SUBSPACES OF BMO(R)...SUBSPACES OF BMO(R" ) 103 where Uf and BMOc. are the spaces of functions of compact support in L00 and BMO, respectively, and the closure is taken in BMO norm

transactions of theamerican mathematical societyVolume 290, Number 1, July 1985

SUBSPACES OF BMO(R")

BY

MICHAEL FRAZIER

Abstract. We consider subspaces of BMO(R" ) generated by one singular integral

transform. We show that the averages along x;-lines of they'th Riesz transform of

g e BMO n Z.2(R") or g e Ve-(R") satisfy a certain strong regularity property. One

consquence of this result is that such functions satisfy a uniform doubling condition

on a.e. v-line. We give an example to show, however, that the restrictions to x-lines

of the Riesz transform of g e BMO n L2(R") do not necessarily have uniformly

bounded BMO norm. Also, for a Calderón-Zygmund singular integral operator K

with real and odd kernel, we show that rC"(BMO, ) Q Lx + K( Zf ), where Lf and

BMO, are the spaces of L°° or BMO functions of compact support, respectively, and

the closure is taken in BMO norm.

1. Introduction. Let BMO(R") be the set of complex-valued functions/ g L\oc(R")

such that

II/IIbmo = sup — [ |/- fQ\ < 00,Q \Q\ JQ

where the sup is taken over all cubes g in R" with sides parallel to the coordinate

axes, fQ = (1/1 gD/ç/, and \Q\ is the Lebesgue measure of g. For us, a singular

integral operator K with kernel ß is an operator of the form

Q(*-y)

'\x-y\-Kf(x)=limf ^f^-f(y)dy,

f-o r,x_y.>t \x y\

where ßs„-> G C00, ü(rx) = ß(x) for r > 0, and /s„iß(x) da(x) = 0 for surface

measure da on 5""1 = {x G F.": |x| = 1}. For /g l)ls¡p<xLp, Kf(x) exists a.e.

(dx), and K is a bounded operator on Lp for 1 < p < oo. Associated with K is the

Fourier multiplier m so that for/ g L2(R"), (Kf)(£) = w(£)/(|), where the Fourier

transform is defined by /(£) = jK-f(x)e2,"x'i dx; m satisfies m(r£) = m(£) for

r > 0 and | G R" - (0); and w|s„-. G C00. For/ G L°°(R"), we define

ft*)-*»/ (Bi^4-eMx..„wU)*,e-D J\x-y\>t \ \x-y\ \y\ I

where \e denotes the characteristic function of the set E. For /eLTiL00,

1 < p < oo, Kf and Kf differ only by a constant. The operator K maps Lx into

Received by the editors March 21, 1984.

1980 Mathematics Subject Classification. Primary 42B20.

Key words and phrases. BMO, singular integral operator.

©1985 American Mathematical Society

0002-9947/85 $1.00 + $.25 per page

101

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

102 MICHAEL FRAZIER

BMO boundedly. Theyth Riesz transform Rrj g {1,...,«}, is the singular integral

operator with kernel ßy(x) = C„Xj/\x\ and multiplier m ■(£) = /'¿,/IH [7, p. 58].

Let H\R") be the set off g L'(R") such that

ll/IU'H!/!L> + Î \\Rjf\\Ll < ».7 = 1

In [1], Fefferman and Stein proved that (//')* = BMO (in an appropriate sense),

and obtained by duality as a corollary that BMO = L°° + L"=1 RjLx, with

(1-1) II/IIbmo ~ inf £ 11^11^:/= go + E Rjgj, modulo constants\J-0 7=1

where ~ means that the norms are equivalent. In [8], Uchiyama obtained (1.1)

constructively for / of compact support, and proved that a certain condition on the

multipliers of a set of singular integral operators Kx,...,Kn implies that BMO =

Y."=xKjLx with an appropriate equivalence of norms. This condition had been

proven necessary by Janson [5].

Using in part techniques developed by Uchiyama in [8], we consider subspaces of

BMO generated by one singular integral operator. The Riesz transforms are the most

natural to consider in this context since one may expect that functions of the form

Rjg for g G L00, or Rjg for g g BMO n L2, when restricted to lines in the Xj

direction, have regularity properties that functions in BMO(R") do not necessarily

have. Results of this type are proved in §3, Theorem 3.2 and §4, Corollary 4.3. (In

§2, we state Uchiyama's results from [8] that are needed later.) Theorem 3.2 is

obtained from Lemma 3.1, which characterizes Äy(BMO n L2) in terms of a

decomposition closely related to Uchiyama's decomposition of BMO n L2 [8, p.

224]. One direction of the characterization follows easily from Uchiyama's work; the

other direction is obtained by solving the equation RjCp = b for <p under appropriate

assumptions on b.

Corollaries of Theorem 3.2, proved in §4, state that if / = R.g, g g BMO n L2, or

/ = Rtg, g g L00, then/has a uniform doubling property on *-lines! namely

(1.2) — f f(xx,...,x„)dXj-— ( f(xx,...,x„)dXj\I\ Ji \J\ jj

< q|g||uMo

for almost all (x1,...,xj_1, xJ+1,...,x„) e R""1, whenever / and J are adjacent

intervals in R of the same length. An example of Kahane [6] from another context

shows that the uniform doubling property for a function/on ^,-lines is weaker than

the uniform BMO property on Xj-lines. Both conditions restrict the singularities of/

along Xy-lines to be at most logarithmic, but the BMO condition further controls the

packing of singularities. Kahane's example is sketched in §4; it is then modified to

show that 3G g BMO n L2(R2) such that esssup^JIFjOXx!, x2)\\BMO(x¡) = oo.

Hence (1.2) cannot be improved to obtain a uniform BMO condition on lines for the

class /^(BMO n L2).

In §5 we show that if K is a singular integral operator with a real and odd kernel

(in particular if AT is a Riesz transform), then

(1.3) #(BMO(.) <z L°° + K{Uf) ,


SUBSPACES OF BMO(R" ) 103

where Uf and BMOc. are the spaces of functions of compact support in L00 and

BMO, respectively, and the closure is taken in BMO norm. The proof is an

adaptation of Uchiyama's main construction in [8]. This suggests that the action of

K on BMO is in some sense close to the action of K on L00.

Finally, in §6 we make some remarks about related problems which are not yet

solved.

Notation. The letter c denotes various constants depending possibly on n and, in

§5, on K. All cubes g, /, or J are assumed to be open and to have sides parallel to

the coordinate axes; xQ and /(g) denote the center and side length of g, respec-

tively. For r > 0, rg is the cube with sides parallel to the axes having center xQ and

side length rl(Q). For/ g {1,... ,n}, Dx is the derivative with respect to x¡, and for

a multi-index a = (ar,...,an), Da is the differential operator d^/dx"1 •■■ 9x°",

where \a\ = af + ■ ■ ■ + an. For/continuous on R", let

ll/llLipi = sup(|/(x)-/(y)|/|x-y|),x+y

and let CX(R") be the space of continuously differentiable functions on R". For

x, y g R*, x ■ y = £*_ixjyj.

Acknowledgements. Most of the ideas for this paper are contained in my

doctoral thesis, directed by John Garnett at UCLA. I thank him for his considerable

encouragement as well as mathematical help. I would also like to thank Akihito

Uchiyama and Alice Chang for their help at various times, and Al Baernstein for

suggestions regarding the manuscript.

2. Background. We state several lemmas from Uchiyama [8] that will be used later.

In fact, the following Lemmas U1-U6 are Lemmas 3.1-3.6, respectively, in [8].

Lemma Ul. Suppose f g BMO n L2(R") and ||/||BMO < 1- Then there exist func-

tions {b,(x)}, and {X,}¡, Xf > 0, where I is taken over all dyadic cubes in R", such

that

(2.1) /(*) = lim E aA(*),A-ôo ml.1 2-*</(/)<2*

(2.2) suppft, ç 3/, jb,(x) dx = 0, \\Dab,\\L~ a C|a|/(/)~H,

(2.3) E KU\ < c\J\ for any cube J

and

(2-4) E*,|/| = c|l/lll>all l

The last equality is not stated explicitly in [8], but follows from the definition of

the {X,} j and Plancherel's Theorem quite easily.

Lemma U2. For {X,}, satisfying (2.3), X, > 0 V/, set

vk(x)= e x,(i + 2k\x- x,\y~l,/:/</) = 2-A


104 MICHAEL FRAZIER

and

Then \fx, y,

(2.5)

(2.6)

(2.7)

and if 1(1) = 2~p,then

(2.8)

e*(*)= E It) %-,(*)■7 = 0 V '

Vk(x) < «*(*) < c>

^(x)<c(2^-y|+l)" + 17,,(y),

e,(x)<c(2^-y|+l)" + 1£,(y),

/ E £*(*)%(*) ¿* < c|/|.

Lemma U3. Letj be a positive integer. Suppose [b,(x)}, dvadu. satisfy

(2.9) supple 2^7, fb,(x)dx = 0, Mu,i<c(2Jl(I))'\

IfX,>OVIandß>a>0, then

(2.10) E M,(*)/:«</(/)</?

<c2''" S>2,|/|1/2

Lemma U4. Suppose j, {b,}, and {X,}, are as in Lemma U3 a«i/ (2.3) ZioWj. For

a > 0, letf(x) = T.nt)<aXfb,(x). Then

(2.11) ||/||bmo < C2>".

Lemma U5. Le/ / andp,(x) e C^R") ¿>e such that

(2.12) fp,(x)dx = 0,

(2.13) |/./(x)|<(l+/(/)-1|x-x/|)"""1

(2.14) | V/>,(*)| < l(I)'l(l + l(iyl\x - x,\y~\

Then 3{ß, f(x)}f=0such that

(2.15) ||j8,JLipl < c{2H(I))'\ suppßjjQVI, fß,,fix)dx = 0

and

(2.16) p,(x)= £ 2-«"+1%J(x).7=0

If p, = (p/0,---,Pi,m) and each component of p, is real valued and satisfies the

conditions (2.12)-(2.14), and a = (a0,... ,am) g Rm + 1 with \a\ = 1 satisfies p,(x) -a

= 0 Vx, then we can take ß, y = (ßfJt.. -,ß"j) such that (2.15) is satisfied for each

component of0tJ,p,(x) = ZJ.02~J("+%j(x) andß,j(x) ■ a = 0 Vx, /.



Lemma U6. Suppose K is a singular integral operator with multiplier m, and b,(x)

satisfies (2.2) for some cube I. Then p,(x) = Kb,(x) satisfies (2.12),

(2.17) \p,(x)\ <S C(K)(l + l(iyl\x - x^)'"'1

and

(2.18) \vpr(x)\ < C(K)l(iy\l + i{I)~l\x - x,\Y"~2,

where \C(K)\ < csup{|Z)am(£)|:|ê| = L H < C(n)}. (In fact, C(n) = n + 4 is

sufficient.) Also it is enough to assume \\Dxb,\\Upf < cl(I)'2 for j = l,...,n instead of

llâMI/«<cw/(/)-iai.

3. Properties of Ä,(BMO n L2). We characterize /^(BMO n L2) in terms of a

decomposition similar to the decomposition of BMO n L2 in Lemma Ul. For

convenience we take/ = 1 and write x g R" as x = (xx, x') for x' g R""1.

Lemma 3.1. Suppose f = Rxg for some g g BMO n L2 with ||g||BMO < 1. Then

there exist functions {a,(x)}r and {Xr},, X, > 0, for I taken over all dyadic cubes,

such that

O..) A»)- * I M,W,k^ccinL2 2-*</(/)<2*

(3.2) \*¿x)\<c(l*l(irl\x-.xA)?-\

(3.3) \D"a,(x)\ < ca/(/)-|a|(l + l(iyl\x - ^l)"""1"'"',

(3.4) ja,{xf,x')dxf = 0 Vx'eR"-',

(3.5) E A2|/| < cl^1 for any cube J

and

(3.6) !>2/|/|<°o.all I

Conversely, if f satisfies (3.1)-(3.6) for some {a,}, and {X¡}/, then f = Rxg for

some g g BMO n L2 with ||g||BMO < C.

Proof. The forward direction follows easily from Uchiyama's lemmas. By Lemma

Ul,

g = lim E * A

with each ¿>, satisfying (2.2), and (3.5) and (3.6) hold. Since Rx is continuous on L2,

we obtain (3.1) by setting a, = Rxb,. Then (3.2), and (3.3) for \a\ = 1, follow by

Lemma U6; for \a\ > 1, D"a, is estimated by taking the derivatives inside the

integral in the expression for a, exactly as in the proof of Lemma U6. The

cancellation property (3.4) holds because â,(£) = /¿1|¿|"1fe/(¿).


106 MICHAEL FRAZIER

The idea of the converse is to use (3.2)-(3.4) to write a, = Rxcp, with appropriate

control of <p,. Let h,(xx, x') = j^xa,(t, x') dt. We claim

(3.7)

and

\h,(x)\^ cl(l)(l + l(iy'\x - x,\)~"

-'• H(3.8) \D«hf(x)\< cal(I)l-la\l + liiy'lx - x,\Y

To prove (3.7), we may assume x, = 0. For x such that xx < 0, let k and m be

positive integers such that -(k + 1)1(1) < xx < -kl(I) and ml(I) < \x'\ <

(m + 1)1(1). By (3.2),

¡(I)_\h,(x)\*p \a,(t,x')\dt^c¿Z

-* (l + ip2 + m2

<-,-^L=-<É/(/)(l + /(/)-l|x-xl|)-.(1 + /rC2 + W2)

If x, > 0, ///(x) = -/" a,(/, x') í/í by (3.4), so the same estimates hold. Then (3.8)

follows from (3.3) and (3.4) similarly. Note that (3.7) and (3.8) imply that h, g L2

and|Vr/,| G L1 n L2.

Define <p, = -T/,j^lRjDx h,. Then <p, G L2 since |v/t,| g L2. Note that $,(£) =

-2ir|||n/(|); therefore (ÄjtjO/)^) = -2TTi£xh,(i;) = à,(£), since £>x Ä, = a/. Hence

/?,<?, = a,. If we can show that E7 A,™, g BMO n L2, then we are done since

*i(l>/<P/) = EMi«P/ = EV, =/•

To see that E, A/rf/ g BMO n L2, we show that Dx h, satisfies the assumptions of

Lemma U5 for l=l,...,n. We have }Dxh,(x) dx = 0 since |vA,|e L1 and

lim)x|_a)A/(x) = 0 by (3.7); the estimates (2.13) and (2.14) follow from (3.8). Hence

for each /, 3{ßfJ(x)}JL0 satisfying (2.15) and Dxh,(x) = I.J=02-J,'n + 1)ßIJ(x).

Therefore, using Lemma U3,

L^,Dx,h,i

< £ 2-J(n+x

L2 7 = 0

EaA,7

<cE 2-"" + 1>2^(Ea2/|/|)y=0 \ / '

1/2

< oo,

by (3.6). Similarly, by Lemma U4,

00

LKDxh, < E 2-JiK+l) LKßuI BMO 7-0 /

2

< C.

Therefore E7 A,/^ h, g BMO n L for each /; hencen

EA/CP, = -Ea, E *A,>«/ = - E ÊA/ZV/ e BMO n Ll- °/ / 7=1 7=1 /

The cancellation property of the Riesz transforms, reflected in condition (3.4),

forces the xraverages of/ G /^(BMO n L2) to have a certain regularity. Let g be a


SUBSPACES OF BMO(R') 107

cube in R", and write g = J X L for J ç R1 and L ç R""1. By Fubini's Theorem,

the average/y = (l/\J\)fjf(t, y ) dt is defined for almost y g L.

Theorem 3.2. Suppose f = Rxg for some g g BMO h L2(R") satisfying ||g||BMO <

1. For Q, J, and L as above, 3c > 0 and a set E ç L of (n — l)-dimensional Lebesgue

measure 0 such that

(3-9) \fjv-fj,\^c Vy,y'eL-E

and

(3.10) \f ^j^ck^ñiogJLSL ify,y'eL-El\i¿) \y - y I

and\y-y'\< \.l(Q).

Remarks. The number i is chosen because the function / -* /log(7) is increasing

on (0, ~). The proof uses only (3.1), (3.2), (3.3) for a = 1, (3.4), and the estimate

X, < c, which follows trivially from (3.5). An example of a function in BMO(R")

which does not satisfy (3.10) is/(x) = log|x'|.

Proof of Theorem 3.2. We may assume g = [-1,1]", so J = [-1,1] and L =

[-1, l]""1. Fory, y' e L fixed, let e = \y - y'\. It is sufficient to obtain |/y - /, | <

ce log} for e < i, since (3.10) implies (3.9). By Lemma 3.1, f=T.,X,a, with

(3.2)-(3.6). We also may assume that 3N so that Xr = 0 whenever /(/) > 2N or

/(/) < 2 ~N. To see this, let fk = E2-*,¡/(/)aí2* A/u/í then lim *_«,/* = fin L2 and, for

each k,fk is continuous and the sum converges absolutely. Let

n

/ = 2

where S > 0 and y = (yl7... ,y„), and similarly define A*-. The result for/A implies

that the averages (fk)Rs and (fk)R\ of / over Rsv and Rsv>, respectively, satisfy

K/a)r? _ (/a)rs,I < celogj with c independent of k or 5. Since lixrxk^,00fk = /in L2,

\fRs — fRs\ < ce logy with c independent of ô. By Lebesgue's Theorem, limÄ_0/Rs =

fj unless y is taken from a set E c L of (n — l)-dimensional Lebesgue measure 0,

and similarly for/y-; hence (3.10) follows.

For / = E, X,a, satisfying X, = 0 for /(/) > 2N or /(/) < 2_w, write / = /,+/,

+ /s + /». where

/i = E A/fl/, /2 = E A/ö/, /3 = E A/Ö,, /4 = E A,a,.HD>X /(/)<1 /(/)<* t</(/)<i

/n3(?=e> iQiQ iQ'iQ

The functions/j and/2 are estimated by Lipschitz estimates. By (3.3) and the fact

that X, < c, if / g J we have00

\fi(*,y)-fi{t.y')\<ct E M'.JO-*/('./)!A=0/(/) = 24

oo

< c E E sup I Vfl/(x)|eA=0/(/) = 2' ïG£

-

2"*< « E E-—tt < e».

k -0a<EZ" (1 +|^|)+ 2


108 MICHAEL FRAZIER

Therefore |( /, )y - (ff)j J < ce. Similarly, if / g /,00

\fzU, y) -h{t, y')\ <cZ E M', j) - «,(*, /)|*-l /(/)=.2_i

/n3£> = 0

< ce E E SUP -A = l /(/)-2-* *eö (l + 2*|x - X,|)

/n3(?= 0

n + 2

oo jk

< ce E E -^TTk-X ,eZ" (1 + |ç|)"

< ce E 2"A = l

ce.

M»2*

Therefore |(/2)y - (/2)AJ < ce.

Using (3.4), we will obtain |(/3)y | < ce log} and K/3)/J < ce log}. Let I Q 3g be

such that /(/) = 2~k < e, with x, = (x¡, x'r), x, g R, satisfying 0 < x, < 1. Let

m g {0,1,2,...,2* - 1} and/? g Z, 0 </? < 4\/ü - 1 (2k - 1), be such that m2~k

<l-x,i<(m + 1)2"* and p2~k < \x', - y\ < (p + 1)2"*. By (3.4), (3.2), and

since \J\ = 2,

r r-x r00-Ja,(t,y)dt <2J \a,(t,y)\dt + 2j \a,(t,y)\dt\J\Jj

<cY,'0(l + Jp2+(m + s)2y + 1 (l + Jp2 + m2)"

If 7 and/? are as above except that -1 < x, < 0, and m g (0,1,... ,2k — 1} is such

that m2'k < x, + 1 < (m + 1)2"*, the same estimate is obtained by the same

methods. If /and/? are as above but 1 < |x, | < 3, say m2'k < x, — 1 < (m + 1)2"*

if x, > 1, or m2~k < -1 — x, < (m + l)2_/t if x, < -1 for some m e

{0,1,2,...,2* + 1 - 1}, then (3.4) is not used since (3.2) implies that

¿nl°Á,'y)dt <cE°(l + Jp2+(m + s)2)'

>-*< c-

(l + //?2 + m2)"

Summing over all / ç 3g with /(/) < e, we obtain

00 -,

Uh,\- L f-j L ̂ ,a,(t,y)A = [log2lAl \J\ J /(/)_2-*

/C3£?

00 1

<c E 2-* E —-—A = [log2l/e] qeZ" (1 + |?| j

WKVT2*

di

00 j

<c E ^2"* < ce log-,A = [log2lA]


SUBSPACES OF BMO(R") 109

where we have used X, < c again, and [x] is the greatest integer in x. With the same

estimate for (f3)j-, we obtain |(/3)y - (f3)j\ < ce log}.

For/4, (3.3) and (3.4) are both used. Suppose e < /(/) - 2"* < 1 and / ç 3g. Let

y* = \(y + /)■ Let /? g Z, 0 < /? < Ayjn - 1 (2k - 1), be such that p2'k < \x', -

y*\ < (p + 1)2"*. Let m g (0,1,. ..,2k - 1} be such that m2'k < xh~ 1 <

(m + 1)2"* if 0 < xh < 1, or such that m2'k < xr¡ + 1 < (m + 1)2"* if -1 < xf¡

< 0. By (3.4),

dt

dt

— f a,(t,y')dt-— f a,(t,y)\J\ jj \J\ Jj

= ï7i"/~ ' a'(i' yï dt - Tri/"1 "'('• x!) àt\J\ ■'-90 |/| •'-00

r°° r°°+ T7ÏJ °t(t,y)dt- —f a,{t,y')

\J\ Jx \J\ Ji

\a,{t, y) - a,(t, y')\dt + 2 / \a,(t, y) - a,{t, y')\-oo •'l

Therefore, by (3.3),

CO

$a,)jt.-{a,)jj<2^í2'k sup \va,(x)\-eí = 0 \x'-y*\<e/2

(m + s)2-k sí\xl-xl¡\^(m + s + l)2-1'

dt.

< c2-*e Ece

=°(l + Jp2+(m + s)2y (l + y!p2 + m2)

If / and p are as above but 1 < |x71 < 3, and m g {0,1,... ,2k+1 - 1} is such that

m2~k < xh - 1 < (m + 1)2'* if xh > 1 or such that m2~k < -1 - x¡ <

(m + 1)2_* if x, < -1, then (3.3) leads directly to the same estimates without use of

(3.4). Therefore, since Xf < c,

T77/ E M,fojO*-?V/ £ ̂ ,a,(t,y')dt\J\JJ „rx_->-* /K/ „r>_.,-*I-7! ̂ ,(/)»r* \J\

IQ3Q

,.z-.(l+|«|)'

/(/) = 2-IQ3Q

ce.

Therefore

|(/4)/, -(h)j,\ < «#{A:: e < 2"* < 1} = celog}.

Theorem 3.2 follows from the estimates on/j, /2, /3 and/4. D

4. Corollaries and examples. Theorem 3.2 implies that functions in Äy(BMO n L2)

satisfy a doubling condition on almost every x^-line. For convenience we take/ = 1

and write x G R" as (x,, x') for xx g R, x' g R"_1.


110 MICHAEL FRAZIER

Corollary 4.1. Suppose f = Rxgfor some g e BMO n L2(R"). Then 3c > 0 and

a set E ç R"_1 of (n — $-dimensional Lebesgue measure 0 such that for x' G E and

/, and I2 adjacent intervals in R with \IX\ = \I2\,

(4.1) — f f(xx, x') dxx - ■— ( f(xx, x') dx,\h\ '■ U2I J,2

< c]|g||BMO-

Proof. We can assume ||g|]BMO = 1. Assume first that Ix and I2 are dyadic

intervals. Let F be the union over all dyadic cubes of the exceptional sets obtained

for/in Theorem 3.2. Forx' <£ F, let L ç R"-1 be a closed dyadic cube containing x'

such that \IX\ = l(L), and write Qx = Ix X L, g2 = I2 X L. By Lemma 3.2, since

x' G F, \fjx - f,f\ < c for a.e. s & L - F. Therefore \fr- - /e | < c, and similarly

I//J- -/eJ < c. Since gx and g2 are adjacent cubes in R" and fe BMO(R"),

l/e, - fQl\ < cI/Ibmo < c [3, P- 223]. Therefore \fi; - ff}\ < c for x' G F. To obtaina single exceptional set E for /t, 72 not necessarily dyadic, apply this result to all

dilations and translations of/by rational coordinates. D

Example 4.2. There exists/ g BMO n L2(R") which does not satisfy (4.1). In R2,

for example, for each dyadic square /, let/, be a function satisfying 0 </,(x) < 1,

supp/, ç 37, ||/,||Lipl < cl(I)-1 and/,(x) = 1 Vx g /. Let/I, = {/dyadic: /(/) =

2"\ 0 < x,t < 1, and / n {x:x2 = 0} # 0}. Let/= E~_0E/e/<(l /,- Then it is not

difficult to show that/ G BMO n L2(R2), but

lim Í f(xx,x2)dxx - f~ f(xx, x2) dxf\ = 00.a,^0\^0 ■'1 /

For details, see [2].

Corollary 4.3. /// = Rxgfor some g g L°° such that \\g\\ L«, <\, then (3.9), (3.10)

and (4.1) hold.

Proof. The estimates hold if f = Rxgm, where gm(x) = g(x)x(x:|xKm)(x) since

gm G L2 n L°° so that /?,g„, and Ä1gm differ by a constant. The result follows since,

by a calculation in [2, p. 48], if K is any compact set in R",

lim (sup|Ä1(g-g„,)(x)|) = 0. n

A proof of Corollary 4.3 based on duality with singular measures is included in

[2].Example 4.4. If f(x) = loglog(l/|x'|) for |x'| < 1/e and/(x) = 0 otherwise, then

/G BMO(R") but/does not satisfy (3.9). Although/Í L™,f belongs to the BMO

closure of L00. Therefore L°° + RXLX, Lx + Rx(BMO n L2) and L00 + Ä^Lf ) are

not closed subspaces of BMO. This observation is relevant to §5 below. The space

L°° + RXLX is not dense in BMO in BMO norm; a proof due to Paul Koosis is

included in [2].

Example 4.5 (Kahane's Example). Kahane's example [6] can be used to show

that there exists a sequence {/„}"=i of functions on (0,1) satisfying a uniform



doubling condition but such that lim„_00||/B||BMO(0il) = oo. For mx, m2,...,mn g

{1,2,3,4}, define

, m, - 1 m-, - 1 m„ - 1I = ——,-+ —-+ • '

4"

mx - 1 mn_x - 1 mn—-h • • • H-—-1--

4 4«-i 4"

and g„„...mn = /„„...„,„ X [0,4-"].Let/1(x1,x2)bedefinedby

/i = Xq, ~ Xq2 ~ Xqj + X<?4-

With sgn / = t/\t\ if / # 0 and sgnO = 0, define fn + x inductively by

4

In+1 ~ J n ' i—i am¡-m„,m,.... ,»!„ = 1

where

ami-m. = sgn/^-Xe^ „,, + XQ„n..m„2 + XQ„v..m¿ ~ XQmy.„J-

Regard/, now as restricted to (0,1). Then clearly

||/J,, = 1 V« = 1,2,3,....

If an = |supp/„|, then an decreases to 0 as n -» oo. That an decreases is clear; that

lim„ J0O an = 0 can be seen by the fact that a simple random walk in one dimension

is recurrent. Since /J f„(x) dx = 0 V«, and since

/ 1 2\1/2

II/IIbmo* sup — / |/ - fQ\Q \\Q\JQ I

we obtain lim,,_,J|/„||BMO(01) = oo since

(A -l\ ll.fl! ^ 11/11 ^ 11/J/.'(0.1) C(4-2) II/„IIbmo(o.i) > c||/J|¿2(01) > c-=— = —.

If Ix and I2 are adjacent intervals of the same length in R, then \(ff), — (/„)/,| < c,

with c independent of Ix, I2, or n, by the main estimate in Kahane's example: see [6,

p. 190], DExample 4.6. There exists G g BMO n L2(R2) such that

(4.3) esssup||Ä1G(x1,x2)||BMO(A1)= °°-

We construct RXG by symmetrizing and smoothing the functions f„(xx, x2) of

Example 4.5 in the x2 direction.

Regarding/, above as a function of (xx, x2) again, let/B(jcl5 x2) be the function

which agrees with fn(xx,xf) for x2 ^ 0 and satisfies f„(xx, -xf) = f„(xx, xf)

V(x,, x2). Define äm m similarly related to am „, . Since

ll/n+i-/Ji.^)<c2:fl,


112 MICHAEL FRAZIER

m, ... m„ = 1

/ = lim,^^/, exists in L2(R2) and a.e. The following sequence of steps can be

carried out to show that/G BMO(R2):

(a) Since the measure of the supports of fn+x - /„ decrease rapidly,

à #-*■<«

for any 4-adic square g.

(b) If g! and g2 are 4-adic squares of the same side length with Qx n g2 =£ 0,

then \fQi — fg\ < c. For Qx and g2 horizontally adjacent, this follows from the

doubling condition for/,. For Qx and g2 vertically adjacent, this follows from the

construction.

(c) An easy observation, noted by Varopoulous, shows that (a) and (b) imply

/g BMO(R2).In this example, we wish to replace the functions äm m with versions which are

smoothed out in the x2 direction. Let \¡/ g C°°(R) be a fixed function satisfying

0 < <K*)« 1 Vx, suppt// ç (-2,2), and i>(x) = 1 if x g [-1,1]. For mx,...,mn g

{1,2,3,4}, let

?>«,..,«„(*i> *i) = X/„M „,Ui)«/'(4"x2).

Let gx = <px — <p2 - (p3 + <p4, and define g„+x(xx, x2) inductively by

4

gn + l=gn+ E b,

where

fem,...m, = Sgng„(x1,0)(-<pmi m ! + <fm¡ m¡2 + 9mi...m„3 ~ «Pm,.. .«,„4) •

Note that sgn g„(xx, 0) = sgn f„(xx, 0) for/, in Example 4.5; in particular,

supp(£m,...m< - 5Mi...mJ S /mi...mii X (x2: 4-"-1 < |x2| < 2 • 4—1).

Hence if g = limn_00g„, then g — /g L°°(R2) by the disjointness of the supports of

the different gn - f„. Therefore g g BMO n L2(R2) since/g BMO n L2(R2).

We assert that g = RXG for some G g BMO n L2; we solve for G by the method

of Lemma 3.1. Let

/xl gx(t,x2)dt-00

and

/xxf>mi...m„(t,X2)dt.

-oo

Note that hm¡Mfxx, x2) = 0 if xx € Im^Mn (set /0 = (0,1)), or if |x2| > 2 • 4—».

Further,

l-P.vA,,...„,„(*!' *2)|

= Dlj(i(4"x2))jrjX^ mi(/) - X/„„ „,„2(0 - X,„„ „,„3(0 + XV.-,(0) *

<^'(4"x2)-4"-4-"-1 < C.



Also

supp/W..m„ C /„„...„,„ X (x2: 4-"-1 < |x2| < 2 • 4-"1},

since \p(x2) = 1 if x2 g [-1,1]. Therefore

oo 4

H = Dxh0+ E E Dxhmx_meUf.n=\ Wi,... >'"„ = 1

By Fourier transform we obtain gx = Rx(-Rxgx - R2Dxfh0) and

K „, =Rii-Ri6m „, -R2Dxhm m).

Therefore

oo 4

g = gi+E E *„,..,,,.-Ji^-Äx« - Ä2ir)./; = 1 m, ... m„ = 1

Since g, He. BMO C\L2,G= -Rxg - R2H G BMO n L2 and /vjG = g. Note that

/(x,,x2)=/„(x1, x2)for4-""1 < x2 < 4"", and hence by (4.2)

esssup||/(x1,x2)||BMO(x1) = °o-

Since RfG -fe L°°, (4.3) follows. D

5. A density theorem. Throughout this section K is a fixed singular integral

operator with kernel ß which is real and odd; it follows [7, p. 39] that the multiplier

m for K is odd and pure imaginary. We will show that

tf(BMOc) ç L00 + K(L?) ,

where the closure is taken in BMO norm. To fix notation for this section, / always

denotes a dyadic cube, / denotes a vector-valued function with two components,

/ = (/o> A). and we define

\f(x)\-(g(x) +/12(x))1/2, ll/lH/oll + ll/,11

for any function space norm || • ||, and K ■ f = f0 + Kfx. Also define

|v/(x)| = |v/o(x)| + |v/1(x)|

and

ff(x) dx = (//0(x) dx, ffx(x) dx

Our method is based closely on Uchiyama's construction in [8]. Uchiyama obtains

uniformly bounded solutions, which satisfy a certain orthogonality condition, to a

certain inversion problem. We are not able to obtain uniformly bounded solutions to

the corresponding inversion problem in our case. Hence, some additional estimates

are necessary.

First we state explicitly the assumptions we require regarding the inversion

problem. We say that a collection of real-valued functions {b,(x)}¡ satisfies (*) if


114 MICHAEL FRAZIER

there exist constants c > 0 and / > 0 such that for all /,

(i) for any a = (a0, ax) e R2 with ax =£ 0, the equations

(5.1) Kü, = b,

and

(5.2) a-U,(x) = 0 Vx

have a solution ü, = (u0 ,ux ), u,: R" -» R2, satisfying

(5.3) \u,(x)\<c[l+ ^ )(l+l(iyl\x-x,\)

(5.4)

|«/(x)|< c

|vw/(x)|< c

-n-X

1 + /(/)"'(! + /(/)-1|x-x,|)-n-2

and

(5-5)

and

fur(x) dx = 0;

(ii) the equation K(ux ) = b, has a solution ux : R" -* R satisfying

(5.6) |Ml;(x)|^c(l + /(/)-1|x-x/|)"""1,

(5.7) |vuX/(x)| ^ c/(/)_1(l + /(/)_1|x - x,|)~"~2

and

(5.8) juXi(x) dx = 0.

Theorem 5.1. Suppose f(x) = linxk^x>Y.2-t^nJ)6i2i.Xrbl(x) (convergence in L2)

with Xj > 0 V/, w/iere //te real valued functions {b,(x)}, satisfy (*), and for some

c> 0,

(5.9) Ea2|/|<c|7| for all cubes J,IQJ

Ea2,|/|< oo,all I

limM—» oc

E a a/(/)>2M BMO

(5.10)

(5.11)

and

(5.12)

F/te/i/ g Lx + K{LX).

Before proving Theorem 5.1, we mention some consequences.

Corollary 5.2. For K as above, A:(BMO(.) £ L°° + K(LX), where the closure is

taken in BMO norm.

3 a cube S such that X, = 0 if 31 n S = 0 .



Proof. Suppose g g BMOc. We may assume that g is real valued. By Lemma Ul,

g = E/ X,a, with (5.9) and (5.10). The functions a, satisfy supp a, C 31, \\Daar\\L*,

< c|a|/(/rH and ja,(x) dx = 0 V/. That (5.12) and the condition

limM_00||E/(/)>2" îai\\BMo = 0 are also obtained by the decomposition in Lemma

Ul for functions of compact support is proved in [8, pp. 232 and 234]. Set b, = Ka,;

since K is bounded on L2 and BMO, Kg = Y.X,Ka, = Y.X,b, in L2. Each b, is real

valued since the {a,}, obtained by Lemma Ul are real valued (since g is real

valued), and since K has a real-valued kernel.

We need only show that {b,}, satisfies (*). Suppose ax + 0. Then (5.1) and (5.2)

can be solved by setting

(5.13)

and

(5.14)

"o,(*)

«!,(*)

m(è)a0 - fljh,(i)

MO

(*)

(*),m(è)a0 - ax

where is the inverse Fourier transform. (This is Uchiyama's solution reduced to

our case.) Since h,(i) = m(£)â,(£), the u0 and ux are obtained from a, by the

Fourier multipliers

-axm{i) -a0m(í)«o(0 and mx(i)

m(£)a0-ax m($)a0 - ax '

respectively, which are C°° on S"^1 (since m(£) is pure imaginary and ax ¥= 0), and

homogeneous of degree 0 (since m is). In fact, since m(-£) = m(£), the same is true

for m0 and w1 and hence w0 and ux may be taken real valued. By [7, p. 75],

3a0, af g R and singular integral operators T0 and TY such that u, = a¡a, + T¡a„

/ = 0,1. An easy calculation and the fact that m is pure imaginary show that

\Dam,(i)\ < c|a|(l + Iflo/fl^WjVÍ g S""1. Also K| < H»!,!!^^.-., < Cfor /' = 0,1.

Hence by Lemma U6, the solutions u, of (5.1) and (5.2) satisfy (5.3), (5.4) and (5.5)

for / = C(n). Solving K(ux ) = br is trivial since b, = Ka,, so that ux = a, satisfies

(5.6), (5.7) and (5.8). D

Corollary 5.3. If f(x)

hold and each b, satisfies

E/ X,b,(x) in L2 with À,»0V/, and if (5.9)-(5.12)

(5.15) supp¿?,c.3/, I^Îlípi * c/(7) 2' and jbi(x', x") dx' = ° Vx'

where x' = (xx,... ,xk) andx" = (xk + x,... ,x„), then-:-

/GL-+ T,Rj(L?)7 = 1

where the closure is taken in BMO norm.

Proof. We may assume each b, is real valued. If k = 1, we need only to show that

(*) holds for K = Rx and {b,}, under the assumption that fb,(xx, x") dxx = 0

Vx", /. For ax * 0, (5.1) and (5.2) can be solved by (5.13) and (5.14) with properties


116 MICHAEL FRAZIER

«i,- \Ri

(5.3), (5.4) and (5.5) as above. We solve RxUf = b, as in Lemma 3.1 by setting

h,(xx,x')= J^b,(t,x')dt. Then by (5.15), supple 3/. Set

- Uxb,+ tRjDXjhtY,

then Rxux ; = b, as in Lemma 3.1. Since supp Dx h, ç 31, ¡Dxh,(x) dx = 0 and

11/^/^/7,11 Lipl < cl(I)~2,Vj,p g {1,...,«}, we obtain (5.6)-(5.8) by Lemma U6.

In general, it is sufficient to apply the case above once we note that if b, satisfies

(5.15), then b, = T.p=xb,, where each b, satisfies suppZ?, ç 31, \\DX b¡ \\LipX ̂

c/(/)-2V7, and

jb,p(xx,...,xp_x,xp,xp+x,...,xn)dxp = 0

V(xx,.. .,xp_x, xp + x,...,xn). To see this, suppose I = \~\"J=xIj and let a(xx) be a

function satisfying suppaixjc 3/,, (l/\Ix\)ja(xx) dxx = 1 and \\dpa/dx{\\L« <

cpl(iy for p = 1,2, 3. . .. Let x = (x2,. . . ,x„) and define ß(x) =

(l/\Ix\)jb,(xx, x) dxx. Set bf(x) = a(xx)ß(x) and b, = b, - bf. Then it follows

from the definitions that fbl¡(xx, x) dxx = 0 Vx, suppZ)7i ç 3/ and \\DX ¿>7 ||Lipl *S

cl(I)1 for j = 1,...,«. Similarly suppô* ç 3/, \\Dxbf\\Upf < c/(/)"'2 for / =

1,... ,n, and

I bf(xf,... ,x„) dx2 ■ ■ ■ dxk = a(xf) Iß(x) dx2 ■ ■ ■ dxk

a(Xf) r=- I b,(x) dXf ■ ■ ■ dxk = 0 Vxj.

Continuing similarly with x2 and bf, after Ac - 1 more steps we obtain b, = E* = 1 br

with the desired properties. D

Since the proof of Theorem 5.1 is an adaptation of the proof of the Main Lemma

in [8, pp. 231-238], we follow the organization and terminology of Uchiyama's proof

quite closely. Estimates which are essentially the same as Uchiyama's will only be

stated or sketched. More detail is given in [2].

Proof of Theorem 5.1. Let e > 0 be given. We will obtain g0 g L°°, gx e Lf

such that ||/- g0 - A^Hbmo < e- We may assume S = [-1,1]". Define {r\k}^_x

and {ek }"__„ as in Lemma U2 for the {X,}, given in Theorem 5.1. Let / > 0 be the

number given by ( * ). A sufficiently large positive integer M, and sufficiently large

positive numbers R and /? will be determined later, depending on e. We inductively

construct the functions {gk(x)}x=_M_x, {<pk(x)}x=_M, {0¡j(x)}?=o,«n^2»,

{jk(x)}x_Mand {ïk(x))f..M, satisfying

(5.16) supp/3;,, ç VI, \vß,j{x)\ <s cä'/'(24(/))~\ /&,,(*) dx = 6,

00

(5.17) K- E 2-jin + 1)ß,J = b, V/such that/(/) <2M,

7=0

(5.18) yk = 4>k + Ík for -M < k < oo,

(5.19)

|ít(*) - $k(y)\ < c{2°' + 2)MR{2l/p-l) + R-x/p)2k\x - y\ if \x - y\ < 2~k,


(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

and

(5.26)

SUBSPACES OF BMO(R" )

|^(x)|<c2<" + 2»w/í<2^-1>e,(x)7,,(x),

supple (x: |x| < 2v/ñmax(2Af~/í,l)},

supple {x: |jc|< 2/r7max(2w-*,l)},

g.w_x - (R,0),

\gk(x)\=R Vx,VA:>-M-l,

117

M

gk(x) - gk-i(x) = E A, E 2-*" + %,f(x) - <p,(x)/(/) = 2-* 7 = 0

\gk(x) - gk(y)\ < c0Rl/»ek(x)2k\x - y\ if \x-y\< 2~k.

As a consequence of the induction, we will obtain

(5-27)

and

(5.28)

E ïj ( x ) converges in L2 as k -* oo

t—M

E £(*)j = -M

< C/?"1^.

First we assume the construction and complete the proof of the theorem. By (5.25)

and (5.18), if k > -M

(5.29) gk(x) - g-M-tW = Ê 2-<" + 1> E \fít,j7=0 /: 2AI>/(/)>2-

- E 2-/<«+1> E Â,7=A/+1 /: 2M>/(/)>2-*

- E *,- E 6-

By (5.10), (5.16) and Lemma U3, the first two terms on the right-hand side of (5.29)

converge in L2 as k -» oo. By (5.27) the last term in (5.29) converges in L2 as

k -* oo. By (5.20), (5.21), (5.9) and Lemma U2, E*=_Miy converges in L1 as A: -» oo.

Since \\gk — gj\\Lm < cR VAc, / > -M - 1, gk - g.M-X converges in L2 as k -> oo

by the argument in [8, p. 233]. Hence E*_M>pj converges in L2 as k -* oo as well.

Set

í(*) = £-a#-i(*)+ lim ,(**(*)-¿-a/-i(*))-k -» oo in L

Then |g(x)| = R Vx g R". By (5.12), (5.16), (5.21), (5.29) and (5.23),

supp(g-(/?,0))ç. {x:|x|<2Vr722M}.


118 MICHAEL FRAZIER

Hence gx g Lx. By (5.29) and (5.17),

K-K-gk= E *A2M>/(/)>2_*

Therefore,

K'g-f- E aa/(/)>2M

/V

E 2-><-+1> E Kkjj=M+l /: 2M»/(/)»2~*:

A A \

+ I %+ E ?,/■- M y—M

E 2-'-+1> E Kkj7 = M+1 /:2M»/(/)

y—w /-M

By (5.11), lhnw_Q0||E/(/)>2« a^Jô = 0. By the boundedness of K on BMO, and

by (5.16) and Lemma U4, we obtain, as in [8, p. 237],

k- E 2-;<-1> E M/jy-A/+l /:2M>/(/)

As in [8, p. 238], from (5.19) and (5.20) we obtain

< cf2-MRl/p.

BMO

*• E Í,- M

< c2(2("+2)MÄ(2'/'-1) + /T1^).

By (5.28), ||/f • E^.m^-IIbmo < ^/T1/'. Let c4 = max(Cl, c2, c3,1). We will choose

R sufficiently large and p so that Rl/p = 8e_1c4. Choose M so that

||E/(/)>2«XAIIbmo < e/4> and so that cf2-MRl/p = Cf2-M(He-1cA)' < e/4. Now

choose R sufficiently large so that

C22<» + 2>»ä<2'//'-i>= c22<" + 2)A/(8e-1c4)2//?^1 < e/8.

Finally pick /> so that #1/'' = 8e_1c4; then c2R~1/p < e/8 and c3R~1/p < e/8. Then

||á! • g - /Hbmo *■ e> witn £o e ^°°> £i G ^°- ÔT future reference, note that our

choices allow us to assume R1/p > 1 and R2l/p •« R.

We begin the proof of the construction. Define g„M-i by (5.23). Assume

{gP}kp-=\M-i, {%)kp-J-M, {ßrjr-ox**UKn<2», {^}*:lMand [tp}kp-J_M, satisfy-

ing (5.16)-(5.26), have been constructed. Let / be a dyadic cube with /(/) = 2 k. Let

a(I) = (a0(I), 0,(1)) = gk-iix,)- Let

Ak = {/: /(/) = 2"* and|fl0(/)/fll(/)| < A1/'}.

Let/3A = {/: /(/) = 2~k and either a,(/) = 0 or ^„(.O/a^J)! > Ä1/p}.

For I e Ak, apply (*)(i), to /, b, and a = a(/) to obtain u,: R" -» R satisfying

(5.1)-(5.5). Since ^(/ya^/)! < Ä1//', by Lemma U5, 3{/J//x)}7oo=0 satisfying

supp/3^ ç 2/r, \\ß,JUpi < cR^p(Vl(I))-\ 0,j(x) dx = 0, 3(1) ■ ßfj(x) = 0 Vx,and «,(*) = E°°_o2";(" + 1)/8,,/x). Note that therefore \ßfJ(x)\ < cR'^ Vx.



For / g Bk, apply (*)(ii), to solve KUf, = b, with uir satisfying (5.6)-(5.8). By

Lemma U5, l{#,,(x)}j°_o satisfying suppß,1,,,,£ VI, ||/3>^!|Lipl < c(Vl(I))~1 and

fß)j(x) dx = 0, such that

00

uu= E 2-"»+1WJ(x).7 = 0

Define ß,J(x) = (0,ß1,J). Note that \ßrj(x)\ ^ c Vx, and \a(I) ■ ßrj(x)\ =

\ax(I)ß] j(x)\ ^ cR}-1/p, since \ax(I)\ < \aa(I)\R~1/p and / g Bk, and |a0(/)| <

|gVi(*/)l-*by(5.24).Note that the ß, • defined in the above two cases satisfy (5.16) and (5.17). Define

M

hk(x)- E a,E 2-<"+I>'/3/,7(x),/:/(/) = 2"A 7 = 0

yk(x) = gk_1(x) + hk(x), gk(x) = R-fÎyaWI

and

(5.30) Vk(x) = gk-i(x) + hk(x) - gk(x)

= [gk-i(x) + hk(x)} 1 -R

\gk-i(x) + hk(x)\

yk(x)

\lk(x)\1\Ux)\-r]-

Then (5.24) and (5.25) hold. Note that hk(x) = 0 and hence, by (5.30) and (5.24),

gk(x) = 0, if |x| > 2jn~ max(2A/-*,l), by (5.12) and the fact that supply ç VI.

From \ß,j(x)\ < cRl/p we obtain

(5.31) \hk(x)\^cR</pVk(x)

as in [8, p. 235]. Similarly, from \vßtJ(x)\ < cRl/p(2Jl(I))-1, we obtain

(5.32) \hk(x)-hk(y)\<csR'^k(x)\x-y\ if |x - y| < Vk.

Therefore, if |x - y\ < 2~k, using (5.26) inductively, we obtain

\yk(x) - yk(y)\ <\gk-i(x) - gk-i(y)\ + M*i - hk(y)\

< {(c0/2)ek_f(x) + ~c5i¿x)}R">2k\x - y\

<|c0e,(x)/?'/'2*|x-y|,

if we choose c0 > |c5. Since \hk(x)\ < cR'/p «: /? = |grA:_1(x:)|, we obtain

\gk(x) - gk(y)\ < l\%(x) -%(y)\< c0ek(x)R'/p2k\x - y\

for |x — y| < 2"k. Hence (5.26) holds.


120 MICHAEL FRAZIER

We consider now çk(x) = yk(x)[\yk(x)\ - R]/\yk(x)\. Write

\%(x)\ -R=\gk^x(x)+hk(x)\-R

_ RL + 2gk_x{x)-hk(x) + ]^(x)l2\1/2

R2 R2R

= «1 + g^M-î-) + MúL\_R + EÁx)R2

gk-i(x)-hk(x)

R

2R¿

+ E2(x),

where the last two equalities define Ex and £2. Since |/l + a — 1 - \a\ < a2 for

\a\ < \, using (5.24) we obtain

(5.33)"

|E2<*)| < H 2̂R

,2 v 2

\gk-i(x) -hk(x)\ \hk(x)\ \ \hk(x)\-L + J-"— ^ c-

R2 R2 ^ R

Hence

- / N ?*(*) gk-i(x)-hk(x) yk(x) _Vk\x) = . , ,. -p- + 7——--¿2 = (l)k + {U)k-

\Ux)\ R \Ux)\We write

(I);gk-i(x) + hk(x) gk-x{x)-hk(x)

\gk-i(x)+hk(x)\ R

gk-i(x)-hk(x) hk(x) gk-i(x)-hk(x) gk_x(x)

R\gk-i(x) + hk(x)\

R R

, gk-i(x)-hk(x) _+-ft-Sk-x(x)

= (III),+(IV),+(V),.

[\gk-i(x)+hk(x)\

l_

R

Now

(IV),

M

M^-Ir- 5- KY,2-^gk_x(x).ß,jx)\\ /:/(/) = 2"* 7 = 0 /

( \ M

^JP- E K E 2-<«+»(g,_1(x) - gk_x(x,)) ■ ß,Jx)K

1

/:/(/) = 2-' 7 = 0

M

+ ~2 E X, I 2-><"+1)(g,_1(x) - gVt(*/))gA-i(*/) • ßrj(x)K

1

/:/(/) = 2"* 7 = 0

+ "TÏ E A, £ 2-<"+1»g,_1(x/)g,_1(x/) • &,,(*)" l:l(/) = 2-k 7 = 0

S (VI), + (VII), + (VIII),.


and


Define $k = (II), + (III), + (V), + (VI), + (VII),, and f, = (VIII),. Then (5.18)

holds. By (5.12) and the fact that sapp'0TJ Q VI, (5.22) holds. By (5.30), $,(x) = 0

if |x| > 2{n max(2w"*, 1), and so (5.21) holds also.

We now prove (5.20). By (5.33), (5.31) and (2.5),

|(II),|=|F2(x)|<^|/;,(x)|2<cä<2^-1>1?2(x)

<c/?<2//^1»e,(x)7,,(x).

Similarly,

\(lll)k\ ^ ^\hk(x)\\\cR^p-êk(xUk(x)

\/v\ i^l- t \ u t M \R-\gk-x(x) + hk(x)\(V)k\ < gA-i(*M*(*)|J--1-——r——

R\gk-i(x) + hk(x)\

\gk-x(x) -hk(x)\ |g,_i(x) -hk(x)\\E2\

* R%-i(x) + hk(x)\ R\gk-i(x) + hk(x)\

^^\hk(x)\\cR^p-»ek(xhk(x),

by our above estimates on |y,(x)| - R and F2. To estimate (VI), and (VII),, we

note, as in [8, p. 235], that if /(/) = 2'k and ß,j(x) ^0,0<j<M, then |x - x,\

< c2M~k, and

(5.34) \gk-i(x) - gk-Ax,)\ < c2<"+1>wÄ'/"e,„1(x)2*-1|x - x,\.

Therefore, for the same x and /,

(5.35) |(g,_,(x) - gk_f(x,)) ■ ß,Jx)\ < c2<"+2'^ê,_1(x)|/Jí,7(x)|

since |x - x,\ < c2w"* for these x and /. Therefore

M

l(VI)*| < j E A/ E 2^<" + 1)/?^2<-2)%_1(x)|/3/J(x)|/:/(/) = 2~A 7 = 0

< c2("+2>wÄ<2//"-1)e,_1(x)T,,(x),

since 1/3, ;(x)| < cfl'7'', by the method used to obtain (5.31). Similarly, using (5.34)

instead of (5.35),

M

l(VII),| < | E A, E 2-^« + 1»2'" + 1»w/?'/"e,_1(x)2^-1|x - jc,||/8,,,(jc)|

< c2<" + 2)A'/*<2//''-1,e,_1(x) £ 2-(" + 1»^ E À,

7 = 0 /:/(/) = 2_*

dis«*./)« 2^*

<c2(" + 2)WÄ<2//''-1'e,_1(x)T),(x),

as in (5.31). Hence (5.20) holds, since ek_x < \ek.


122 MICHAEL FRAZIER

To establish (5.19), we prove corresponding estimates for <p, and f, and use (5.18).

Following [8, p. 236], for \x - y\ < 2~k, write

$k(x) - <Pk(y) = \yk(x)\'\\yk(x)\ - R)(yk(x) - y,(y))

+yk(y)RlUx)l~lUy)l\yk(x)\\yk(y)\

-(ï)*+(n)t.

Since ||y,(x)J - R\ < cR</p, |y,(x) - y,(y)| « c0ek(x)R'^2k\x - y\ and |y,(x)| >

R/2 (since \hk(x)\ •« R), we obtain

|(ï),| < cR(2l/p-1)£k(x)2k\x - y\ for \x - y\ < 2_*.

As in [8, p. 237],

|(n)*| < 2||g,_1(x) + hk(x)\ - \gk-i(y) + hk(x)\\

+ 2\\gk-i(y)+hk(x)\ -\gk-i(y) + hk(y]

= (m)k + (rv)k.

Now

(M)*-

I2 \ V2

2 v 1/2

2

L + 2gk_f(x)-hk(x) + |//,(x)|

\ Ä2 R2

_ RL 2gk_1(y)-hk(x) \hk(x)f

\ R2 R2

< ji\(gk-x(x) - gk-x{y)) -hk(x)\

< cR(2l/p-1)ek_f(x)2k\x - y\r,k(x) < cÄ(2//"-1)e,(x)2A:|x - y\,

by (5.26), (5.31) and a Lipschitz estimate for the function /(/) = (1 + /)1/2 for /

sufficiently small. Similarly,

|(IV)*| = 11 +2gk-i(y) -hk(x) \hk(x)

l2\!/2

R:

-Rll +

R-

2gk-x(y)-hk(y) | \hk(y)\2, 1/2

j£

R2 R:

j\gk-i(y)-{hk(x)-hk(y))\ + ^\\hk(x)\2 ~\hk(y)\

-|W*I+|(VI),



By (5.32) and (2.5) we have |(VI),| < cR(2,/p-1)2k\x - y\. Now

l(V),l . j

+

M

E a, E 2-n"+l){gk-Ay) - gk-Ax,)) \h,j(x) - frjy))/:/(/) = 2_t 7 = 0

R

M

E A, E 2-<"+1>g,_1(x/) -(ß,Jx) - ßrj(y))/:/(/) = 2"A 7 = 0

- (VII), + (VIII),.

If ß,j(x) and ßrj(y) are not both 0, then \y - x,\ < c • 2¥~k, since |x - y| < 2""*;

so by applying (5.34) we obtain

M

|(VII),| < cRu/p-^"+2)M E a, E 2-^ + 1)|/3/J(x) - ^.(y)!/:/(/) = 2"* 7 = 0

< c/?(2//^-1)2(" + 2)%(x)2*|x -y|,

by the calculation in estimate (5.32). Since g,_1(x/) • ß,,(x) = 0 Vx whenever

I e A,., we have

Kviii)a| = £M

EX/E2-("+%(/)(/3/1,(x)-/8/1,y(y))/<eb* y-o

since ßKj = (0, #,) for / g /?,. Since |öl(/)| < R1'1^ and ||#,,l|Lipl < c(2>/(/))"1

for 7 g 73,, and by (2.5) and the proof of (5.32),

M

|(VIII),| < cR-1/" E 2-'(" + 1) E A,|x -y|7 = 0 /:/(/) = 2"A

dis«*,/) «2^*

< c/?-1/^tj,(x)2a|x - y| < c/?-1//72/i|x - y\ for |x - y| < 2"*.

Hence

I<Pa(*) - My)\ < c(2<" + 2)A/Ä<2//"-1» + R~^p)2k\x - y\

if|x-y|< 2~k.

Whenever |x — y| < 2~k, we have

M

E A,E 2^" + 1'g,_1(x/)g,_1(x/).(/i/,7(x)-/3/,/y))ImBk y-o

|f*(*)-f*(»l-¿

«Te*"1" £ X/E 2-<" + 1»|/i/1,y(x)-/3^(y)|/efi» 7=0

< cR~1/p2k\x y\,

since g,_!(x,) • ß,f(x) = 0 for / g .¿„ and for / g 73,, g,„i(*/) = (a0(/), a,(I)),

with !«!(/)[ < R1-1^ and ^ = (0, ß)/). The estimates for ¿, and fk give (5.19), by

( J. lo).


124 MICHAEL FRAZIER

We now verify (5.27) and (5.28). We have, for k > q > -M,

1k

E I-</+i R2

M

/eu:

M

*,L2-^ + 1X-Ax,)ax(I)ßlS, 7 = 0

< cR-^p E 2-^(" + 1»7 = 0

E/Gil*,

&-i(*/)aiU)M/\y

Note that Ift-iOâ^/)/*2-1''! < c for / G U*

E f,s-q+l

< CÄ"1/P

! /3Í, so by Lemma U3

i1/2

A2,|7|

By (5.10), then, (5.27) holds. Similarly, by Lemma U4,

M

E I--M

^cR~1/pzZ 2~Ji"+1)BMO 7 = 0

E/eur_.M;

?.-i(*j)«iU)«2-1/p

A A1,,

< cR-1/p.

This establishes (5.28) and hence Theorem 5.1. D

6. Problems and remarks. The work of Janson [5] and Uchiyama [8] gives the

complete result that a collection {Kj}f=x of singular integral operators satisfy

E;_ , KjLx = BMO if and only if

'-Ai) m At) '(6.1) rank

m.

mA-a) mÂ-i)2 V| g S""

However, little is known about the subspace Y.Pj_xKjLx of BMO when (6.1) fails.

For the Riesz transforms, Corollary 4.3 implies that any f e T.p^xRjLx, p < n,

satisfies a regularity condition on appropriate /»-dimensional subspaces of R" that is

not satisfied for all BMO functions. One would hope to obtain some similar

interpretation, whenever (6.1) fails, of the fact that T.p=xKjLx # BMO. The study

of the Riesz transforms was facilitated by their cancellation property; to proceed in

the general case, one would presumably require a geometrical interpretation of the

failure of (6.1).

However, even for the Riesz transforms the results are far from complete. Our

regularity conditions hold for/g Rj(BMO fl L2) as well as RXLX; boundedness

played a role in Corollary 4.3 only in removing the L2 restriction. Further, only the

condition X, < c, rather than the full strength of the BMO packing condition (3.5),

is used in proving Theorem 3.2. Hence our regularity conclusions hold for larger

classes of functions and cannot characterize 7?,(BMO n L2) or RXLX. The decom-

position result in Lemma 3.1 is a complete, but not explicitly geometrical, char-

acterization of /?x(BMO n L2); it is not clear whether a complete geometrical

characterization is to be expected. Example 4.6 shows that we cannot expect much

stronger regularity along lines.



Since our regularity results hold for /?x(BMO n L2) as well as RXLX, it is natural

to ask if there are additional geometrical properties that hold for RfLx but not for

/?,(BMO n L2). Corollary 5.2 suggests that any such distinction may be subtle.

Explicit examples of functions/ g Lx + /^(BMO n L2) such that/ G L°° + RfLx

would be helpful, but are not apparent since our only criterion for excluding / from

L°° + RfLx is the failure of our regularity condition, which holds for Lx +

Rx(BMO n L2) as well. Our function G g BMO n L2(R2) in Example 4.6 is

relevant: if RfG e Lx + RfLx, then a uniform BMO condition on x,-hnes does not

hold for Lx + RfLx; if not, then RfG distinguishes RX(L2 n BMO) from Lx +

RfL°°. Either case will give information as soon as it is determined which case holds.

A further question regarding each subspace Y c BMO under consideration is the

determination of dist(/, Y) = inf{||/- g||BMO: g g Y} for fe BMO. When Y =

Lx, Garnett and Jones [4] have complete results. For K as in §5, Corollary 5.2 is

obviously equivalent to the statement that

dist(/, Lx + K(LX)) = dist(/, Lx + K(BMOj).

For the Riesz transforms, our conditions on lines should play a role in this problem.

As noted above, some of our results must apply in greater generality than

presented here. Since the method of the decomposition of BMO (Lemma Ul)

appears quite general, one suspects that further information about the action of

specific operators on appropriate spaces can be obtained by similar methods.

REFERENCES

1. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193.

2. M. W. Frazier, Functions of hounded mean oscillation characterized by a restricted set of martingale or

Riesz transforms, Ph.D. Thesis, University of California, Los Angeles, 1983.

3. J. Garnett, Bounded analytic functions. Academic Press, New York, 1981.

4. J. Garnett and P. Jones, The distance in BMO to Í.00. Ann. of Math. (2) 108 (1978), 373-393.

5. S. Janson, Characterization of //' by singular integral transforms on martingales and R", Math. Scand.

41 (1977), 140-152.

6. J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseign. Math. (2) 15 (1969), 185-192.

7. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press,

Princeton, N. J., 1970.

8. A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition o/BMO(R"), Acta Math. 148

(1982), 215-241.9. _, A constructive proof of the Fefferman-Stein decomposition of BMO on simple martingales,

Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II (Beckner, et al., eds.), University of

Chicago Press, Chicago, 111., 1981, pp. 495-505.

Department of Mathematics, Washington University, St. Louis, Missouri 63130


Documents

SUBSPACES OF BMO(R)...SUBSPACES OF BMO(R" ) 103 where Uf and BMOc. are the spaces of functions of compact support in L00 and BMO, respectively, and the closure is taken in BMO norm