]

[3]

[4]

[5]

[ 6]

[7]

[8]

[9]

184

REFERENCES

ASKEY R., NOIlm i.ne.quatj..tl..u 601t .6ome oltthogonat .6eJti.u, Bull.e­tin of theAMS; 72, (1966), pp. 808-823.

ASKEY R. and WAINGER s., Mean c.onvellgenc.e 06 expan.6i.on.6 i.n La:,· gueltlte and Heltmi.te .6eJti.e.6, American J. of Math., 87, (1965), 695-708.

BENEDEK A. and PANZONE R., On mean c.onveltgenc.e 06 Foulti.elt-Be.6 .6et .6eJti.u 06 negati.ve oltdelt, to appear. -

GENEROZOV V.L., Matemati.c.hull.i.e Zametll.i., 3, (1968), pp. 683-692;.Mathematical Notes ~f the Acad. of Sc. of the USSR, 3 , (1968), PP. 436-441.

HARDY G: H. and LITTLEWOOD J. E., Some molte theoltem.6 c.onc.eltni.nq' Fo-ulti.elt .6 eJti.u and F oulti.elt powelt .6 eJti.u, Duke Math. J. 2 , (1936), 354-382.

KACZMARZ S. and STEINHAUS' H., Theoue delt Oltthogonatlte;£hen New York, (1951).

LAVRENTIEV M.A. and SHABAT B.V., Method.6 06 the theolty 06 6un£ ti.onl> 06 a c.omptex valti.abte, Moscow, (1958).

MERLO J. C. On o·llthogonat potynomi.atl> and .6ec.ond oltdelt Unealt opeltatoll.6 , Annales Polonici Matematici, XIX, (1967), pp.69-79.

MUCKENHOUPT B., Mean c.onveltgenc.e 06 Jac.obi. l>euel>, Proc. of the the A.M.S., vol. 23, (1969), pp. 306-310.

[10] NEWMAN J. and RUDIN W., On mean c.onvellgenc.e 06 oltthogo.nat .6e­Iti.u, Proc. of the A.M.S., 3, (1952), pp. 219-222.

[II] OKIKIOLU G.O., On c.elttai.n exten.6i.onl> 06 the Hi.tbeltt Opeltatoll, Math. Ann., 169,. (1967), pp. 315-327.

[12] POLLARD H., The mean .c.onveltgenc.e 06 ollthogonat '.6elli.e.l> , I , Transactions of the A.M.S., 62, 11947), pp~ 387-403.

[13] ----------, The mean c.onveltgenc.eo 6 olttho gonat .6 elti.e.6, II , Trans. A.M.S., 63,. (948), 355~367 .•

[14) ----------, The mean c.onveltgenc.e 06 'olLtho gonat l> elLi.el> , II I , Duke Math. J., 16, (1949), pp. 189-191.

[15] T1TCHMARSH E.C., Ei.gen6unc.ti.otL expanl>i.onl> al>l>oc.i.ated wi.th .6ec. ond-oltdelL di.66eltenti.at equati.on!>, Oxford, (1946).

[16] WATSON G.N., A tlLeati.l>e on the theolty 06 Be!>!>et 6unc.ti.onl>, Cambridge, (1952).

[17] WING M., The mean c.onvelLgenc.e 06 oltthogonat !>elti.u, Amer. J. of Math., LXXII, (1950), pp. 792-808.

[18] ZYGMUND A., TILi.gonometlLi.c.at !>elLi.el>, N.ew York, (1955).

[19] RUTOVITZ D., On the Lp-c.onveltgenc.e 06 ei.gen6unc.ti.on expan.6i.on.6, The Quat. J. of Math., vol. 7, N°25, (1956),24-38.

Universidad Nacional del Sur. Recibido en agosto de 1970. BahLa Blanca, Argentina.

Revista de la Uni6n Matemstica Argentina Volumen 25, 1970.

SUBESPACIOS IN Y EL CALCULO FUNCIONAl DE SZ.-NAGY Y FOIAS

por Domingo A.Herrero*

Vedieado al p~o6e~o~ Albe~to Gonz~lez Vom~nguez

I. INTRODUCCION.

Sea K un espacio de Hilbert complejo y separable con producto in­terno (.,.)y norma II.II K y consideremos el espacio de Hardy Hi de las funciones (anallticas) de cuadrado sumable en aD (D =

= {z: Izl<1}) con valores en K. Es decir, Hi es el espacio de las (clases de equivalencia de) funciones F:aD + K tales que:

1) (F(e ix), <1» E H2 (espacio de Hardy esca~az'), para to do <I> E K ,

2)

con la relaci6n de equivalencia usual F1 - F2 si Y s6lo si

2 HK es un espacio de Hilbert complejo y separable con producto in-terno

( I ix ix / F ,G ) = (F (e ), G (e )) dx 211 an

y las funciones Hi pueden extenderse de manera natural a funcio­nes anallticas en D con valores en K (indicaremos la extension a­nalitica de F E Hi mediante F(z); ver detalles en 5).

2 Un 8ubespacio inva~iante de HK es, por definicion, un subespacio cerrado M tal que SM eM, siendo S el operador "multiplicacion por e ix1 ••

* Este trabajo ests parcial mente subsidiado por Grant GP-14255 de la National Science F(~ndation.

186

El subespacio (obviamente cerrado) K = ~ es invariante bajo S# el adjunto de S. Sea T el adjunto de S#, considerado como opera­dor en K; es facii verificar que T es una contracci6n completamen­te no unitaria de K. La afirmaci6n inver-sa es tambi€n cierta; dio. cho en forma mas precisa: "Toda contracci6n completamente no unit.!! ria de un espacio de Hilbert separable y complejo es unitariamente

equivalente a un operador T del tipo descrito, operando en el com­plemento ortogonal de algun subespacio invariante". (ver: 8, capf­tulo III).

Sobre esta base, BelaSz.~Nagy y Ciprian Foias, han desarrollado el siguiente calculo funcional:

Sean K Y T como antes, y seaU E Hoo (espacio de las funciones ana­Ifticas uniformemente acotadas en D); para F E K, definimos

(1) u(T)(F) = P(uF)

siendo P la proyecci6n ortogonal de H~ sobre K. (En particular, se tiene que: TF = P(zF) = P(SF)).

La aplicaci6n u(z) + u(T) es un homomorfismo de' algebras de Hoo en

el conjunto de los operadores acotados en K, con las siguientes propiedades (8, cap. III):

(1) u(T) es l1mite fuerte de. polinomios en T, Y

lIu(T) II .;;; Ilu(z) II""

2 (11.11 indica la norma de un elemento de HK, 0 bien, como en este caso, la norma de un operador en K).

(i i) Si {u } es una sucesi6n de funciones uniformemen-n

te acotadas que convergen, en casi todo punto (c. t.p.) d~ aD a u(eix)(valo~eslimites de u(z)), en

tonce's un(T) converge fuertemen te a u (T) •

(iii) Si li(z) = u(z), entonces u(T)'" = li(T*).

(aquf T* indica el adjunto del operador en K).

Las propiedades de u(T) dependen obviamente del subespacioinvaria!!. te M, y han sido objeto de estudio por varios autores. Por ejemplo,

187

D. Sarason (7) se ocupo del caso en que el "espacio base" K tiene

dimension uno', lcigrando, el siguiente resultado: "Sea L un opera­

!iorlineal y continuo en K; entonces, L conmuta con T si y 5610

si existe una fl1l'l.ci6n u E HOOtal que L = u(T)".

El resultado.mlis general de es:te tipo file estu!iiado, por variosa!!

tores, entre ellos SZ.-Nagy y Foias (9; ver tambien 2 y 4).

En (3), Paul A. Fuhrmann anali za las propiedades es,pectrales de

u(T) para el caso en que dim K = N < 00.

Algunos resultados de (3) a dmiten una extensi~n bastante natural,'

no para el caso mlis genetal, pero 51:' al menos para un espacio K

de infinitas dimensiones y un subespacio invariante de tipo res­

tringido, a saber:

2 DEFINICION. Sea M un subespaaio invariante de HK para e~ aua~

2 e:r:iste una funai6n (es.aa~ar) in.terior q ta~ que oHK eM En ta~

aaso. diremos que M es un subespaaio IN.

II. ANALISIS ESP~CTRAL.

Los subespacios IN han sido estudiados en la tesis del autor (6),

y de aUi tomamos el siguiente re,sultado:

PROPOSICION 1. Si M es un subespaaio IN. entonaes e:r:isten un oP!:

rador interior U(es deai'1'. U es una funai6n definida en a. t.p. de

aD. aon va'Zo'res en Zos operadores unitarios de K. ta~ que

U(e ix )</> E Hi • para todo ~ E K) y una funai6n interior q. taZes que:

(i) M 2 = U~ , Y U esta un:lvoaamente dete'rminado por M.

a menos de un faator unitario aons'tante a dereaha •

(ii) qH~ eM. y pq E Hoo • para toda fun&i6n interior p

2 taZ que pHK eM.

El 9perador interior U es el operador IN de M y q es la funcion

interior minimal (FIM) del subespacio. Algunos de los resultados

que siguen pueden probarse exactamente como en (3), de 'modo que £ mitiremos su demostraci6n.

188

LEMA 2. Sea M = UH~ un subespacio inva:riante de H~ y U su ope:ra­

do:r inte:r"io:r (no necesa:riamente IN). Si u(T). (u(T"'). :resp.) tie­

ne inve:rsa aeotada en K. entonees e:r;iste c5 > 0 tal. que

-1 -1 donde IU (z) UK se define igua1.. a ee:ro si U(z) no es inve:rtib1.e

en K. pa:ra todo ZED. (3. teo:r. 2.3)

LEMA 3.

LEMA 4.

(i) Sean M y U eomo en e1. 1.ema 2 y sea M = UH 2 donde - - -1 K' U(z) = U",(z). y K~M • Entonees. T ope:rando en K es uni ta:ri amen te equiva1..ente a Sll ope:rando en K.

(ii) Si U es un ope:rado:r IN eon FlM q. entonees U es un 0-

pe:rado:r IN eon FIM q.

Sean M y U eomo en e1. l.ema 2 y sea A E D.

(i)I E ~p(T"') si y s61..0 si e1.. nae1.eo de U"'(l) es distinto

de (0). Las autofunciones (no:rma1.isadas) de T'" tienen

l..a fo:rma [(1-llI2)1/2/(1_Iz)1~. siendo ~ un veeto:r uni

ta:rio de K pa:ra e1.. eua1. U"'(l)~ = 0 •

(ii) 1 E !:p (T) si Y s61.o si ker U(l) f O. Las autofuncio­

nes (no:rmal.isadas) de T tienen la fo:rma V~. siendo ~

un veeto:r unita:rio de K pa:ra el eual U(l)~ = O. Y V e1.

siguiente ope:rado:r inte:rior: si R es la proyeeei6n 01'­

togonal de K sobre ker U(l),b(z,l) es el facto:r e1.eme~

tal de B1.asehke (I/lll)(l-Z)/(l-Iz) (I/lll se define i gua1.. a (-1) si l = 0). Y

B(z) = (I-R) + b(Z,l)R

entonces

La demostraci6n de este lema esti contenida en los resultados de (6, cap. III).

189

TEOREMA S. (a) Sean M y U aomo siempre. Los siguientes enunaia­

dos son equivalentes:

(i) M es un s'\.tbespaaio IN.

(ii) El nualeo de la apliaaai6n j: u(z) -+- u(T) es distinto

de (0).

(iii) E:r:iste una funai6n "no invertible" u E Hoo tal que

u(T) es invertible en K y [u(T)]-l veT) para aierta

funai6n v E Hoo•

(b) Supongamos que vale (i) y que q es laFIM de M;

entonaes la aondiai6n

(2) in f { I u ( z) I + Iq ( z) I : ZED} ;;;. 6 ( 0 > 0 )

es neaesaria Y sUfiaiente para que u(T) sea invertible. y que di­

gha inversa pueda e:r:presarse en Za forma veT). aon v E Hoo•

NOTA. Luego veremos que u(t) puede ser invertible aan cuando no se satisfaga (2).

Demostraai6n. (a) (i) es equivdente a· (ii). Es facil verifi­car que, para una funci6n p E Hoo

, las siguientes condiciones son equivalentes:

1) peT) = 0 ~ 2) pF E M, para todo F E K; 3) pHic M .

Por otra parte, ker (j) es invariante bajo multiplicaci6n por e ix

y w* -cerrado en H 00. Por 10 tanto (ver 5 ,cap. IV), 0 bien ker (j) = (0), 0 bien ker (j) = qH OO

, para alguna funci6n interior q. (Si q es constante, entonces es obvio que K = (0); supondremos que no es este el caso).

En consecuencia: ker (j)

igual a q. qHOO

, (0) si y s6lo si M es IN con FIM

(iii) - (i) y aondiai6n neaesaria en (b): Supongamos que u, v E Hoo

, u no invertible y [u(T)]-l = veT).

Dado que j es un homomorfismo, tenemos que

190

I = u(T)v(T) = v(T)u(T) = uv(T) , en K

es decir: F = PF = P(uvF), y por tanto 0 2 F E K, 10 cual implica (l-uv)HK C M.

P(l-uv)F, par,a toda

Ahora bien, dado que Uno es invertible, 1-uv '" O. Si P es el fac­tor interior de (l-uv), entonces

pH~ = i:lausura [(l-uV)H~l c M

de donde s'e deduce que M es un subespacio IN, cuya FIM, q, divide a p. Luego, existe g E H~ tal que

uv + qg = 1

y de aqul se sigue la validez de (2), en caso en que uno es inver tible. Si u es invertible en H~, entonces (2) se satisface trivial

-1 -1 mente poniendo 0 = II u II ~ •

En este caso [u(T)1-1 = (u- 1)(T), como es facil verificar; esto de­

muestra que si u es invertible, 'entonces la condicion (2) de (b) es

tambien suficiente.

Condiai6n sufiaiente en (b). Sea M un subespacio IN con FIM q Y

sea u E H~ tal que se cumple (2).

De acuerdo al teorema de ~a aorona. de Garleson (1), existen dos funciones a',b E Ho!> tales que

(3) 'au + bq en D .

Aplicando j a la identidad (3), se tiene

1= (au+bq)(T) = au(T) = a(T)u(T) = u(T)a(T)

(en efecto, bq(T) = b(T)q(T), y q(T) = 0, por las observaciones

que hemos hecho al principio de la demostracion); por 10 tanto

[u(T)I-1 = afT)

(i) .. (iii) Sea A E D tal que q CA) '" 0, y pongamos u(z) = Z-A; es evidente que u(z) es una funcion no invertible de" H~, Y que vale

191

(2). Sin embargo, u(T) es invertible en K y su inversa puede es-cribirse en 1a forma a(T), para cierta funci6n a E H~. Q.E.D.

Si dim K = N < ~ , entonces e1 determinante de U satisface las condiciones:

1)

2)

2 ' (det U)HK eM,

para todo zED, tal que q(z) I O.

Se tiene asf e1 siguiente

COROLARIO 6 (teor. 2.3, en 3). Si dim K < ~, entonaes Za aondi­

rJi6n (2) de Z teorema E, (b) es neaesa:ria y sUfiaiente para, Ja existenrJia de [u(T)I-l. Mas aun, en (2). q(z) puede reempZaaarse

pOl' det U(z).

Si u(T) es invertibZe en K, entonaes su inve:rsa puede ex:presar,se

en Za forma v(T), aon v E H~

Usando e1 teorema 13 de (5, cap. VIII) y los resultados de (3), tenemos:

COROLARIO 7. Sea U un operador IN y u E H~. Se tiene que:

(i) Si 1 E D, Y U(l) no es inOertibZe en K, enton­

aes U(l) perteneae a'Z espeatro de u(T).

(ii) Si, 1 E aD~ U(z) no puede aontinuarse anaHtiaa­

mente a z = 1 " Y u puede ,e'xtenderse, en forma" '

~o~tinua a D U (ll , entnnaes u(l) perteneae aZ

espeatro de u(T).

TEOREMA 8. Sea U un operador IN. Si upuede ser aontinuamenteex­

tendida a Za a'Z.ausura de D. entonaes

LEMA 9.

r(u(T)) = u(r(T))

2 Sea M un subespaaio invariante (auaZquiera) de HK, y

192

u E If", u iF O. Entonces ker u(T) # (0) si y s6lo si existe una fu!!.

ci6n F E Hi - M. Y una funci6n interior p tales que pF E M y

jiu E Hoo •

Demostraci6n. Si F Y P satisfacen las condiciones del enunciado, es evidente que PF -# 0 y Que p no puede ser constante; ademas, PF y P tambi~n satisfaeen dichas condiciones. Por 10 tanto, podemos suponer directamente que F es una funci6n de K, F # O. Entonees

u(T) (F) P(uF) = 0 si Y s610 si uF EM,

y esto equivale a decir que pF E M, para algun divisor interior p de la funei6n u (p no eonstante). Q.E.D.

TEOREMA 10. Sea M un subespacio IN con FIM q, y u E HW Entonces,

ker u(T) ~ (0) si y s6lo si u y q admiten un factor i~terior co­

mun, no trivial.

Demo8traci~. Es claro que si la funci6n p del lema 9 es elegida minimal con respecto a la eondici6n pF E M, entonces: pq E Hoo , 10 eual prueba el enuneiado directo.

En el otro sentido: si jiq , jiu E Hoo , con p interior, y p , 1, en­toneS's:

1) Si p q entonces u(T) = 0 y ker u(T) = K •

2) Si p , q sea Mp {F E H~:PF EM} ; entonces Mp es un_subespaeio.invariante (IN) que contiene a M, y su FIM es igual a pq (ver 6, cap. II). Por 10 tanto, Mp es estrietamente mas grande que M, y si F E M -M , entonees PF es una funei6n de K p tal que PF , 0 , y pPF E M. En eonelusi6n: uF = (jiu)pF E M, y por tanto, ker u(T) , (0). Q.E.D.

COROLARIO 11. Sea M un subespacio IN con FIM q, Y u E Hoo, u # O. Entonces ker u(T*) # (0) si y s6Zo si u y q admiten un factor inte

rior comun, no trivial.

Demostraci6n. Usar el lema 3 y e] teorem1 a 10.

COROLARIO 12. Sea M un subespacio IN. Entonaes e l espectro "res:!:.

dual" de u(T) es vac{o, para todo u E Hoo . (igual demostraei6n que en (3, cor. 2.10)).

193

III. EJEMPLOS.

Para terminar, ilustraremos los resultados con dos ejemplos. En el primero de ellos veremos que el coroZario 6 no puede extenderse al caso en que K es de dimension infinita, ni aun cuando el opera­dor U pertenece al tipo mas restringido de operadores anal!ticos para los cuales se puede definir el determinante.

Probaremos que existe un operador U y una funcion interior u, no constante, tales que:

1) U es IN , Y FIM de U = determinante de U(z) de Blasch)s.e,

q(z) es un producto

2 ) j u (z) j + II U- 1 (z) II ~ 1 ;;. (1 /4) , (z ED) ,

3) inf {juCz)j + jq(z)j: zED} = 0,

4) uCT) es invertible en K (obviamente, el operador inver­so de uCT) no puede expresarse en la forma veT), con v E Hoo ).

Sea {~n }:=l un sistema ortonormal y completo en K, y definamos U como el operador "diagonal" tal que

n = 1,2, ...

donde los ceros de los factores de Blaschke elementales satisfacen

la condicion 0 < r 1 < r 2 < ••• < rn < 1 , y q(z) = n:=lb(z,rn) es un producto de Blaschke convergente.

En estas condiciones, 1) se cumple trivialmente.

No es dif!cil verificar que (2,Zema 3.2) si D±R son los c!rculos cuyas circunferencias pasan por {-l,±iR,+1} ,0 < R < , entonces IIU- 1 (z) II~ 1 ;;. R , para todo z en el conjunto: D-{D+R n D_ R} .

Sea u(z) = n:=lb(z,Ak ) un producto de Blaschke cuyos ceros corre~ ponden a puntos de la circunferencia que pasa por los puntos {-1,C3/4)i,+1} y satisfacen la condicion

194

o < Re A 1 < Re A 2 < ... < Re A k .,. 1

Es inmediato que, si An tiende a 1 suficientemente rapido, enton­

ces la condici6n 2) sesatisface, independientemente de .c6moes­

ten distribu1dos los ceros, r n , de U(z).

Ahora procederemos a e1egir estos Ceros.

00 mk . Sea ~ = Re Ak ' p(z) =TIk=lb (z,Rk ) , donde {mk } es una suce-

si6n no decreciente de enteros positivos que tiende a +00 10

bastante despacio para que

(es decir, el producto p(z) es convergente).

Por otra parte, no es dif1cil ver que 1a sucesi6n {mk } puede ele­

girse de modo que valga 3), es decir,

inf {iu(z)i + ip(z)i: izi<R} + 0, cuando R + 1.

ffik ffi k Para cada k fijo, reemplazamos b (z,~) por ITh=lb(z,Rkh ), donde

los valores ~h estan elegidos de modo tal que

• •• < <: 1 ,

y si q(z) '" ffik TTk=l ITh=lb(z,Rkh ), la condici6n 3) sigue valiendo pa

ra u(z) y q(z).

S6lo resta verificar 4). Para esto, observemos que

(sumas di rectas ortogonales), donde

g (z) = (.1 - r2) 1/ 2/ ( 1 - r z) r

AS1, si F E K, entonces F

IIFII2 = I:=li c n i 2 < 00', y

195

u(T) (F)

Es decir •. ·u(T) es un operador normal en K. diagonalizable con res­

pecto al sistema ortonorJ.Ilal y completo {gr CPn }:=l ; el n-esimo coe ficiente de la diagonal es n

De esta igualdad y de 1). deducimos que u(T) es invertible. y U[u(T)l-lU ~ 4 • 10 cual prueba 4).

Vayamos al segundo ejemplo. Si M es un subespacio IN con FIM q. entonces se sigue de los resultados de (8. cap. III; ver propiedad

(ii). en la introduaai6n) y de la demostracion del teorema 5 que si qr(z) q(rz) (z E D). entonces

limqr(T) = q(T) = O. (r ... 1) •

en la topologia fuerte. Ahora bien. aqui qr(T) debe entenderse en

el sentido del calculo funcional (es decir. "via" la proyeccion p. etc.). i.Es posible darle un sentido mas directo a la expresion an terior? .

Esto es posi·ble. al menos para los productos de Blaschke. Si

donde A1 .A 2 ••..• AN son los distintos ceros de q en D. entonces

q(z) = m(z)TT~=l(z-Ak)' donde m(z) es cierta funcion invertible de

H~. y no es dificil ver que qr(T) converge a q(T) = 0 en la topol£ gia de la norma; se obtiene asi el siguiente resultado

COROLARIO 13. Si M es un subespaaio. IN auya FIM es e l produato de

Blasahke fini to q (z) (aomo fue definido mas arriba), entonaes N mk

ITk=l (T-A k) 0 , siendo este produato un polinomio en T (aomo '2

perador en K) en el sentido usual.

196

m Mas aun, TT:=l (Z-Ak) k es eZ poZinomio "minimaZ" de T.

Inversamente, si M es un subespaaio invariante y T, eZ operador ad

junto de SH restringido a K = uL, satisfaae una eauaai6n poZin~mi: aa, entonaes M es un subespaaio IN y s·u FIM es un produato de BZa~

ahke finito aoneatado aon eZ poZinomio minimo de T de Za manera in

diaada.

La demostraci6n completa de este corolario fue obtenida en (6) u­sando argumentos directos; de (6, cap. III) tambien se puede extr~ er el siguiente resultado:

PROPOSICION 14. Sea M un subespaaio IN auya FIM es eZ produato ae 00 mk

B"Lasahke q(z) = TTk=lb (Z,Ak), y sea K e"L aompZemento ortogona"L

de M. Entonaes K admite Za desaomposiai6n

(suma direata aerrada),

mk donde (Tk-A k) o (Tk indiaa "La restriaai6n de T a Kk ), para

k = 1,2, ...

Si, ademas, b(T,Ak) (~/IAkl)(Ak-T)(I-AkT)-l (entendido aomo eZ N mk

"Limite uniforme de "La serie de TayZor!), y TN = TTk=lb (T,Ak),

entonaes IITNU ,.;;; 1 , para todo N,

lim TN = 0 , (N + 00)

en "La topoZogia fuerte.

197

REFERENCIAS

[ 11 CAlLISON LINNAB.T. I ntu,potaU.on b!l bou.nde.d anatyt.lc. 6u.nc.Uon.6 and the c.OII.ona p'ILobte.m. Ann. cof Math. (2) 76 (1962), 547-559.

[2] CLAIll( DOUGLAS N. On c.ommu..t.lng c.ontlLac.t.lon.6, (comunicaciiSn pe,!

aona].) •

[3] 'UHlMANN PAUL A., On the. c.olLona the.olLe.m and .It.6 apt.lc.at.lon .to

4pe.c..tILat plLobte.m4 .In H.ltbe.ILt 4pac.e.. Trans. Amer. Math. Soc.,

132 (1968). 55-66.

[4] ---~------------. A 6u.nc..t.lonat c.atc.u.tu.4 .In H.ltbe.1L.t .6pac.e. b44e.d on ope.lLatolL vatu.e.d anat!lt.lc. 6,u.nc.t.lon.6. Israel J. of

Math., 6 (1968), 267-278 • . [5] HILSON HINIlY, Le.c..tU.1Le.4 on .lnvalL.lant 4u.b4pac.e.4. Academic Press,

Princeton, New Jersey, 1964.

[6] HIIlIlBIlO DOMINGO A., I nne.ic. 6u.nc.t.lon- ope.lLa,tOlL.6, t rabaj 0 de te­

aia, Universidad de Chicago, 1970.

[7] SAIASON DONALD, Ge.ne.lLat.lze.d .lnte.ILpotat.lon .In H~. Trans. Amer.

Math. Soc., 127 (1967), 179-203.

[8] SZ.-NAGY BELA Y FOIAS C., Anat!l.6e. halLmon.lqu.e. de.4 opilLate.u.1L4 de. t'e..6pac.e. de. H.ltbe.lLt. Akademiai l(iado, Budapest, 1967.

(9) --------~----------------, Commu.tant4 de. c.e.IL.ta.ln opilLate.u.1L4, Acta Scientiarum Math. (Szegell). XXIX (1968), 1-17.

Universidad de Chicago.

Chicago, Illinois.

Recibido en lRarzo de 1970.

PUBLICACIONES DE LA U. Me A.

Revist,}, de la U. M. A. - Vol. I (1936-1937); Vol. II (1938-1939); Vol. III (1938-H):l9); Vol. IV (1939); Vol. V (1940); Vol. VI (1940-1941); Vol. VII (1940-1941); Vol. VIII (1942); Vol. IX (1943); Vol. X (1944-1945).

Rp.vil;ta de 10, U. M. A. Y 6r!}ano de la A. F. A. - Vol. XI (1945-1946); Vol. XII (19-16-1947); Vol. XIII '(1948); Vol. XIV (1949-1950).

}{e~:ista de 10, U. M. A. Y cle la A. F. A. - Vol. XV (1951-1953); Vol. XVI (1954-HI55); Vol. XVII (1955); Vol. XVIII (1959); Vol. XIX (1960-1962); Vol. XX (1962); Vol. XXI (19G3); Vol. XXII (1964-1965); Vol. XXIII (1966-1967).

LOS volumenes III, IV, V y VI comprenden los siguientes fasclculos separados' Nq 1. GINO LORIA. Le Matematiche in Ispagna e in Argentina. - N9 2. A. GON-

ZALEZ DOMINGUEZ. Sabre las be1"ies de funciones de Hermite. - NQ 3. MICHEL PETRO­VICH. Remarque.s arithmetiques sur 1,ne equation differentielle du premier ordre. -NQ 4. A. GONZALEZ DOMINGUP:Z.Una nueva demostraci6n del teorema limite idel Cdlcu-10 de Probabilidades. Condiciones necesa"ias y suficientes para que una funci6n sea integral de Laplace. - NQ 5. NIKOLA OBRECHKOFF. Sur la sommation absolue por 10, transformation d'Euler des series divergentes. - NQ 6. RICARDO SAN JUAN. Deriva­cion e integraci6n de series' asint6ticas. - NQ 7. Resolucion adoptada por la U. M. A. en la cuestion promovida pOl' el Sr. Carlos Biggeri. - NQ 8. F. AMODEO. Origen y desarrollo de la Geometria Proyectiva. -- NQ 9. CLOTILDE. A. BULA. Teoria y cdlculo de los momentos dobles. - NQ 10. CLOTILDE A. BULA. Cdlculo de superficies de fre­cuencia. - NQ 11. R. FltUCHT. Zur Geometria aUf einer Flache mit indefiniter Metrik (Sobre la Geometria de una superficie con metrica_ indefinida). - N'" 12. A. GON­Z.'\'LEZ DOMINGUEZ. Sobre una memoria del Prof. J. C. Vignaux. - NQ 13. E. TORAN­zos. Sobre las singularidades de las curvas de Jordan. - NQ 14. M. BALANZAT. F6,'­mulas integrales de la intersecci6n de conjuntos. - Nil 15. G. KNIE. El problema de varios electrones en 10, mecdnica cuantista. - Nil 16. A. TERRACINI. Sobre 10, existen­cia de superficies cuyas liMas principales son dadas. - N'" 17. L. A. SANTALO. Valor medio del numero de partes en que una figura convexa es dividida por n rectas arbi­tral·ias. - NI' 18. A. WINTNER. On the iteration of distribution functions in the cal­cu,lus of probability (Sobre la iteraci6n de funciones de distribuci6n en el '--Ilculo de probabilidades). - N'" 19. E. FERRARI. Sobre la paradoja de Bertrand. - ~'" 20. J. BABINI. Sobre algunas propiedades de las derivadas y ciertas primitivas de los polino­miosde Legendre. N'" 21. R. SAN JUAN. Un algoritmo de sumaci6n de series diver­gentes. - NQ 22. A. TERRACINI. Sobre algunos lugares geometricos. - NQ 23. V. y A. FRAILE Y C. CRESPO. El luga,' geometrico y lugares de puntos dreas en el plano. -NQ 24. - R. FRUCHT. Coronas de grupos y sus subgrupos, con una aplicaci6n a los determinantes. - NQ 25. E. R. RAIMONDI. Un problema de probabilidades geometri-cas sobre los conjuntos de tridngulos. .

En 1942 la U. M. A. ha iniciado la pUblicacion de una nueva ,serie de "Memorias y monografias" de las que han aparecido hasta ahora las siguientes:

Vol. I; NQ 1. - GUILLERMO KNIE, Mecdnica ondulato1'ia en el espacio curvo. NQ 2. - GUIDO BECK, El espacio fisico. NQ 3. ~ JULIO REY PASTOR, Integrales par­ciales de las funciones de dos varia.bles en intervalo infinito. NQ 4.· - JULIO REY P AS­TOR, Los ultimos teoremas geometricos de Poincare y sus aplicaciones. Homenaje 'pos­tumo al Prof. G. D. BIRKH0FF.

Vol. II; NQ 1. - YANNY FRENKEl" C"iterios de bicompacidad y de H-comple­tidad de un espacio topol6gico accesible de FnJchet-Riesz. NI' 2. - GEORGES V ALIRON,

. Fonctions entieres. Vol. III; NQ 1. - E. S. BERTOlVlEU Y C. A. MALLMANN, FUllciO'liam'iento de un

generador en cascadas de alta tensi6n.

Ademas han aparecido tres cuadernos de Misceldnea Matemdtica.

INDICE Volumen 25, Numeros 1 y 2, 1970

Une propiete des racines de l'unite J. Dieudonne,.,. ::. ; ...... ; ... : ... ;.. .. .. 1

~nriched Semantics-Structure (Meta) Adjointness Eduar.do Dubuc ...................... ,.. 5"

Ideals and universal repre~entations of certain Co -algebras '

Horacio ,Porta ....... : ............. -. . . . . 27

Uniqueness of distributions A. P. Calder6n ....................... '.. ~ , .

Some comments on the sp~~i:ral theorem M. H. Stone .... '....................... ffl

A geometric observation about linear partial differential operators

W. Ambrose ."............................ 77

Uniform approxiniiition to bounded analytic functions

T. W. Gamelin and John Garnett ........ 87

Probabilidades sobre cuerpos convexos y cilindros L. A. SantaI6'.......................... 95

A remark oli'Sidon sets R Kaufman 105

On uniqueness conditions for the initial value problem for the differential' equation y'. = f (x,y)

S. c. Chu and J. B. Diaz ................ 109

On the boundary values of holomorphic functions Stephen Vagi ...... , ..... 01 •• • • • • • • • • • • 123

Singular integrals and multipliers on homogeneous spaces

R. R. Coifman and M. de Guzman ...... 137 Multiplier transformations of functions on Su(2) and l: 2

R. R. Coifman and G. Weiss ............ 145

Note on mean convergence of eigenfunction expansions A. Benedek and R. Panzone ..... ~ ... ~ .. ; . 167

Subespacios In y el caIculo funcional de Sz.-Nagy y Foias Domingo A. Herrero ................... 185

Reg. Nac. de la Prop. Int. N9 1.044.738

g ,~ t:'E o " lJ !!'

<

eo 'E'" " u u

::I 011.1

'" 8 ;..

TARIFA REDUCIDA

CONCESION N9 9120

FRANQUEO PAGADO

CONCESION N9 3626

OFFSET. ANDREANI & CIA, ALSINA 223. BAHIA BLANCA