Functional analysis of subelliptic operators on Lie groups

Citation for published version (APA):Elst, ter, A. F. M., & Robinson, D. W. (1992). Functional analysis of subelliptic operators on Lie groups. (RANA :reports on applied and numerical analysis; Vol. 9214). Eindhoven University of Technology.

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EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science

RANA 92-14October 1992

Functional analysisof subelliptic operators

on Lie groupsby

A.F.M. ter Elst and Derek W. Robinson

Reports on Applied and Numerical AnalysisDepartment of Mathematics and Computing ScienceEindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe NetherlandsISSN: 0926-4507

Functional analysisof subelliptic operators

on Lie groups

A.F.M. ter Elst1 and Derek W. Robinson

Centre for Mathematics and its ApplicationsSchool of Mathematical SciencesAustralian National UniversityGPO Box 4Canberra, ACT 2601Australia

October 1992

AMS Subject Classification: 47A60, 43A65, 22E45, 35J30, 42B20.

1. Home institution since 1-8-1992:

Department of Mathematics and Compo Sc.Eindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe Netherlands

Abstract

Let G be a Lie group with a left Haar measure dg and L the action ofG as left translations on Lp(G; dg). Further let H = dL(C) denote anm-th order subelliptic operator associated with L. We establish thatfor a large family of holomorphic functions f one can use the Cauchyrepresentation to define functions f(H) of the operator H and thatthese functions are bounded operators if p E (1,00). Similar conclu­sions hold on the spaces formed with respect to right Haar measure,or in any unitary representation. As an application we show thatthe Cn-subspaces of the left regular representations, or the unitaryrepresentations, have natural interpolation properties relative to thecomplex method.

1 Introduction

The theory of general subelliptic, or subcoercive, operators H associated with a continuousrepresentation of a Lie group G was developed in two previous papers [EIR2] [BER] and theaim of this paper is to establish that these operators have good function-analytic properties.Specifically we examine operators H associated with the left regular representation of Gon the Lp-spaces formed with respect to left, or right, Haar measure. We then prove thatfor a large class of holomorphic functions f one can use complex analysis to define f(H)and the resulting operators are bounded on all the Lp-spaces with p E (1,00).

Holomorphic functional analysis of the type we consider has been developed for a largeclass of operators, including generators of holomorphic semigroups, by McIntosh and Yagi[Mc~ [McY] [Yag]. Their theory makes essential use of earlier work by Kato [Kat3] [Kat1][Kat2] and Lions [Lio] on fractional powers of operators and interpolation spaces. Sub­sequently, the general theory has been applied to elliptic partial differential operators byDuong [Duo] and our analysis is comparable to the discussion in Duong's Chapter 7. Themain tool is a version of the Calderon-Zygmund approach to singular integration. It isbased on interpolation between L2-estimates and weak L1-estimates. In our case, however,the required L2-estimates can be derived for an arbitrary unitary representation. But firstwe describe the general framework for semigroup generators. Basically we adopt McIn­tosh's formalism modified in a manner suitable for application to subelliptic operators.For an extension to operators of type w we refer to [CDMY].

Let H be the generator of a holomorphic semigroup t 1-+ St = e-tH on a Banach spaceX which is strongly, or weakly·, continuous. Suppose that S is holomorphic in the sectorA(Oa) where 0 < Oa ::; 1r /2 and

A(cp) = {z E C\{O}: largzl < cp}

for all cp E (0, 1r]. Then if 0 E (O,Oa) there exist M ~ 1 and w ~ 0 such that II SZ II ::; M ew/zl

for all z E C with Iarg zl < O. Therefore replacing H by H + vI with v ~ 0 suitablylarge we may assume that S is uniformly bounded uniformly in the sector A(O), i.e., thereis an M ~ 1 such that IISzl1 ::; M for all z E C with Iarg zl ::; (). It then follows thatthe resolvent (->.1 +Ht1is defined and satisfies bounds 11(->.1 +Ht11l ::; MI>'I-1 forall non-zero>' E C with Iarg >'1 ~ 1r /2 - O. For example, if H is a positive self-adjointoperator on a Hilbert space then the semigroup t 1-+ St = e-tH is holomorphic in the openright half-plane A(1r/2) and IISzlI::; 1 for all z E C with largzl::; 1r/2.

Next for 0 < cp ::; 1r let

F'{J = {f: A(cp) ---. C : f is bounded and holomorphic }

Then F'{J is a Banach space with respect to the norm

Ilflloo = sup{ If(z) I : z E A(cp)}

It is also convenient to introduce the subspaces

<P'{J,5 = {f E F'{J: If(z)l::; clzl5(1 + Izl)-25 for some c > 0 and all z E A(cp)} ,

1

where 8 > 0, and to set

~"" = U ~"".6 .6>0

Then if f E ~"" with c.p E (11"/2 - 8,11"] one can define an operator f(H) by

f(H) = (211"i)-1 f d>' f(>.)( ->.1 +Ht 1

Jrx

where r x is the contour determined by the function

(1)

if t E [0, (0)

if t E (-00,0] ,

with 11" /2-8 < X < c.p. The integral in (1) is norm-convergent, independent of the particularchoice of contour and the operator f(H) is bounded.

If f E F"" one can use a similar algorithm to define f(H) but the contour integralin (1) is not necessarily norm-convergent and the resulting operator is not necessarilybounded. Therefore f(H) is defined by interpreting the integral in (1) in the strong, orweak", topology according to whether S is strongly, or weakly", continuous. The domain off(H) is then taken to be the subspace of X on which the integral is convergent. It followsthat the operators f(H) constructed in this manner are closed. Now we define H to havea bounded functional calculus over F"" if all the operators {f(H): f E F",,} are boundedand if f 1--+ f( H) is continuous as a map from the Banach space of holomorphic functionsF"" into the Banach algebra £(X) of bounded operators on X, i.e., if there is a c > Osuchthat

IIf(H)11 ~ cllflloofor all f E F"". For example, if H is again a positive self-adjoint operator on a Hilbertspace then it is readily established that H has a bounded functional calculus over F"" forany c.p E (0,11"]; the resulting operators f( H) coincide with the usual functional definition

in terms of the spectral family EH of H.

McIntosh and Vagi [McI:1 [McY] [Vag] have given a variety of criteria for a generator tohave a bounded functional calculus and we will apply some of their results to subellipticoperators. In this latter context the existence of the functional analysis is a property of therepresentation of the Lie group to which the subelliptic operator is associated. Throughoutthe sequel we adopt the notation of [Rob] modified as in [ElR2] and [BER]. In Section 2 wefirst discuss the holomorphic functional analysis for subelliptic operators associated witha general unitary representation. Then in Section 3 we consider similar problems in thecontext of the left regular representation of the group acting on the Lp-spaces. Finally, inSection 4, we apply our results to the complex interpolation theory of the Cn-subspaces ofthe unitary representations and the regular representations.

2

2 Unitary representations

Let ('H, G, U) denote a (continuous) unitary representation of G on the Hilbert space 'Hand H = dU(C) the m-th order subelliptic operator corresponding to a subcoercive form Cand a fixed basis for the Lie algebra of G. It follows from Theorem 6.3 of [EIR2] that Hisclosed on the natural domain 'H'm and it generates a holomorphic semigroup 8 with a sectorof holomorphy A(Oil) which contains a representation independent subsector A((0 ) with00 E (0, 1r /2]. The value of 00 is determined by the principal coefficients of H and if thesecoefficients are real and symmetric then 8 is holomorphic in the open right half-plane, Le.,Oil = 00 = 1r/2. Now if the real part of the zero-order coefficient Co of H is sufficiently largethe semigroup 8 is contractive uniformly in a subsector A(0) of the universal sector A(( 0 )

of holomorphy, i.e., 118z 11 :5 1 for all z E A(O). This latter statement follows from Theorem6.3 of [EIR2]. The fifth statement of this theorem establishes that for each 0 E (0,00 ) thereis an w ~ °such that 118z11 :5 ew1zl for all z E A(O). Therefore replacing H by H + vI withv = w / cos 0 the corresponding semigroup is uniformly contractive in the subsector A(0).But with this normalization of Co it follows that H is accretive and since H is a generatorit must then be maximal accretive. This suffices, however, to ensure that H has a boundedfunctional calculus over Frp for any 'P E (1r /2 - 0, 1r].

The last statement can be inferred from the results of Kato, McIntosh and Vagi. First,if IHI = (H* H)1/2 denotes the usual self-adjoint modulus then D(jHI) = D(H) = 'H'm and

II Hxll = I11Hlx11for all x E 'H'm. Since H is maximal accretive and IHI is positive self-adjoint it follows from[Kat3] that for each I E [0,1] one has D(H'Y) = D(IHI'Y) and there exists C-y ~ 1 such that

c~lIIIHI'Yxll :5 II H'Yxll :5 C-yIlIHI'Yxll

for all x E D( H'Y). Similar bounds follow by the same argument for the fractional powersof the adjoint H* of H and then H has a bounded functional analysis over Frp, for each'P > 1r/2 - 00 , as an immediate consequence of the theorem in Section 8 of [Mc~. Thisresult can be extended to Frp with 'P > 1r /2 - Oil ~ 1r/2 - 00 if one replaces H by H +vIwith a suitable v ~ 0. The argument is similar to the above. If 0 E (0, Oil) then 8 satisfiesbounds 118z 11 :5 Mew1zl for all z E A(O) and hence after replacing H by H + vI withv = w/ cos 0 the semigroup is uniformly bounded. But H + vI is still maximal accretiveand the foregoing reasoning still applies.

One can, however, deduce much more from the criteria given by McIntosh and Vagi.

Theorem 2.1 Let H = dU(C) be an m-th order subelliptic operator associated with theunitary representation ('H, G, U). Further suppose that the holomorphic semigroup 8 gen­erated by H has holomorphy sector A(Oil) and that the real part of the zero-order coefficientof C is sufficiently large that 8 is uniformly bounded in the sector A(0) where 0 E (0, 011)'Then one has the following.

I. The operator H has a bounded functional analysis over Frp for each 'P E (1r /2 - 0, 1r].

3

II. The imaginary powers {Hit: t E R} form a strongly continuous group.

III. If c.p E (1r/2 - 0, 1r] and f E ~ 'P then there exists Cf > 0 such that

100

dt C1Ilf(tH)xIl2 :5 cfllxll2

for all x E 1i.

IV. If [1i, ,q'Y denotes the complex interpolation space between 1i and the subspace K,then

D(H'Y) = [1i, 1i~m]'Y/n = D((H*P)

for all, E (O,n) and n E N.

Proof Statement I follows from the above discussion and Statements II and III are aconsequence of the theorem in Section 8 of [Mel]. Statement IV also follows from this lattertheorem, which establishes that D(H'Y) = [1i, D(Hn )]'Y/n and D((H*P) = [1i, D(Hn*)]'Y/Mtogether with the observation that D(Hn) = 1i~m = D(Hn*). These latter identificationsare established in Theorem 6.3.1 of [EIR2]. The second uses the fact that the formal adjointis closed on 1i'm and hence H* = Ht. 0

Statement III of the theorem is of interest when applied to the function z 1-+ Zl-'Y e- z

where, E (0,1). Then one has

100

dt r 1(t1-'YIIH1-'YStxll)2 :5 cllxl1 2

for some c > 0 and all x E 1i, or, equivalently

100

dtr 1 (t1-'YIIHStxll)2:5 c1IH'Yx11 2

for all x E D(H'Y). Now the real interpolation spaces (1i, D(H) )'Y.2;K defined by the Peetre[{-method have an equivalent norm

x 1-+ Ilxll'Y = Ilxll + (100

dt r 1(tl-'YIIHStxllrr/2

(see [BuB], Theorem 3.4.2). Hence D(H'Y) is continuously embedded in (1i, D(H))'Y.2iK

and one reproduces one direction of [EIR1] Theorem 7.2.V.

The foregoing results apply directly to G acting by left translations L on the Hilbertspace L 2 (G; dg). But if G is a subgroup of a second Lie group G1 they can also beapplied to the action L of G on L2(G1) = L2(G1; dg). Alternatively, the argument in theproof of Theorem 3.7 of [EIR3] to obtain the L2 case from the L 2 case can be used todeduce that D(IHI) = D(H) = L~;m(Gtl. Then, as a result of the Garding inequality,D(IHI'Y) = D(H'Y) with equivalent norms if the real part of the zero-order coefficient of Cis large enough. The same reasoning applies to the dual operator. Therefore we have thefollowing conclusion.

Corollary 2.2 Let H = dL(C) be an m-th order subelliptic operator associated with therepresentation of the Lie subgroup G of the Lie group G1 acting by left translations L onL2(G1 ). If the real part of the zero-order coefficient of C is sufficiently large then theconclusions of Theorem 2.1 are again valid.

4

3 Lp-spaces

We continue to examine two Lie groups G and G l with G a subgroup of Gl and now considerthe representation of G by left translations L on the Lp-spaces, Lp(Gt} and Lp(Gl ), formedwith respect to left, and right, Haar measure over Gl • In addition we now assume that Gl

is connected. Our aim is to establish that each m-th order subelliptic operator H = dL(C)associated with L and a fixed basis al, ... , ad' of rank r for the Lie algebra of G hasa bounded functional calculus on these spaces if p E (1,00). The crucial case is whenG =G l but it is subsequently useful to consider the more general situation G C Gl • Sinceall analysis takes part on the connected component of the identity of G we may as wellassume that G is connected.

First note that the interpolating semigroup S generated by the closures H on the variousspaces has a representation independent subsector of holomorphy A(Oo), by Theorem 5.2of [EIR2], and in particular S is holomorphic in this subsector on each of the spaces.Therefore, replacing H by H + v I with v sufficiently large, one may assume that S isuniformly bounded on each of the spaces uniformly in a subsector A(0) with 0 E (0,00 ),

i.e., one has bounds IISzllp-+p S M, IISzllp-+p S M for all Z E A(O) and all p E [1,00]. Nowthe main result we prove is the following.

Theorem 3.1 Let H = dL(C) be an m-th order subel/iptic operator associated with lefttranslations L by the group G acting on the spaces Lp(Gl ), and Lp(Gl ) and let 0 E (0,00 ),

If p E (1,00) then H is closed and there is a va ~ 0, independent of p, such that theoperators H + vI, v > va, have a bounded functional analysis over Flp for each r.p E

(11" /2 - (}, 11"].

Proof First remark that it is not at all obvious that the operators H are closed but thisis established by [BER] Theorem 2.2.

The next step in the proof is to establish the result for all f E ~lp,6 with 8 sufficientlylarge. Then the general result follows by approximation. The proof for the special classof f is based on the L2(Gt}-result established in the previous section and a weak Li(Gl )­

estimate which follows by the methods of singular integration theory. This estimate is themost difficult part of the proof and it is established from analysis of the kernel associatedwith f(H). Finally we deduce the Lp-statement from the Lp-statement. The details of theproof are very similar to the arguments used in [BER] to prove Theorem 2.2.

It follows formally from the definition of f(H) that its action should be given by

f(H) = L(KJ) = Ldg KJ(g) L(g)

where the kernel K J is defined on G\ { e} by

KJ(g) = (211"it l I d>.f(>.)R_)..(g) (2)Jrx

and R).. denotes the kernel of the resolvent (>'1 +Htl • We begin by examining propertiesof K J for f E ~lp and for this purpose it suffices to consider the operators acting on theLp-spaces over G, so for a moment we consider the case Gl =G.

5

Ifone replaces H by H+vI with v sufficiently large, one may assume that the semigroupS satisfies the bounds IISzllp.-.p ::; M for all z E A(O) and all p E [1,00] where the normsare the operator norms on Lp(G; dfJ). Then one has the resolvent bounds

(3)

for all non-zero A E C with Iarg z I~ 7r12 - 0 and all p E [1,00]. In particular this impliesthat the kernel R_>. E LI(G; dg) because the Ll-norm is given by

IIR_>.lh = II(-AI +Htllloo-+oo .

Therefore if f E (><p with c.p E (7r12 - 0, 7r] it follows that KJ is well-defined by (2) and KJ E

LI(G; dg). Then if v is large enough it follows from Corollary A.2 that KJ E Li(G; dg)where p > 0 is such that .6.1(g) ::; eplgl' for all 9 E G with /. I' the modulus on G associatedwith the algebraic basis and .6.1 the modular function on GI • Hence K J is the kernel ofthe operator f(H).

Secondly, following Duong [Duo], we argue that if f E (>cp,e5 with 6 > 2D'lm thenKJ E Loo(G; dg). The proof of this observation begins by noting that (/+H)-e5 is boundedon Loo(G; dg) for all 8 > 0 and hence its kernel RI,e5 E LI(G; dg) because

IIRI,e5111 = 11(/ +Ht e5 l1oo-+oo .

But if 8 > D'im then RI,e5 E Loo(G; dg) by the resolvent estimates of Theorem A.l inthe appendix. Therefore if 8 > 2D'lm then (/ +Ht e5/2 is bounded from L2(G; dfJ) toLoo(G; dg) and one has

11(/ +Ht e5/2 112-+OO = IIRI,e5/2112::; (II RI,e5/211t1I RI,e5/21Ioo)I/2 < 00 .

Next by a duality argument (/ + Ht e5/2 is bounded from LI(G; dfJ) to L2(G; dfJ). Nowsince f E (>cp,e5 one may choose fl E Loo(G; dg) such that

f(z) = (1 +zte5 ft(z) .

But fl(H) is bounded on L2(G; dfJ) by Corollary 2.2 and

f(H) = (/ + H)-e5/2 fl(H) (/ +Ht e5/2

Therefore f(H) is bounded from LI(G; dfJ) to Loo(G; dg). Consequently,

IIKJlloo = IIf(H)lli-+oo < 00

and KJ E Loo(G; dg).

Since KJ E LI(G; dg) n Loo(G; dg) it follows automatically that KJ E L2(G; dg) andthis is crucial for the principal estimate.

Proposition 3.2 There is an M > 0 such that

IlI({g E GI : 1(J(H)u)(g)1 > ,})::; Mllflloollulli,-l

for all f E (>cp,e5 with c.p E (7r12 - 0,7r] and 6> 2D'lm and for all u E LI(GI; Ill) where IIIdenotes right Haar measure on GI.'

6

Proof The action of the operator f(H) is determined by the kernel KJ E L t nLoo and wefirst decompose it into a local part and a global part. Let x: G -+ [0,1] be a COO-functionwith compact support in a bounded symmetric neighbourhood 0 of the identity e of Gand suppose that X = 1 in a second neighbourhood 0 0 C 0 of e. Then one can decomposef(H) as a sum Tj + Rj where Tj = L(XKj). But it follows from Corollary A.2 in theappendix that one has bounds

where I . I' is the canonical modulus on G associated with the algebraic basis used inthe definition of the subelliptic operator H and the positive parameter c is a linearlyincreasing function of the scalar term II. Thus if II is sufficiently large K j (1 - X) E Lf, (G) =Lt (G; ePlgl'dg) with p > 0 so large that dt (g) ~ eplgl' for all 9 E G where dt is the modularfunction of Gt . Since

where the norms of Rj are now relative to the Lp-spaces over Gil it follows that R f is abounded operator on Lp(Gt ) with norm bounds continuous in f, uniformly in p E [1,00],i.e., one has estimates

IIRjllp-+p ~ cllJlloo ,in particular this is the case if p = 1 and p = 2. Hence to establish the weak-Li(Gt}estimate on f(H) it suffices to establish an estimate of this type for Tj. This is achievedin several steps by the reasoning used for an analogous problem in [BER].

We begin by considering the action of Tj on the Lp-spaces over G, i.e., we effectivelyassume that G = Gt . Then the first step is to derive a local version of the required estimate.Hence we introduce the operator 8j from Lp (02; J.lr) to Lp (02; J.lr) by 8 ju = Tj(X'u) whereX': G -+ [0,1] is a COO-function with support in 0 C G and J.lr is the restriction to 0 2 ofthe right Haar measure on G. Then

wherekJ(g; h) = X(gh-t)KJ(gh-t)X'(h) .

Now the tactic is to deduce the local result as a corollary of Theorem 11.2.4 of [CoW].This requires verifying three properties of 8 f and kf. First one needs 8 f to be bounded onL2(02; J.lr). But this follows from Corollary 2.2 since

Moreover, one has bounds 118j 11 2-+2 ~ cllJlloo. Secondly, one requires that the kernelkf E L2(02 002

; J.lr ® J.lr). But this is evident because K f E Loo and hence kf is bounded.Finally, one needs an estimate on the variation of Kf in the second variable. But for thisit suffices to consider a particular O.

7

Lemma 3.3 Let,., > °and 0 = B~-l'7 = {g E G: Igl' < 2-1,.,}. Then there is an M > 0,

independent of the choice of f, such that

f dlJ-r(g) IkJ(g; h) - kJ(g; ho)1 $ Mllflloo}O(h; ho)

for all h, ho E 0 where O(h; ho) = {g E B~ : Igha11' > 2Ihha1I'}.

Proof The proof is a slight variation of the argument used to establish an analogousproperty for the sequence of operators Tj in the proof of Theorem 2.2 of [BER]. In theestimates on the Tj uniformity in j is essential and in the current context the uniformityin f is crucial. Therefore we repeat some of the details in order to display this uniformity.

First by right invariance of IJ-r one has

f dlJ-r(g) IkJ(g; h) - kJ(g; ho)1 $ f dlJ-r(g) IkJ(gho; h) - kJ(gho; ho)1}O(hiho) }02(h;ho)

for all h, ho E 0 2 where 02(h; ho) = {g E B~rj : Igl' > 2lhha1 l'} and where we also denoteby IJ-r the restriction to B~'7 of the right Haar measure on G. Now to bound the integrandwe choose an absolutely continuous path ,: [0, 1] -+ G from ho to h with tangents in thedirections at, ... , ad" Then

where the Ai are left derivatives with respect to the second variable. We fix , such that

where c E (0,1). Then

for all t E [0,1]. Therefore

1 (d' )1/2( d' )1/2Ikj(gho; h) - kJ(gho; ho)1 $ i dt t;'i(t)2 t; I(AikJ)(gho; ,(t))12

$ 2d'lhoh-1I'sup{I(AikJ)(gho; ,(t))l: i E {l, ... ,d'}, t E [0, I]}

and our next aim is to bound the last factor. But

(AikJ)(g; h) = (Bix)(gh -1) KJ(gh-1)x'(h) +X(gh-1) (BiKJ)(gh-1) X'(h)

+ X(gh-1) KJ(gh-1)(AiX')(h)

where Bi denotes the right derivative in the direction ai. It follows, however, from theestimates on R). given in the appendix that

8

Since

Igho,(tt11' 2 Igl' - Iho,(tt11'

2 Igl' - (1 +c:)lhoh-1 1' 2 2-1(1 - e)lgl'

for 9 E fh(h; ho) one then has an estimate

for all 9 E 02(h; ho) if c: is small enough. Next to handle the right derivative of KJ wenote that by Lemma 7.3 of [EIR2] one has

(Biep)(g) = (L(g-l)AiL(g)ep)(g)

= L: Ci.~(g)(A~ep)(g)~eJr(d')

1~I;tO

where the real-valued functions Ci.~ satisfy bounds lci,~(g)1 ::; M(lgl')I~I-le(1lgl'. But theresolvent bounds in the appendix give bounds

and hence one has

for all 9 E 02(h; ho). Combination of these various estimates then gives

and, consequently,

IkJ(gho; h) - kJ(gho; ho)1 ::; a'llflloolhoh-11'(lg\')-(D'+l)

for all 9 E 02(h; ho).Finally, setting s = Ihoh-1 1', one has

[ dpr(g) IkJ(g; h) - kJ(g; ho)1 ::; a'IIflloo sup [ dpr(g) s(lg\')-(D'+l)lfl(hi ho) 6$1/ 126$lgl'$21/

::; Mllflloo

where M > 0 is a finite constant independent of f. The last estimate is established as in~E~. 0

The operator SJ' defined with 0 = B~-l1/ acts on the spaces Lp(B~; Pr). Thereforeone can now apply Theorem II.2.4 of [CoW] to SJ and deduce that it satisfies a weak-Liestimate. But since the L2-bound on Sf and the bounds of Lemma 3.3 are uniform for f

9

with 1111100 ~ 1 it follows that the resulting Li-estimate is also uniform, i.e., one has anM > 0 such that

ILr( {g E G: I(Sju)(g)1 > ,}) ~ Mililiooliulli ,-1for all I E (Pcp,6 with <p E (1r/2 - 0,1r] and 6> 2D'/m and for all u E Ll(B~'1; ILr).

At this stage one can remove the assumption that G = Gl by the reasoning of [HER].The basic idea is that Gl '" G x R dl -d locally and the right Haar measure ILIon Glcorresponds to the product of ILr and Lebesgue measure on R d1-d. Repetition of thearguments of [HER] then give the local weak-Li(Gl ) estimates

ILl({g E Gl ; I(L(XKj)u) (g)1 > ,}) ~ cililiooliulln-l

for all I E (Pcp,6 with <p E (1r/2 - 0,1r] and 6 > 2D'/m and for all u E Ll(GI ; ILl) withsupport in a ball of radius 4-lTJ centred on the identity if TJ is small enough. Then, however,one can use right invariance to extend the bounds to all u E L l (Gl ; ILl) with support ina ball of radius 4-l TJ centred at an arbitrary point h E Gl . Moreover, the bounds areuniform in h. Finally, the bounds can be extended to global bounds by use of Lemma 2.4of [HER]. One then has

ILl({g E Gl ; I(Tju)(g)1 > ,}) ~ cililiooliulln-l

for all I E (Pcp,6 with <p E (1r /2 - 0,1r] and 6 > 2D' /m and for all u E Ll(Gl ; ILd·These bounds combined with the earlier Ll-bounds on Rj immediately yield the boundsof Proposition 3.2. D

Now we can complete the proof of Theorem 3.1.

If v is sufficiently large I(H +vI) is defined and satisfies a bound

III(H + vI) 112--+2 ~ cililiooon L2(G l ) for all IE Fr.p, with c independent of I, by Corollary 2.2. If, however, I E (Pr.p,6

with 6 > 2D' /m then I(H + vI) satisfies the weak-Li(Gt} estimate of Proposition 3.2.Therefore I(H +vI) is bounded on Lp(Gd for all p E (1,2] and I E (Pcp,6 with 6 > 2D' /mby interpolation and in addition one has bounds

III(H + vI)llp--+p ~ cililioo .This result then extends to all I E Fcp by McIntosh's convergence theorem, [McI] Section 5.Since similar arguments apply to the formal adjoint Ht of H the desired boundednessproperties of I(H + vI) on Lp(GI ) for p E (2,00) follow by duality.

Finally boundedness on the Lp ( Gd-spaces follows from boundedness on the Lp(Gd­spaces as follow. since ~~/pAi~~llp = Ai - p-IbJ we consider the m-th order subcoerciveform Cby

C= L L c~( -prh1b"Y~EJm(d') "YEJm(d')

(1J,"Y)ELb(~)

10

~~/P(-\I +Ht\? = (AI +lit l ~~/P<p

for all <p E C~(Gd. Moreover, the previous results apply to li. So if f E Ftp,6 and<p E C~(Gd one obtains

Let H = dL(C) and li = dL(C). Arguing as in the proof of Lemma 2.1 of [BER] oneobtains that

Ilf(H)<pllp = II~~/p f(H)<pllp

= Ilf(H)~~/P<pllp

:5 cllfllooll~~/P<pllp

= cllfllooll<pllp .By Mcintosh's convergence theorem and density the theorem follows. o

Now one easily obtains two of the three other conclusions of Theorem 2.1.

Corollary 3.4 Let H = dL(C) be an m-th order subelliptic operator associated with the

left translations L by the group G acting on the spaces Lp(GI ) where p E (1,00). If thereal part of the zero-order coefficient of C is sufficiently large then one has the following.

I. The imaginary powers {Hit: t E R} form a strongly continuous group.

II. D(H"I) = [Lp , L~;nm]"I/n = D((H*P) for all 'Y E (0, n) and n EN.

Similar statements are valid on the L p(G I )-spaces.

Proof Statement I follows from Theorem 3.1 by the same argument as in Section 8 of[Mel].

In Theorem 2.2 of [BER] the equality L~;nm = D(Hn) has been proved for all n E Nif the zero-order coefficient of C has a sufficiently large value, depending on n. Then byTheorem 1.15.3 of [Tri] and Statement lone has

for all 'Y E (0, n) and n E N. The proof on the Lp-spaces is similar.

(4)

oThe interpolation properties (4) immediately gives the following comparison of domains

for different subcoercive operators.

Corollary 3.5 Let CI and C2 be subcoercive forms of order ml and m2, both of step r.If Hi = dL(Ci) are the operators on Lp ( G I ) with p E (1,00) associated with the algebraic

basis of rank r of the Lie algebra of the subgroup G ~ G I then

for all t E [0,00). Similar identities are valid on the L p(Gt}-spaces.

11

4 Interpolation of Cn-subspaces

The complex interpolation properties of subcoercive operators give some interesting resultsfor interpolation between the Cn-subspaces associated with a general set all"" ad' ofelements of the Lie algebra g of G. It is no longer necessary to assume that al, ... ,ad' isan algebraic basis.

One can again derive results for unitary representations or the regular representationson the Lp-spaces with p E (1, (0). First recall that (11., K)-y,qjK denotes the real interpolationspaces defined by the Peetre K-method.

Proposition 4.1 Let (11., G, V) be a unitary representation of the connected Lie group Gand 11.~ the Cn-subspaces associated with a family al, . .. ,ad' of elements of the Lie algebrag ofG. Then

11.~ = [11.j,11.n(k-i)/(I-i) .

for j, k, 1 E N with j < k < 1 and with the convention 11.~ = 11..More generally

(11.j,11.;)-y-i,2jK = [11.j, 7-£~]b-i)/(l-i)

for / E (j,1).

Proof We may assume that all' .. ,ad' are linearly independent.

Let G' denote the connected subgroup of G with Lie algebra g' generated by the subbasisall" ., ad' and let H be the sublaplacian associated with the subbasis. Now we can applyTheorem 2.1 to the representation (11., G', V'), where V' = Via', and to the subcoerciveoperator H. Then if v is large enough the operator (vI + H) has a bounded functionalcalculus. Therefore (vI +H)is is uniformly bounded if s E [-1,1] and by Theorem 1.15.3of [Tri] one has

[D((vI + H)O), D((vI + H)"Y)]{I = D((vI + H)O(I-{l)+-y{l)

for 0 ~ b < / < 00 and 0 < () < 1. But 11.~ = D((vI + H)n/2) by [EIR2], Theorem 6.3.Therefore

7-£~ = D( (v I +H)k/2)

= [D((vI +H)i/2),D((vI +H)I/2)](k_i)/(I_i) = [11.j,11.n(k-i)/(I-i)

for j < k < 1. Moreover, if j < / < 1 then

(11.j,7-£;)-y-i,2;K = (7-£,11.;)-y,2;K

= D((vI +H)"Y/2)

=[D((vI + H)i/2), D((vI + H)I/2)]b_i)/(I_i) = [11.j,11.nb-i)/(I-i)

where the first two identifications follow from [EIR1] Theorem 3.2 and Lemma 7.1. 0

12

The proposition establishes that the real interpolation space with q = 2 play a dis­tinguished role for unitary representations. It is unclear whether there are analogousidentifications for all the complex interpolation spaces associated with the Cn-structureof the regular representations. Nevertheless the first conclusion of Proposition 4.1 can beextended to these representations acting on the Lp-, or Lp-, spaces over G, with p E (1,00),by a similar argument based on Theorem 3.1.

Corollary 4.2 Let aI, ... ,ad' be elements of the Lie algebra g of the connected Lie groupG and L~jn(G), L~jn(G) the corresponding Cn-subspaces. Then

L~;k(G) = [L~;j(G), L~iG)](k-j)/(l-i)

if j < k < I and p E (1,00). Similar identities are valid on the Lp(G)-spaces.

It is possible that complex interpolation results similar to the first statement of Propo­sition 4.1, or to the statement of Corollary 4.2, are valid for general representations butthe above proof which relies on special regularity properties of the unitary representationsand the regular representations does not shed any light on this question.

A Resolvent estimates

Let S be the semigroup generated by the closure of the m-th order subelliptic operatorH = dU(C) determined by the subcoercive form C = (Ca)aEJm(d') with coefficients Ca E C.Then for v E C with the real part Re v sufficiently large the fractional powers (vI + H)-Sof the resolvent are defined for all 8 > 0 by the Laplace transforms

(5)

Since S satisfies bounds IIStl1 ~ M ewt for some M ~ 1, some w E R and all t ~ 0 it followsthat the resolvents are indeed defined and satisfy norm bounds

whenever Rev> w. More generally if a E J(d') with 10'1 < m8 one has the bounds

(6)

on the left derivatives of the resolvents.

It follows from the definition of subcoercivity [EIR2] that to each subcoercive form Cthere corresponds an angle Oe E (0, 7r /2), determined by the principal coefficients of C, suchthat each ofthe forms eieC = (e ie

Ca )aEJm(dl), 0 E (0, Oe), is subcoercive. Hence the closures

13

of the operators eiO H = dU(eiOC), 0 E (0, Oe), generate continuous semigroups SO whichgive a holomorphic extension of S to the sector A(Oe) = {z E C : Iarg zl < Oe } by theidentification Sei6t = sf for t > 0. If 0 E (0, Oe), the holomorphic extension automaticallysatisfies bounds IISzli $ Mew1zl for all z E A(O) with M ~ 1 and w E R independent of z,but dependent on O. Consequently, the fractional powers (vI +Ht6 of the resolvent aredefined for all II E C such that Re(ei'Pv ) > w for some cp E (-0,0) by

(vI + Ht6 = (ei'PvI + ei'PH)-6ei6'P

= r(h't1e

i6'P100

dte-ei<Pvtt6-1Si

Now setr(cp; w) = r +(cp; w) u r _(cp; w)

where r ± denote the half-planes

Then the resolvents and their fractional powers are defined by the above method for allv E L\(0; w) where

A(O; w) = U r(cp; w)'PE[O,O]

Moreover, one has bounds, analogous to (6),

(7)

for all v E A(O; w) where p(v; A) denotes the distance of v to the boundary of A.

Next remark that the semigroup S has a kernel K which has a holomorphic extensionto the sector A(Oe) and which satisfies 'Gaussian bounds' uniformly throughout each ofthe subsectors A(0) with 0 E (0, Oe). Specifically for each a E J(d') there exist a, b > °and w E R such that

(8)

for all Z E A(O) and 9 E G. Consequently the operators (vI + H)-6 have kernels Rv.6determined by the semigroup kernel K and the appropriate Laplace transform, e.g.,

(9)

(10)

for Re v > w. But as one has bounds on the derivatives of K one can derive bounds onthe corresponding derivatives of the Rv•6 for all v E A(O; w). One finds

I(AaRv.6 )(g)1 $ a100dt e-ptrl+.6e-b((lgl,)mt-l)l/(m-l)

for all 9 E G and v E A(O; w) where f3 = (mh' - D' - lal)jm and P = p(v; A). Estimatingas in the proof of Theorem III.6.7 in [Rob] these bounds can be reformulated as follows.

14

Theorem A.1 For each a E J(d') and () E (0, ()c) there exist a, b > 0 and w E R, suchthat

I(AcrRv,s)(g) I :s; ap(D'+lcrl-mS)/m Flcrl,s(pl/mlgl') e-bpl/mjgl'

for all 9 E G\{e} and all v E ~((); w) where p = p(Vj ~) and

x-(D'+k-mS) if D' + k > m8

(11)

1 + log+ X-I if D' + k = m8

1 if D' + k < m8

with log+ y = logy ify ~ 1 and log+ y = 0 ify:S; 1.

Proof Before beginning the proof we briefly comment on the structure of the bounds(11). First the factor p(D'+lcrl-mS)/m is a reflection of the behaviour given in the normbounds (6) of the dependence of the resolvent derivatives on p. The remaining termsdepend only on the rescaled distance 9 1-+ pl/m Ig I'. Secondly, the factor Fk,S(pl/m Ig I') givesthe small distance behaviour and this depends on the relative size of the local dimension,the order of the operator, et cetera. Thirdly, the exponential factor e-bpl/mlgl' dictates thelarge distance behaviour and this is dimension independent.

The proof consists of estimating the bounds (10) and there are three distinct cases toconsider, (3 < 0, (3 > 0 and (3 = O. We examine them in this order.

Case 1 : (3 < O. A change of integration variable in (10) gives

I(A O RII,s)(g) I :s; a(lgl,)m/3100

dt rI+/3 e-f(t)

wheref(t) = p(lgl')mt +brl/(m-l)

In particular one has the estimate

for all 9 E G with

(12)

a' = a100dt rI+/3 e-bt -

1/(m-l)

independent of v E ~((),w). This establishes the required estimate for pl/mlgl' :s; 1.

Next we consider the case pl/m Igl' ~ 1. Since f has a unique minimum at the pointt = ((m - l)p(lgl,)m /br(m-l)/m the integral in (12) can be estimated in two parts, 1< theintegral over (0, to] and I> the integral over [to, 00) where to = (pl/mlgl,)-(m-l). One has

I <100

dt rI+Pe-p(lgl')m t>-

to

and

15

ThereforeI < e-p(lgl,)mto100

dt cl+fJ < _a-It fJ e-p(lgl,)mto •>- - fJ 0

to

But if n is any integer greater than -(m - 1)(3 then one has bounds

tg(//mlgl,)n ~ nle-neepl/mlgl'

for each e > O. Moreover, p(lgl,)mto = pl/mlgl'.Consequently, by choosing e = 2-1, oneobtains bounds

for all 9 E G.

Next consider 1<. A change of integration variable t -+ t-(m-l) gives

where t l = tol/(m-l) = pl/mlgl' and, = (m - 1)(1 - (3) - m > -1. Suppose, ~ O. If nis any integer greater than " then fY ~ tn ~ nlcneet for all t ~ t l ~ 1 and e > O. Hencesetting e = b/2 one obtains bounds

1< ~ 2n (m -1)nlb-n100

dte- bt/2 ~ ae-b'pl/mlgl'h

for suitable a,b' > O. Secondly, suppose, E (-1,0). Then

1< ~ (m - 1)100

dt e-bt = (m - l)b- Ie-bt1tl

Thus one again obtains bounds

Since similar bounds were already established for pl/m Igl' ~ 1 this completes the discussionof Case 1.

Case 2 : (3 > O. Since pt = b((Igl')mtijl f/(m-l) if t = b(m-l)/mlgl'p-(m-l)/m we set

to = p-(m-l)/m Igl' and consider the integral in (10) in two parts, 1< the integral over (0, to]and I> the integral over [to, (0). Then

b(1 1,)mt-1)1/(m-l) lto l+fJ tI < ~ e- 9 0 dt C e- p

o

~ e-bpl/mlgl'100

dt cl+/3e- pt .

But the latter integral is finite for p > 0, because (3 > 0, and by a change of variabless = pt one obtains bounds

I < ap(D'+lol-m6)/me-bpl/mlgl'<-

16

as desired.

Alternatively

and the bounds follow from another change of variables.

Case 3 : f3 = O. Now (10) gives

I(AQ'Rv,o)(g) I ~ a100

dt r 1e-/(t)

where f(t) = pt + b((lgl,)mt-lr/(m-l). Therefore, integrating by parts, one has

I(AQ'Rv,o)(g)/ ~ a100

dt logtj'(t)e-/(t)

= a100

dt log(pt) j'(t) e-/(t)

= ato100

dt log(ptot) j'(tot) e-/(tot) ,

with to = pl/m-1 Igl'. Now we divide the integral into a first part over the interval (0, 1]and a second part over [1, co). Then

1< ~ to11dt log(ptot) j'(tot) e_b'(pl/mlgl')t-1/(m-l)

and a change of variable t -+ r(m-l) gives

1< ~ to(m - 1)100

dt r mlog(ptor(m-l») j'(tor(m-l») e-bpl/mlgl't

But f'(t ot-(m-l») = p(1-b(m-1)-ltm), so If'(tor(m-l»)1 ~ cptm for somec > 0, uniformlyin t ~ 1. Therefore one has an estimate

1< ~ cllptolog(pto)ll°O dt e-bpl/mlgl't +C2pto100

dt e-bpl/mlgl't log t

and this immediately yields a bound

1< ~ c3(1 + Ilog(pl/mlgl')l)e-b'pl/mlgl'

Next consider

I> = to100

dt log(ptot) f'(tot) e-f(tot)

~ to100

dt log(ptot) f'(tot) e- ptot

= to100

dt log(ptot) f'(tot) e-bpl/mlgl't

17

But now f'(tot) = p(1 - b(m _1)- lr m!(m-I)). Therefore one has estimates

I> $ cllptoIOg(pto)/l°O dt e-bpl/mlgl't +C2pto100

dt e-bpl/mlgl't log t

, b Ilml I' [00 b Ilml I't$c1Ilog(pto)le- P 9 +C2ptOJ1

dtlogte- P 9 •

Moreover

log t = log(ptot) -log(pto)

$ ptot - 1 -log(pto) $ e-1el1:ptot +1 +Ilog ptol

for all e > O. Hence choosing e sufficiently small one obtains a bound

Therefore combining these estimates one finds bounds

18

with X E (1r /2 - 8, ep). The bounds of the theorem then give bounds on K f and itsderivatives.

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(13)

{

teiX if t E [0,00)r x(t) =

-te-ix if t E (-00,0] ,

The zero-order coefficient Co of C contributes a multiplicative factor t 1-+ e-cot to thesemigroup S. Therefore the values of a and b in the kernel estimates (8) can be assumed tobe independent of Co and w can be taken as a linearly decreasing function of Re Co. Thusif Re Co is sufficiently large the parameter w is negative and the kernel K is exponentiallydecreasing. Moreover, D.(8; w) = r(8; w). Now if f is bounded and holomorphic in thesector A(ep) with ep E (1r/2 - 8,1r] one can define the kernel K f on G\{e} by

for all e > O. Therefore by redefining the values of a and b the bounds (13) can bereexpressed in the desired form. 0

where R II = RII,1 denotes the kernel of the resolvent (vI +Htl and the contour r x isdetermined by the function

for all 9 E G. Finally if x $ 1 then log X-I = log+ X-I but if x ~ 1 then

1 + Ilogx-ll = 1 + log x $ x $ e-1el1:x

Corollary A.2 Let a E J(d'), () E (0, ()c) and'P E (11"/2,11-]. If the real part of the zero­order coefficient CQ of C is sufficiently large then there exist a, b > 0 independent of CQ anda c > 0 linearly-dependent on Re CQ such that

I(AO K, )(g)1 ::; allflloo(lgl't(D'+loDe-bcl/mlgl'

for all f E FV' and all 9 E G\{e}.

Proof By elementary geometry one obtains bounds p = p(lI; ~) ~ lwl +TllIl for 11 E rxwith T > O. In particular pl/m ~ 2-1 (lwI 1/ m+ Tl/mI1l1 1/ m) and pl/m ~ Tl/mill/ l/m. Nowthere are three cases to consider corresponding to the three cases of the theorem.

If D' + lal > m then the theorem gives bounds

I(AO RII)(g) I ::; a(lgl')-(D'+lol-m)e-b'llIll/mlgl'e-b"l..,jI/mlgl'

with b' = 2-1 bT1/ m and b" = 2-1b. Therefore

I(AO K f )(g)1 ::; 2allfII00(lgl't(D'+lol)e-b"lwll/mlgl'100

dll (lgl,)me -b'11I11 /mlgl'

and the desired bounds follow immediately. The other two cases D'+!al =m and D'+lal <m are very similar and we omit the details. 0

References

[BER] BURNS, R.J., ELST, A.F.M. TER, and ROBINSON, D.W. Lp-regularityofsubelliptic operators on Lie groups. Research Report CMA-MR30-92, The Aus­tralian National University, Canberra, Australia, 1992.

[BuB] BUTZER, P.L., and BERENS, H., Semi-groups of operators and approxima­tion. Die Grundlehren der mathematischen Wissenschaften 145. Springer-Verlag,Berlin etc., 1967.

[CoW] COIFMAN, R.R., and WEISS, G., Analyse harmonique non-commutative surcertains espaces homogenes. Lect. Notes in Math. 242. Springer-Verlag, Berlinetc., 1971.

[CDMY] COWLING, M., DOUST, I., McINTOSH, A., and YAGI, A. Banach spaceoperators with a bounded Hoo functional calculus, 1992. Draft version.

[Duo] DUONG, X.T., Hoo functional calculus of elliptic partial differential operators inLP spaces. PhD thesis, Macquarie University, Sydney, 1990.

[EIR1] ELST, A.F.M. TER, and ROBINSON, D.W., Subellipticoperators on Lie groups:regularity. J. Austr. Math. Soc. (Series A) (1992). To appear.

[ElR2] --, Subcoercivity and subelliptic operators on Lie groups. Research ReportCMA-MR4A-92, The Australian National University, Canberra, Australia, 1992.

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[ElR3] --, Subcoercive and subelliptic operators on Lie groups: variable coefficients.Research Report CMA-MR16-92, The Australian National University, Canberra,Australia, 1992.

[Kat1] KATO, T., Fractional powers of dissipative operators. J. Math. Soc. Japan 13(1961), 246-274.

[Kat2] --, Fractional powers of dissipative operators, II. J. Math. Soc. Japan 14(1962), 242-248.

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[Lio] LIONS, J .1., Espaces d'interpolation et domaines de puissances fractionnairesd'operateurs. J. Math. Soc. Japan 14 (1962), 233-241.

[Mel] McINTOSH, A., Operators which have an Hoo functional calculus. In JEFFERIES,B., McINTOSH, A., and RICKER, W.J., eds., Miniconference on operator theoryand partial differntial equations, vol. 14 of Proceedings of the Centre for Mathe­matical Analysis. CMA, ANU, Canberra, Australia, 1986, 210-231.

[McY] McINTOSH, A., and YAGI, A., Operators of type w without a bounded Hoo

functional calculus. In DOUST, I., JEFFERIES, B., LI, C., and McINTOSH, A.,eds., Miniconference on operators in analysis, vol. 24 of Proceedings of the Centrefor Mathematical Analysis. CMA, AND, Canberra, Australia, 1989, 159-172.

[Rob] ROBINSON, D.W., Elliptic operators and Lie groups. Oxford MathematicalMonographs. Oxford University Press, Oxford etc., 1991.

[Tri] TRIEBEL, H., Interpolation theory, function spaces, differential operators. North­Holland, Amsterdam, 1978.

[Yag] YAGI, A., Coincidence entre des espaces d'interpolation et des domaines de puis­sances fractionnaires d'operateurs. C. R. Acad. Sc. Paris 299 (1984), 173-176.

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