Topics in harmonic analysis, related to the Littlewood-Paley theory

TOPICS IN HARMONIC ANALYSIS Related to the Littlewood-Paley Theory BY

ELIAS M. STEIN

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1970

Copyright© 1970, by Princeton University Press ALL ItlOHTS RESERVED

L.C. Card: 72-83688 S.B.N.: 691-08067-4 A.M.S. 1968: 2265, 4201, 4750

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

PREFACE This monograph contains essentially the material presented in a course I

given during the spring semester of 1968 at Princeton University. My purpose in these lectures was two-fold: First, to give a new approach to that part of harmonic analysis which for .tl:re-sake of simplicity we refer to as the "Littlewood-Paley Theory." The techniques that are used lead to a wide generalization of results hitherto restricted to

an

and other special

contexts. My second aim was to give the interested student a rapid (although admittedly sketchy) introduction to various areas in analysis, in particular some elements of Lie groups, almost everywhere limit theorems in the context of martingales, and complex interpolation of operators. If I have succeeded in my two aims it is because the main tools used in Chapters III and IV come from martingale theory and interpolation theory, while interesting examples of the results may be obtained in the setting of compact and semi-simple groups. I am deeply indebted to R. Gundy for several enlightening conversations and to C. Fefferman for his great care and effort in preparing the lecture notes.

CONTENTS Preface •••....•••..•..•.•.•....•... '· . . . . . . . . • . • • • • • • • . .

v

Introduction . . . • . • • • • • • . . • . . . . . • . • •.. . . . . . . . . . • . • . • • . • . • .

1

Chapter I. Lie Groups (A Review) §1. Compact groups .•..........•.....•....-............

5

§2. The Peter-Weyl theorem ........................ , . . 12 §3. The Peter-Weyl theorem (Concluded) ..........•••... 15 §4. Lie groups; examples .••••.••.........••.•••.••... 20 §5. Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §6. Universal enveloping algebra •.•................•••. 28 §7. Laplacian ......••••.•..••..•.•.....•..•.••...... 33 Chapter II. Littlewood-Paley Theory for a Compact Lie Group §1. The

heat~iffusion

semi-group ...................... 38

§2. The Poisson semi-group; the main theorem . . . . . . . . . . . 46 §3. Proof of Theorem 2. . • . . . . . . . . . . . . . . . . . . • . . . . • . . . . . 50 §4. Applications: Riesz transforms, etc.

. . . . . • . . • . . • . . . 57

Bibliographical remarks .••..••.......•••.•......... 64 Chapter III. General Symmetric Diffusion Semi-groups §1. General setting .....•.•..•.•.........•.....•....... 65 §2. Analyticity of these semi-groups .........•••......... 67 §3. The maximal theorem .•.••.•••.......•............. 73 §4. A digression: L 2 theorems ••••.......•.•.•.••••••.. 82 Bibliographical remarks . . . . . . . . . . . . . • . . . . . . . . . . . . . 88

vii

viii

CONTENTS

Chapter IV. The General Littlewood-Paley Theory §1. Conditional Expectation and Martingales .....••••.....••. 89 §2. The inequalities for martingales ................. , . . . . . . 94 §3. An additional "max" inequality ••..•...•....•••..••••• 103 §4. The link between margingales and semi-groups

, , .. , .. , .. 106

§S. The Littlewood-Paley inequalities in general ..........••. 111 §6. Denouc_ement ........•.•... , •..........•.•.•..•••..... 120 Bibliographical remarks .....••.•..•.....•....••.•••••. 122 Chapter V. Further Illustrations §1. Lie groups ..•.•••..•...........•.••..•.•....•..•••.. 123 §2. Semi-simple case .......•....•••..•.••......•... , ..... 128 §3. Sturm-Liouville ••.••..••.....••....•••••..••••..• , .••• 136 §4. Heuristics ••..•.............•.•••..•..•...••••........ 137 Bibliegraphical remarks •••..••••...........••••.•..•• 141 References

143

TOPICS IN HARMONIC ANALYSIS Related to the Littlewood-Paley Theory

INTRODUCTION

Background

We shall use the phrase "Littlewood-Paley theory" rather loosely, to denote a variety-of related results in classical hannonic analysis whose extension to a general setting is our mai.n goal. In its one-dimensional form the theory goes back to the 1930s and may be said to contain the Hardy-Littlewood maximal theorem, Hilbert transforms, the work of Littlewood-Paley in (161, and capped in effect with the multiplier theorem of Marcinkiewicz [35}. 1 This theory may be described in terms of the Poisson integral, which in R 1 is given by the family of transfonn ations f(x) .... .!. 11

f

oo

-oo

f(x- y)dy t2 + Y2

u(x, t)

As is well known the behavior of the hannonic function u(x, t) closely reflects the behavior of its boundary values. Now the main point of the socalled "complex method" is to pass to the analytic function F(z) whose real part is u and to exploit complex function theory to study F and thus f . An example of this far-reaching idea arises in the Hilbert transfonn, which gives in effect the passage from the real to the imaginary parts of the boundary values of F. The next stage of the development of this type of analysis which culminated in the 1950's, saw the primacy of complex function theory give way to real-variable methods, and this led to an extension of many of these

1

See also Zypund [20, Chapters 14 and 15].

2

INTRODUCTION

results to Rn 2 . Characteristic of these techniques were various "covering lemmas" and certain singular integral transforms, whose kernels had a quite explicit description, all rather specific to Rn.

Main results Our approach is essentially different from the above two and allows for a very 'general formulation of an essential core of the subject; it can be applied, in particular, to various new and interesting situations. Our starting point is the semi-group of Poisson integrals, that is we assume that we are dealing with a family of operators !Ttlt> 0 , defined simultaneously on Lp(M)

1 ~ p ~ "", with the'property thatTtl. Tt2 = Ttl+t2.

To= I,

and in. addition to the usual measureability in t , it satisfies the following basic assumptions: (I) (U)

Tt are contractions on Lp(M), i.e.,

II T~ll p :5 II fll P'

1 S P :5 ""·

Tt are symmetric, i.e., each Tt is self-adjoint on L 2(M).

(Ill)

Tt are positivity preserving, i.e., T~ ~ 0, if f?: 0.

(IV)

Tt(l) = 1.

We refer to the above as symmetric diffusion semi-groups. The task we set ourselves is to develop, as far as is possible, the analogues of the Littlewood-Paley theory in the context of these semigroups. The interest in this arises from the multiplicity of examples of symmetric diffusion semi-groups and the consequence this theory has for the eigenfunction expansions of their infinitesimal generators. Besides the usual Poisson integrals for R 0 a variety of examples may be found in Chapter 2, Section 2 of Chapter 3, and in Chapter 5. The main tools that are used are three-fold:

2 See the bibliographic indications at the end of Chapter

2.

3 It is to be noted that the assumptions (I)- (IV) are to some extent

redundant.

3

INTRODUCTION

(i) The spectral representation in L 2 (M) of Tt as Tt = [ 000 e-At dE{A.),

for an appropriate spectral family E(A). This is, of course, the direct substitute for the Fourier transfonn in Rn. (ii) Connections of the semi-group Tt! and certain auxiliary martingales

and ergodic theorems. (iii) Convexity properties of holomorphic families of operators which allow one to mediate between (i) and (ii). '\

A curious fact that should not be ov-erlooked'J,s that the theorems for martingales required in the technique (ii) were to a significant extent already anticipated by Paley in his paper[ 17] dealing with the Walsh-Paley series. Among the results we obtain are: The maximal theorem (in Chapter 3), to wit II sup ITtf(x~ lip< Apllfll t

>0

-

p

,

In Chapter 4 we prove the Littlewood-Paley type inequality

r. ~~F'I dr1/

1(

2

Apll'ip •

l

HI T

£

=

A. 2 ¢>1 are orthogonal.

We shall use the following spectral theorem for Hilbert-Schmidt operators. THEOREM. Let T be a self-adjoint Hilber-Schmidt operator given by a kernel K(x, y), let A. 1 , ~' •.• be the non-zero eigenvalues, counted according to their multiplicies (the multiplicity of an eigenvalue is the dimension of its eigenspace). Then I A. 2 < oo: Let ¢>1' ¢> 2 , ••• be an orthonormal sequence of eigenvectors such that Ti

ij(x) (see statement (4) of the Peter-Weyl theorem). Now f(x) while

f at G

= faE G

f(axa- 1)da ,

1 ) da = d- 1 8. . )( (x). Details are left to the reader. lj>~(axa1J a 11 a

These observations show that although the theory of compact topological groups is not essentially commutative, there is an important commutative part of the theory, namely the study of central functions and characters. Fourier analysis will be especially concerned with this part of the theory.

Exercise for the reader: Show, using Schur orthogonality, that a f.d. representation is uniquely determined by its character.

Section 4. Lie groups; examples. The purpose of the next three sections is to sketch some portions of the theory of Lie groups which we shall need later on. Useful references are: 1. Nomizu, Lie Groups & Differential Geometry, [ 4]. 2. Chevalley, Theory of Lie Groups, [1]. 3. Pontrjagin, Topological Groups (1st and 2nd editions), [6] and [7]. 4. Helgason, Differential Geometry and Symmetric Spaces, [2].

21

§4. LIE GROUPS; EXAMPLES

A Lie ~roup is a group G, which is also a COO-manifold, such that the group operations (a, b) c G x G .. ab £ G and a .. a- 1 are CC"-functions. It can be shown that every Lie group has a real-analytic structure which

makes the group operations real-analytic . . Two Lie groups G1 x G2 are isomorphic (G 1 ~ G2) if there is a group isomorphism j: G1 -. G 2 (onto) which is also a diffeomorphism. G1 and G2 are locally isomorphic (G 1 "' G2 ) ir' there are neighborhoods N1 f G1 and N2 f G2 of the identity and a diffeomorphism j: N1 ... N2 (onto) such that a. If x, y, x · y c N1 then j(x)j(y) = j(xy) ; b. If x, y, x · y £ N2 then j- 1(x)j- 1(y) = j-1(xy). For example, the real line R 1 is locally isomorphic to the circle group. We shall use the notion of local isomorphism to classify compact connected Lie groups. First we give the easy part of the classification. THEOREM.

In each equivalence class of locally isomorphic connected

Lie groups, there is a unique simply-connected group G. Every group G in the equivalence class is of the form G = G/Z, where Z is a discrete central subgroup of G.

Conversely, G = G/Z is locally isomorphic to

G

if Z is a discrete central subgroup.

G is called the rmiversal covering group of

G.

The theorem is proved by picking any group G of the equivalence class and setting

Gequal to the universal covering space of

G. It is very

easy to impose a group structure on G. Details may be found in Chevalley [1). Notice that the fact that Z is central is immediate from the observation that the fibre Z is a discrete normal subgroup of G. For if z £ Z is arbitrary, aza- 1 will be close to z if a belongs to a small enough neighborhood of l.in G. On the other hand aza- 1 £ Z, which implies that aza- 1 = z for a

close to 1, since Z is discrete.

Gis connected and is

therefore generated by any neighborhood of 1. Hence aza- 1 = z for any a

£

G. Thus Z is central.

22

LIE GROUPS (A REVIEW)

Having determined the structure of each equivalence class of connected Lie groups, we are left with the immensely more difficult task of classifying (connected) Lie groups up to local isomorphism. For compact groups, the solution is given in terms of the following: 1. The circle group T 1 = R 1 /Z; more generally, then-torus Tn = T 1 x ... x T 1 (n factors). These are the only compact connected abeJian Lie groups (see Chevalley [l], p. 212-213). 2. The group SO(n)

(n ~ 3) of all orthogonal n-dimensional matrices

of determinant + 1. SO (n) is called the special

ortho~onal ~roup.

When n = 2k, SO (n) is called a Dk group; when n = 2k + 1, SO (n) is called a Bk group. 3. The special unitary group SU (n) (n ~ 2), the group of all unitary n-dimensional complex matrices of determinant 1. This gives the A -series of groups, An-t= SU(n). 4. The symplectic group Sp(n), the quaternionic analogue of the real group SO (n), and its complex version SU (n). More explicitly, let Qn denote quatemionic n-space, a vector space over the quaternion field Q. Recall that for each quaternion a= a 1 + a 2 i + a 3 j + a 4 k, the quatemion conjugate a is defined as a 1 - a 2 i - a 3 j - a 4 k. Using the quaternion conjugate, we can define the "inner product" on Qn by (a, b)=~= 1

ae he

for a= {al··· an}, b = {bl··· bn} ( Qn.

Sp(n) consists exactly of those transformations of Qn, linear over Q, and preserving the inner product. Sp(n) is also designated Cn. S. The exceptional groups E6 , E7 , E 8 , F 4 , G2 , which we cannot describe here. The classification of compact connected groups is given by the following result. THEOREM. Every compact connected Lie group is locally isomorphic to a finite product of the groups listed above. The proof is too difficult and long to be included here.

§4. LIE GROUPS; EXAMPLES

23

A few remarks are in order. First of all, the product of basic groups, mentioned in the theorem is unique, except for the following redundancies:

1. SO (3) "' SU (2)

~

2. S0(4)"' S0(3)

X

Sp (1) S0(3)"' SU(2)

X

SU(2)

3. SO (5) "' Sp (2) 4.

so (6)

"'

su (4)

Secondly, it is possl'~e to show that except for list have compact

univers~l

-rn, all the groups in our

covering groups. In particular, Spin (n), the

spinor group, defined as the universal covering group of SO (n), is compact. It can be shown that SO (n) = Spin (n)/Z 2 •

Finally, the reader is entitled to an explanation of the cryptic symbols Ak, ~· Ck, Dk, E 6 , E 7 , E8 , F 4 , G2 . The notation is based on the concept of rank. The rank of a compact Lie group is the largest integer k such that the group has a k-torus embedded in it. Equivalently, the rank of a group G is the highest dimension of any abelian subgroup of G. The groups Ak, ~· Ck, and Dk all have rank k, E 6 has rank six, and so forth. The importance of the classification theorem for us, is that it gives us a bird's eye view of what Fourier Analysis on compact Lie groups might be like. For among the basic groups 1-5, Tn is the setting for classical Fourier series, and on SO (n) much of the classical theory has been carried over.

Section 5. Lie algebras We shall (eventually) introduce the Lie Algebra of a Lie group G.

Re-

call first that if M is an n-dimensional C"" -manifold and p is any point of M, the tangent space at p, T p(M) is the n-dimensional vector space of all linear functionals L on C00 (M) which satisfy L(fg) = L (f) g (p) + f (p) L(g). L(fg) = L(f)g(p)+ f(p)L(g) . A vector field is a linear mapping X: C00 (M) .... COO(M) which satisfies the

24

LIE GROUPS (A REVIEW)

condition X(fg) = (Xf)g + f(Xg). This is equivalent to the usual definition. The bracket operation assigns to any two vector fields X and Y, a vector field [X, Y] defined by [X, Y](f) = X(Yf)-Y(Xf). The bracket operation is bilinear, anti-symmetric, and satisfies Jacobi's identity [X, [Y, Z]] + [Z,[X, Y]] + [Y, [Z, X]]

= 0.

The space of all vector fields is infinite-dimensional, and so too big for our purposes. We therefore restrict our attention to the space of all left invariant vector fields. The vector field X on G is called left inva-

riant if for any a£ G, A.(a)X = XA.(a), where A denotes the left regular representation of G. In other words, a left-invariant vector field commutes with left translations. The space of left-invariant vector fields is closed under the bracket operation. Every vector field X on a Lie group G determines an element X1 of the tangent space toG at 1, defined by X1 (£) = (X(£))(1) for f

£

C""(G).

It is easy to show that X -o X 1 is an isomorphism of the space of left-

invariant vector fields on G, onto the tangent space T 1(G). This isomorphism induces a bracket product [ · , · ] on T 1 (G). T 1 (G) with its bracket product, is called the Lie algebra g of the Lie group G. The process which defines the Lie algebra is essentially differentiation. In particular, a vector field, being a section of the tangent bundle is really nothing but a first-order differential operator on G. We shall define a process, analogous to integration, which takes us from the Lie algebra back to the Lie group. A family {cptl is called a one-parameter group of diffeomorphisms of

the n-manifold M if

1. cf>t is a diffeomorphism of M onto itself, for each t, -oo < t < +oo. 2. The map (t, p) ... cf>t(P) taking R 1 x M into M is smooth. 3. cf>t

0

cps

=

cf>t+ s and

cp 0

is the identity.

We can associate a vector field X to each one-parameter group {cptl, by setting

25

§s. LIE GROUPS

(Xf)(p) = lim f(cpt{p))- f (p) t ... 0

for f

E

C""(M), p

E

t

M.

The converse problem is not, in general, solvable. That is, given a vector field X, ther,e may not be any one-parameter group l!f>tl which satisfies (Xf )(p) = ~ f 0 if>t(p)l t= 0

•

But there does exist a local one-

parameter group for which it holds. More precisely, given any point p c M, there is a neighborhood N f M of p, an E > 0, and a family lcfot II tl =5 E of mappings, defined only on N, such that l '. cfot is a diffeomorphism of N into M, for each t (j tl < E ). 2'. (t, q)

-o

cfot(q) is smooth.

3'. If !t11 < E, !t 2 1 < E, !t1 + t 2 1 exp(t 1 X1 + ··•·+ tnXn) f G; the system is called the canonical co-ordinate system with respect to the base f

xl ... xn. The following formulas are useful for the application of canonical coordinates. 1. (Taylor's formula) If f: G -+ R is real-analytic, and x f G, then 00

f(x · exp tX)

=I

t~ (Xnf)(x), for X in a neighborhood of zero in g.

n-o n.

2. For X, Y f g , exp tX · exp Y = 2 = exp(t(X+ Y) +.!_[X, Y] + O(t 3 ))

2

3. Fc;>r X, Y E g, (exp tX)(exp tY)(exp tx)- 1(exp tY)- 1 = exp (t 2 [X, Y] + O(t3 ))

as t ... 0.

as t ... 0.

§s. LIE GROUPS

27

Formula 1. follows if we apply Taylor's theorem to the auxilliary function F(t) = f(x · exp tX), keeping in mind that (d/dt)f(x·exptX)\t=O = (Xf )(x) by definition of the exponential.

To prove formula 2., we select any real-analytic function f defined on G. Using formula 1., we can easily check that the Taylor expansions of .

2

t ... f(exp tX · exp tV)··and t ... f(e~p(t(X+ Y) + ~ [X, Y])) agree up to second order.· Since·.f is essentially arbitrary, we conclude that 2

exp tX · exp tY and exp(t(X+ Y) + t2 [X, Y]) differ by a third-order quantity, in the obvious sense. Formula 2. now follows from the fact that the exponential map is a local diffeomorphism. Finally, fonnula 3. is a trivial consequence of formula 2. Details of these hastily sketched proofs are left to the reader or to the cited literature. Let us now consider some examples. First of all, let G = GL(n, R), the general linear group. Although G has two connected components, the Lie algebra and exponential map still make sense for G. Since G is an open subset of the vector space M(n, R) of all real n x n matrices, we can identify the tangent space at I

£

G with M(n, R). Thus, the Lie alge-

bra of G is canonically isomorphic to M(n, R), as a vector space. To comthe exponential map, we shall find all one-parameter groups in G. If t .... N (t) is a one-parameter group, then of course N (s + t) = N (s)N (t). Differentiating this equation in s, we obtain

~t! (t) dt

=

I

AN(t), where A = dN (t) dt t= 0

£

M(n, R).

This differential equation, together with the initial condition N (0)

= I,

has only one solution. 00

N(t)

= etA

=I

tn An . n=O n!

Therefore the one-parameter subgroups of G are t ... etA, A £ M(n, R).

28

LIE GROUPS (A REVIEW)

An easy computation shows that the tangent vector (i.e., element of the Lie algebra) induced by etA is just A. So the exponential map must be given by exp(tA)

= etA,

i e., exp(A)

= eA,

whieh justifies the name

"exponential." Fonnula 2. above, now shows that the bracket operation on the Lie algebra M(n, R) is just [A, B] = AB - BA. Our next example is G = SO (n), the special orthogonal group. Since G is a suhgroup of Gl (n, R), it follows from the first example, that the Lie algebra is a subalgebra of M(n, R) (given the natural bracket product [A, B] "" AB - BA). In particular, the one-par~eter subgroups of SO (n) are all of the form t -. etA. But from elementary linear algebra we know that eA is orthogonal if and only if A is skew-symmetric, i.e., (Ax, y) = - (x, Ay), So the Lie algebra of SO(n) consists of all real skew-symmetric n x n matrices, under the natural bracket product; and the exponential map is exp(A) ... eA. Both for GL (n, R) and for SO (n), we see vividly that the exponential map is a local diffeomorphism. For comic relief, we consider the examples G = ~ and G .. Tn, the n-torus. Since Rn and ~ are locally isomorphic, they have the same Lie algebra. We leave to the reader the task of verifying that the Lie algebra of Rn is Rn with the bracket product [X, Y)

=0,

and that the expo-

nential is the identity map. The exponential map for Tn is the natural projection of Rn onto Tn. Note in passing that in Tn, some one-parameter groups are closed, while others are not. In Rn, however, all oneparameter subgroups are closed.

Section 6. Universal enveloping algebra As we have already seen, the Lie algebra of a group G consists exactly of all left-invariant first-order differential operators on G which anihalate constants. We shall now study the universal enveloping algebra of G, which is just the (non-commutative) algebra of all left-invariant differential operators on G.

§6. UNIVERSAL ENVELOPING ALGEBRA

29

To be precise, let M be an n-manifold. A kth·order differential operator on M is a linear mapping D: CI)O(M) -+ CI)O(M) which can be written in terms of local co-ordinates (x1 · · · xn) for M defined in a neighborhood of p l M, in the form (Df)(x 1 •·· xn) for t

£

=

Ilal ~k aa(x 1 ••• xn) ::.~ (x 1 •·· xn)•

CI)O(M), where aa are fixed cr"' -functions defined in a neighborhood

of p. A differential qperator D on a Lie group G is said to be left- invaril61t if D(Aaf)

= Aa(Of)

for any a £ G, f £ C""(G); where Aa is the left-

regular representation. The left-invariant differential operators on G fonn an algebra (seldom commutative), which we denote by ~(G). For the third time, we note that every X belonging to the Lie algebra g is a left-invariant differential operator, i.e.~ belongs to ~(G). It can be shown that g generates the algebra ~(G). In fact we shall prove a far stronger result: Regard g as a vector space, and let Tg denote the (non-commutative) tensor algebra of g, i.e., Tg ...

Ik=

0 sk

g. Equivalently, Tg is the algebra of real polynomials

in the non-commuting variables X1 · • • Xn, where the Xi form a base for g. Let g(G) denote the two-sided ideal in Tg, generated by all expressions X® Y- Y ®X- [X, Y) where X, Y £g. The quotient algebra Tg,1(G) is written U (G) and is called the universal enveloping algebra of G. We shall exhibit a canonical isomorphism of ~(G) with U (G). THEOREM: g

f Tg-+ U (G) is an injection of g into U (G). If we iden-

tify g f U (G) with g f ~(G), then the resulting correspondence extends uniquely to an isomorphism of U (G) onto ~(G).

Proof: We can easily define an algebra homomorphism from Tg into ~(G), by mapping X® Y ®

••• ®

W into the differential operator

f-+ X(Y(···(Wf)) ··· ), for X, Y, ... , W £ g . By definition of the bracket product, our homomorphism sends X s Y- Y s X- [X, Y) to 0 .

30

LIE GROUPS (A REVIEW)

Hence we obtain a homomorphism j of algebras from the quotient U (G) into :D(G). LEMMA: Let X1, ... ,Xn be a base for IJ, and let m = (m 1, ••• ,mn) be a multi·index, lml = ~~= 1 lmkl· Define X(m) f U(G) to be the coefficient of

t~l

··· t:;'n in the expression

I~~ ! (X 1t 1 + ••:.·+ Xn tn)l ml

(the t's are

supposed to commute with each other and with the X's, but x. and x. do •

J

1

not commute). We set X(O) = 1. Thus X(m) is a "symmetrization" of m m X1 1 ··· Xn n . Then the elements X (m) span the vector space U (G). Proof: We must show that each monomial Z = Xk Xk · ·· Xk 1

2

e

l

U (G)

is in the span of the X(m). This is clear for f ~ 1, and we use induction on f, the degree of Z. Suppose that appearing in Z are m1X1 's, m2 X2 's, ... , mnXn's. Set m = (m 1, m2 , ••• , mn), so that lml =f. It is not difficult to see that Z

=

X (m) +

lower~egree

terms, in U (G). (This is because

X (M) is a linear combination of all monomials obtained by rearranging the order of the Xk. within

z.

But any such monomial Z' differs from Z by

1

terms of degree lower than that of Z. For example, consider the monomial Z'

=

xk2 xk1 xk3 xk4 xjs ... xkf . Then

which has degree f- 1. The case of a general Z' is handled in the same way.) Since the lower degree terms are in the span of the X(m) by indue· tive hypothesis, we have proved that Z is in the span of the X (m). This completes the proof of the Lemma. Now consider j(X (m))

E

QED

:D (G). We show next that these elements are

linearly independent. This will show that the X(m) form a base for the vector space U (G), and that the map j is injective. To prove the linear independence, we use canonical co·ordinates for the base X1 ·· • Xn of g. Let f: G .. R be any real-analytic function.

31

§6. UNIVERSAL ENVELOPING ALGEBRA

Then in the spirit of formulas 1., 2., and 3. above, we can show that F(tl' ... ,tn)

=f(exp(t 1X1 +····+ tnXn)) •Imtm(jX(m)f)(x).

Hence, in

canonical co-ordinates (t 1 , •.• , tn), the elements jX(m) E ~(G) correspond to the operators al ml ;attl at:2 ... i and the latter are obviously

at:n

linearly independent. So the jX(m) are also linearly independent. To finish the prdOf of the theorem, we have only to show that j is onto. Nothing could be·simpler. First of all, a left-invariant differential operator D is uniquely determined by the functional f ... (Df)(l). On the other hand, a monent's thought !eveals that we can write (Df)(O)

=

I

aaXJf

\al::;k

o ••• o

1

(where aa are fixed constants, and Xka

E

Xf a (0),

11

g) for any differential operator

i

D. So if D is left-invariant, ~

Df"'

I

1

lkisa Thus D f j (U (G)), so that j is onto.

QED

The center Z (G), of the universal envoloping algebra ~ consists of all those differential operators D which are bi-invariant, i.e., commute with both the left and the right regular representation. For suppose that D is hi-invariant. To show that D E Z (G), we need only check that D commutes with all X f g, since g generates the universal enveloping algebra. Any X

f

g may be written in the form (Xf) (x)

= d~

(x · y (t ))\t= 0 , where

y(t) is the one-parameter group corresponding to X. Therefore (X

o

D)f(x)

= _E_ dt

= _c!_ D (f(x · y(t))l

(Df)(x · y(t))l t=O

dt

t=O

,. D ... 0 for k ~ 1). The space @ consists exactly of all finite linear combinations of the cpi. If f =I ai cpi l@ (ai finitely non-zero) then of course M =I- ILiaicpi. This suggests a definition for Tt = etfl: for f = I a i cpi

l

@, we set

Ttf =I e-ILitaicpi . Obviously IITtfll~ = Ii le'""1Litail 2 .$ Ii lai1 2 = II fll ~ for f l @, since "-i ~ 0 and t ~ 0. Thus Tt extends from @, to a bounded linear operator on L 2 (G) of norm 1-the extension is also denoted by Tt. The reader may check that !Ttl is a strongly continuous semigroup of self-adjoint operators on L 2 , Ttl = 1; and if f

l

@, the function

41

§1. THE HEAT-DIFFUSION SEMI-GROUP

u(x, t) = (Ttf)(x) belongs to C00 (G x (0, t)) and satisfies the heat equation. Hence, the L 2-theory of the operators Tt is well in hand. Next, we show that Tt is a positive operator. From this fact the Lpproperties of Tt will be easy. Define the resolvent R(.\, 11) to be the operator (.\1 -11)- 1, for .\ > 0. The analysis of R(A, 11) trivializes here, for f "" I ai ¢i(x) t li;, R(.\,11)£ =~ - 1- ai¢i(x) ; (.\+~t;>

which shows when.\> 0, R(.\,11) is a bounded operator on L 2 (G). This is typical of the usefulness of li; in avoiding all technical difficulties. A standard fact from semigroup theory is that each Tt is positive if

and only if R(A, 11) is positive for .\ > 0. On the one hand, for

(A)

f£

&,

since we can write f = ~ ai ¢i(x) (finite sum) and then equation (A) reduces to

~ e-~tita·¢·(x) 1

1

... lim

n .... oo

So if R(.\,11) is positive, then (

~(

n/t ) n a·¢·(x) , (n/t)+ILi) 1 1

f R( ~. 11))n is positive, which implies

that Tt is positive. Similarly, that T\?.. 0 implies R(A, 11) ?.. 0, follows from the identity (B)

R(.\,11)£ •

foo e-.\t Ttf dt

(f f ff,),

0

which in tum comes from the same kind of routine computation as (A).

&. Let f f li;, and suppose that f?.. 0 and R(.\,11)£ = g £ li;, Then .\g-l1g = f ?.. 0. We must We can now show that R (.\, 11) is positive on

show g?.. 0. If this did not hold, then at the point x 0 £ G at which g is a minimum (recall that G is compact) we have g(xo> < 0. On the other hand,

42

LJTTLEWOOD-PALEY THEORY FOR A COMPACT LJE GROUP

!ig(x0 ) ~ 0 since g takes its minimum at x 0 • Hence A.g(x 0) - !ig(x0 ) < 0, contradicting f ~ 0. This completes the proof that R(A., ~) and therefore Tt is positive. We have used tacitly the fact that R(A,

til f is real if f is

real, which we leave as an exercise to the reader. Since the positive operator Tt:

& -. & maps

1 into 1, it follows that

Tt extends to a positive Tt: C(G) -. C(G) of .norm 1. For fixed t > 0 and x0

E

G, .the positive linear functional f

f

C(G) ... (Ttf)(x0 ) is of the

form

where ~~

o

is a positive measure with total mass 1, by the Riesz repre-

sentation theorem. On the other hand, the operator Tt is hi-invariant, since for f E &, Ttf

=I

means that Ttf(xol = (f

tn!inf/n!. Therefore IL~ (E)

* l!i)(x 0 )

0

= IL~ (x01 E),

which

for every f E C(G). Since

we have verified property i) in the statement of the theorem. Property iii) follows from property i) and the density of 6; in Lp. It remains to check properties viii) and ix). Now ix) is clear in the case f E &, from which we deduce by a routine limiting argument that u (x, t) (Ttf)(x) satisfies the heat equation, for any f

£

Ll' once we have proved

viii). To prove viii) we require a simple form of the Sobolev lemma; which we state as an a priori inequality: Let f

£

=

-NII 2

~

K

1\(l + IYI)N f(y)ll 2

I 111Yiaf(y)ll 2 = K I II lal ~N lal ~N

:a

a

£112

Next, we shall extend our Lemma, and transfer it to the setting of the compact Lie group G: LEMMA.

(a)

Let f

€

C00 (G). Then

llflloo

~

N

~ ll~ff\12

A

f= 0 where A depends only on G, and N is any integer > n/2. To prove this, first observe that (b)

where Xk belongs to our basis for the Lie algebra g . For II M 11 2 11 f ]\ 2 2: -(M,f)

=

Iij aij(Xjf, Xif) 2: CIIXkfll 22 by the strict positivity of (aij).

Combining (b) with the inequality 2ab ~ a 2 + b 2 , we obtain II Xkfl\ 2 ~ C (\1M 11 2 + II f 1\ 2). Repeated application of this inequality yields N

IIP(X 1 ·•• Xn)f11 2

~ C~ ~ ll~ffll 2 f= 0

for P(X) in the universal enveloping algebra of G, of degree

~

N. To com-

plete the proof of (a), we have only to show that II fll 00 ~ IPf All P(X 1 • ·• Xn)~l 2 for some finite A of P(X) of degree at most N. If f has small enough

44

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

support, this follows from our Sobolev lemma and an application of canonical co-ordinates-the general case then follows from a partition of unity argument. Thus, inequality (a) holds. '

Let us apply the above estimates to the study of the operator Tt and

ist eigenvectors cpk. Since !1~

=-

p.i cpi and ~ cf>iil 2 = l, inequality (a)

shows that supxeG lcf>i(x)l .S C(l + IP.ii)N for any N > '!:: dim G. Consider any f

= I ak cpk(x) e &, and let u (x, t) = Ttf(x) = I e'iLkt ak cpk(x). Then

tie u (x, t) "' I

('iLk)e e "'iic:t ak cpk(x), so that

ll!ieu(·,t)il~ =Ip.~ee- 2 P.ktlakl 2

.s Cc 2ei

lakl 2 = ct- 2ellflli

(We make use of the elementary inequality p. 2ee- 21Lt _s

cc 2e,

valid for

~ 0 ). We have proved an a priori estimate for llt1eu ( · , t )11 2 • Since (a/at) u (x, t) = !1 u (x, t) for f £ &, we also obtain the a priori inequality

p., t

where C and M depend on k and

e alone.

By the Sobolev lemma and

familiar limiting arguments, u(x, t) = (Ttf)(x) belongs to C"(G x (0, oo)) for every f

l

L 2 (G), and the map f .... u(x, t) is a continuous operator from

L 2 (G) to C (G 00

X

(0, oo)).

Property viii) above is now easy to prove. For, as a very special case of what we just showed, we have Tt: L 2 (G) .... L00 (G) is a bounded operator. The usual duality argument shows that Tt is also bounded as a mapping from L 1(G) into L 2 (G). But then Tt = Tt/ 2 Tt/2, the composition of continuous operators from L 1 .... L 2 and from L 2 .... C"'. Thus, for any f £ L 1(G), Ttf(x) l COO(G x (O,oo)), which completes the proof of Theorem 1. QED We can even give an explicit representation for the operator Tt, in terms of the docomposition L 2 (G) = IalA E9Ha. As usual, let us select a unitary representation lcpfj (x)l of class a, for each a £A, and use as

Y: c/>ij a Ia,i,j• where our orthonormal base {cpil' the family of vectors I da·

45

§1. THE HEAT-DIFFUSION SEMI-GROUP

the factor d~ ... (degrees a)lf;. is put in to normalize vectors. The eigen· value of !':! corresponding to

d~ cpfj is >.a in our previous notation. Hence

Tt(d~ cp~j) • e-Aatd~ ¢\j, so we can write formally, Ttf(x)

(c)

=

f

(Ia e->.at da Xa(xy-I)) f(y)dy ,

G

since

-A t 'h. a d a

a (x) cf>·. lJ

I-l,J,a . e

'h. cf>·. (y) = lJ

· da

~

k

a

-.>.a t d

e

a

~

k· .

l,J

cf>·alJ.(x) ¢·aJl.(y-1 )

Ia e-'Aat da Xa(xy- 1), which follows because lcpfj(x)l is unitary, with character Xa·

We shall verify that this identity holds, not furs formally,

but literally, by showing that the series

converges in the strongest possible sense. Namely any order partial derivative, with respect to x and t, of the series Ia e-Aatda Xa(x) converges absolutely absolutely and uniformly for x

E

G, t

> o > 0. In fact, since f .... l':!e Ttf(l)

is a bounded linear functional on L/G) for each f and since (c) holds for· mally, we have

aEA for each N > 0. Since t

> 0 is arbitrary, da ~ l, and

:xa

has ck-norm

at most some fixed power of 'Aa (again by our form of the Sobolev lemma), we conclude that sup

~

.-4

s~chOaEA

e

-'As a da11Xa11 k C(G}

for each k, which implies the desired conclusion.

(x)dx,

where the supremum is taken

over all positive CO" functions ¢> such that II¢> II q

1

g(f) 2 (x)¢>(x)dx

(*)

G

:S Ap[

•

f

f(x) 2 ¢>(x)dx +

G

:S 2. Assume II f II P = l.

:S I. Our plan is to prove

f

f*{x)g(f)(x)g(¢>)(x)dx]

G

From this inequality part II is not hard. The reason is that the left-hand side involves g(f) 2 , while the right-hand side involves only g(f).

f*(x)

and g(¢>) are under control, the latter by part I and the fact that II¢> llq :S L So the size of g(f) will be controlled. The crucial step in proving (*) is to make the estimate

(**)

J

(g(f)(x)) 2¢>(x)dx =

G

1 A!J

/oo 0

:S

tl 'f u(x,t)1 2 ¢>(x)dxdt

G

0

tl 'fu(x,t)1 2¢>(x,t)dt

G

where ¢> (x, t) is the Poisson integral of ¢>. To prove (**) we will use a "subharmonicity" argument. We remark first, and this is basic, that our Laplacian !':! is hi-invariant, and so commutes with the Xj. Thus any function of it also has this property, and so in particular for Tt = et!:l, t > 0, and the Poisson semi-group. This commutitivity is obvious on the formal level and immediately justifiable on the subspace

&. The passage to general

Coo

functions is then by

a routine limiting argument Since I Ptl is a semi group, u (x, s 1 + s 2 ) = Pslu (x, s 2 ); this shows that Xju(x, s 1 + s 2 ) Hence Xju(x, t)

=

S

dU au as (x, s 1 + s 2 ) = P 1 ~x, s 2 ).

X;u(x, t/2) and

a~

P lX;u (x, s 2 ) and

= pt/ 2

S

u(x, t)

=

Pt/ 2 :tu(x, t/2).

53

§3. PROOF OF THEOREM 2

)

Next, note that since I~ ui 2 = Iij aij (Xiu)(Xju) + ( g~ 2, we can, by a change of bases in the Lie algebra, assume that

i~ui2 =I (X.u)2+(aa u)2. i

t

1

So if we prove that IXju(x, t )1 2 ~ pt/2q Xju(x, t/2)1 2) and similarly for

(t)l ~ M~-tM~ .

QED.

Now let f be a simple function with II flip= l. To show that IIU(t)fltp ~ M~-t M~, we need only prove that

I/'Jn

(*)

(U(t)f)gdxl

~ M~-tM~

for every simple function g for which II g II q ~ l, where q is the exponent conjugate to p. Inequality (*) follows from the three-lines lemma if we set up the right functi~n Ill. In fact, we can write f "' F f and g = G i- where F, G ~ 0 and lf(x)l fz

=F

= li-(x)l =

l for almost all x. Put

p ( 1-z + _.!.) Po p1 f

and

gz

=G

q ( 1-z + .!.._) Qo q 1 g.

where q0 and q 1 are the exponents conjugate to p0 and p 1 , respectively. Obviously ft = f, gt = g; II fzll , II gz II ~ l for Re z = 0, and Po

qo

\\fz\lp 1 , \\gzllq 1 ~ l for Rez = l. Hence the bounded analytic function tl»(z)

=

1m

(U(z)fz)gzdx

satisfies l«(z)l ~ (norm of U(z) as an operator on Lp 0) ~ M0 if Re z

=

and similarly \«(z)l ~ M1 if Rez .. 1. The three-lines lemma now shows that

This completes the proof of (*).

QED.

More detailed expositions of convexity theorems may be found in Stein [22], Dunford-Schwartz Linear Operators [9, Chapter 6, Section 10] and in Zygmund Trigonometrical Series, Vol. II, Chapter XII, [20). Notice that the above argument has to be patched up for p difficult.

= + oo ;

the problem is not

0,

71

§2. ANALYTICITY OF THESE SEMI-GROUPS

Now we can return to semigroups and finish off the proof of Theorem 1. Recall that we have defined a family of operators {Ttl for complex t in the right half-plane Ret

~

0, satisfying the properties

(a)

I\Ttfll 2 :5l\fll 2 (all t intherighthalf-plane).

(b)

If f, g f L 2 then t

-o

~ ( Ttf) g dx is an analytic function of t,

bounded in Ret > 0 and continuous in the closure.

We want to interpolate between (a) and (c). So let 71 > 0 be arbitrary, let

-77/2 < (} < 77/2, and define U(z)f

=

T71ei(Jzf. By (b) above {U(z)l is

an analytic family of operators, in the sense of the hypothesis of the above convexity theorem. Furthermore, (a) shows that II U (z)f 11 2 :5 II f 11 2 for Rez ~ l, and (c) shows that i[U(z)f[[ 1 :5 llfli 1 :5 llfll 1 for Rez = 0. Therefoie, by the convexity theorem, IITtfllp :5 llfllp (l < p < 2) where t = 71e

t

i0(2-2)

p . Hence liT flip :5 llfllp whenever \arg t I < !! (2- ~) • 1L - 2 p 2

0-1

~

p

-

1[)

'

for 71 > 0 and (} c [- ~, ~] are arbitrary. We can prove an analogous result for 2 < p < +oo by interpolation between L 2 and L00 •

It remains to show that

is bounded, analytic, and continuous in the closure of the sector Sp, for f

£

Lp• g l Lq. This follows from (b) when f and g are simple, so that

letting I fkl (respectively

I gkl) be a sequence of simple functions tending

in LP (respectively Lq) to f, (respectively g), we find that Jm(Ttf)gdx is the uniform limit of the analytic functions

QED.

72

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

As an application of Theorem l, we can show that if f

t

Lp (l < p < +oo)

then for almost every x, t ... Ttf(x) is a very smooth function on (0, "")· This is essential is we are to define a Littlewood-Paley g-function involving

dfat Ttf(x).

LE'MMA: Let f

t

LP(m), 1 < p

< +"".

For each t, we can redefine Ttf

on a set of measure zero, in such a manner that for every fixed x, Ttf(x) is a real-analytic function of t

f

(O,oo),

. Proof: By Theorem 1, the function t -+ Ttf t Lp extends to an analytic

Lp-valued function on the sector Sp. Now, our definition of an analytic function ell from a region 0 s;; C to a Banach space B was that for each continuous linear functional L on B, z-+ L(ell(z)) is a complex-valued analytic function on 0. But it is a standard fact of functional analysis that this definition of analyticity is equivalent to any other "reasonable" definition imaginable, for example. ell is continuous and satisfies Cauchy's integral formula; lim .1.z -+ o

If z 0

ell (Z+.1.z)- ell(z)

t

Az

exists in the norm topology on B.

.1.(z 0 , E)

s:; 0, then ell has a power series expansion 00

Cll(z) ""

!.

bk(z-z 0)k where ~

f

B

k= 0

valid in .1.(z0 ,

€)

and such that

Ik= 0 j~j rk < +oo

for any r < €.

We shall use the last definition of analyticity. So for any t 0 Ttf =

Ik=

0

> 0,

fk(t- t 0 )k, for all t in some neighborhood Mt 0 , 2€), and

Ik'... o jjfkJipek < +oo. Each fk isanequivalenceclassoffunctionspick a particular representative, which we also denote fk. For t

i

we can modify Ttf on a set of measure zero, in such a manner that 00

Ttf(x) =

I

k=O

fk(x)(t- t 0)k

(every x).

ll (t 0 , E),

73

§2. ANALYTICITY OF THESE SEMI-GROUPS

This makes sense, because Ik'= 0 £ k I fk(x)l ~ + oo for almost every x, as follows from Ik=O£kllfllp < +oo. Now cover (0, oo) with countably many neighborhoods .1.(t 0 , £). The rest is quite tirival, and details are left to the reader.

QED.

Section 3. The maximal theorem. Our first main result is the following: MAXIMAL THEOREM: Let the semigroup !.Ttl satisfy (I) and (II}. Then (a) the maximal function, defined by f*(x}

=supt> 0 ITtf(x)l,

satis-

fies the inequality

1 < p (b) if f

£

~

+oo ;

Lp("l), then lim Ttf(x) = f(x) a.e.

(1 < p 0

I!. J s

s Ttf{x)dt]

0

S Mf(x) + g1(f){x) so that

by the Hopf-Dunford-Schwartz ergodic theorem and the L 2 -boundedness of the g-function. This completes step 1 of the proof. Step 2: For f ! Lp and k

~

0 define

f*0 is the same as the maximal function f*. We shall show that for each k ~ 0,

llfkll2 S AkilfJJ2 ·

The proof copies the argument of step 1. In fact, integration by parts shows that

76

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

so that by Htllder's inequality,

where

The first term on the left-hand side of this inequality has already been estimated in step 1, where we showed that

So in order to complete step 2, for the maximal function ~ , we need only

show that hi011 2 ~ A llfll 2. This follows from the spectral theorem in a manner analogous to the proof in step 1 for II g1(f) 11 2 ~ A II f 11 2. Thus II~ 11 2 ~ A II f 11 2 . The general case II £k 11 2 ~ Akll f 11 2 follows by induction. We begin the inductive step by computing t /

0

sa k+l

Tsf(x)ds

ask+l

by parts. Details are left to the reader.

Step 3: So far, we know that for p > 1 · (think of p very near to 1), and that

II

k sup tk ak Ttf(x)ll2 t >o

at

~ Akll£112

foreach k 2:0.

Clearly, in order to interpolate between these inequalities, we should

77

§3. THE MAXIMAL THEOREM

search for an analytic family of operators f1 which act on functions of one variable t

£ ( 0, oo ),

such that

1 t

I 1(f)(t) =

f(s)ds,

and

0

The formal computation

In(f)(t) =

Jtf 0

s... Jr f(r) dr ... ds dt "'

0

0

1

~!

f

\t- s)n-l f(s)ds

0

suggests a reasonable definition for fl: (a

£

C).

If Rea > 0 then the integral defining Ia converges absolutely for

f

£

L 1(0, ""), but if Rea

~

0, the integral need not be defined. This is as

it should be, for 1-k is supposed to be the k-th derivative. fl(f) is

called the a·th fractional integral of f. To justify our definition of fractional integration, we shall prove the LEMMA:

Let f be in

defined for Rea

> 0,

L1

n C""

on (0, oc), Then the function a .... fl(f)

has an malytic continuation to all of C. F~rther­

more, the functional equations Ia If3f = ~ + f3c and I 0 f = f hold. Pr 0. Write

~

is analytic,

78

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

~(f)(t)

f

= - 1r(a)

f

'h

(1-s)a- 1 f(st)ds + _1_ 0 r(a)

1

(1- s)a-lf(st)ds

¥..

=CD+®. There is no difficulty at all in continuing term plex line a

f

C. Tenn

@

CD

into the whole com-

is not so simple, because of the singularity

of (1-s)a-! at s • l (Rea :S 0). But we can evaluate

@

formally by

integrating by parts: (1)

@

=

1 1 / (1-s)ll_ -1, this expression makes good sense-using it as a definition of

@ , we obtain

an analytic continuation of~ into the region Rea > l.

If we integrate by parts (fonnally) once again (i.e., in

(1)~.

we get a defi-

nition of Ma valid for Rea > -2. By successive integration by parts, we continue Ma into the region Rea > - k for any k > 0. Thus Ma (and therefore also Ia) may be continued throughout C. The semi group relation Ia 1{3 = Ia + f3 follows from a routine computation for Re a, Re

f3 :;: :

0, and so holds for all a,

f3

f

C, by virtue of the

analyticity of ~. To show that 1° is the identity, consider JU (a > 0), let a ... 0, and apply a routine "approximation of the identity" argument. Details are left to the reader.

QED.

In particular, since lkl-k =identity, we have 1-kf =(ak;atk)f. The operators JU and Ma are fraught with applications to our maximal functions. For, the inequalities between which we are trying to interpolate can be rephrased (l

and

< p < +oo),

79

§3. TilE MAXIMAL THEOREM

We are trying to show that

Two obstacles stand in the way of interpolation: (a) To interpolate using the given family of operators m : f -o sup I~(T 0 f)(t)\, a t > 0 we need inequalities not merely for m_k + iy for any y. (b) The operators

ma

'"1 and '"-k• but for '"1 + iy and

are unfortunately non-linear.

Neither of these problems is very serious. Let us first take Rea > 0.

'"cp a

=sup _ l _ t> 0 \1\a)\

< sup I 1\Re a) - t >0 1\a) "' I if

cp

1\Re~l

~ 1 f\t-s)a- 1¢(s)dsl

\till

0

l

I·

11\Re a)\

m

1\a)

· -l

f

jtal

t R 1 (t- s) e a- \¢ (s)\ds

o

1¢1 , Rea

is a decent function on ( 0, oo ), If we use the fact that jr(X+iy)\ -

e

-E.Irl 2

·IYI

(x-~)

. ..j2TT

as Y-+.±""

(see Titschmarsh, Theory of Functions, p 259) and apply the last inequality to the function if>(t)

=

Ttf(x), we obtain the inequality ilma II p< Ke"llm al11m··Rea II p

for any a and p, where the constant K' is uniform in Re a , provided Re a varies inside a bounded set.

i/

Therefore, 11m 1+ II p .$ Kpen'\YI II f II p for any p (l < p < +oo) and any real y; and 11'"-k+i/11 2 .$ Ake"IY\11£!1 2 foranypositiveinteger k and any real y, similarly, if we use (1) and the integration by-parts that follows it.

80

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

To handle (b), we linearize our operators as follows: For any reasonable function t(x), mapping our basic measure space

Clll, dx) into (0, oo),

define an operator T~(*) on Lp by setting

T~(*)f(x)

= r(~)

J

t(x)

t(x)-a

T 8 f(x)ds .

0

Obviously ITat(*)f(x)l

s ma(f)(x)

for any function. By our inequalities for

ma, (1)

and (2)

where Ap and Bk are independent of the function

t( · ). Since the opera-

tors T!(•) are linear, we may immediately apply the convexity theorem (of the previous section) to inequalities (1) and (2). To do this, set U(z) =

where a= a(z) '"'0-z)(-k) + z.

eZ 2 Tt(·)

a

The result is IITJfllpS Kllfllp

where p is determined by the equations

.. (3)

-1 p

1·0 + 0-0)(-k) - 0. The "constant" K, whatever else it may depend on, is independent of t( ·).

Any p (1

< p < oo) arises from equations (3) for some values of k

and p 0 > 1, for we have only to pick k very large. Therefore, we have shown that for every p (1

ClR, dx) ... (O,oo),

< p < oo) for every measurable function t(·):

the inequality

81

§3. THE MAXIMAL THEOREM

(4)

is valid, with Kp independent of t(·). Now we are (essentially) done. We have merely to pick our function t(·) in such a way that ITtf(x)j ~ Y. supt > 0 ITtf(x)j for each x. By inequality (4), ~supt > 0 !Ttf(x)l!p :S 2Kpl!fllp' In other words,

II£*~ p S Apllfll p

{l

< P < oo),

which is exactly what we wanted to prove. It remains only to show that

limt ..

0+ Ttf

= f almost everywhere

(f ( LP, 1 < p 0. Hence lim sup !Ttf(x)-Hx)\ .S lim sup t .. O+

t .. O+

\T~f-Tsf)(x)l

+lim sup \Tt(Tsf)(x)- Tsf(x)\ + !Tsf(x}-f(x)l t .. 0+

S sup t >0 =

IT 1U- Tsf)(x)j

+ ITsf(x)- f(x)j

(f- T 8 f)*(x} + jTsf(x)- f(x)j .

So

!I lim sup t

>0

!T~(x)- f(x)l !1 2 < 21!(£- T 8 f)* -

\b

S Kl!f-T8 f\1 2

(by the L 2-boundedness of the maximal function) .. 0 as s .. OT, by strong continuity of !Tsl on L 2 • This proves that lim sup

t .. 0+

=0

ITtf(x)- f(x)l

in other words Ttf .. f a.e. as t .. OT, for f

l

a.e.,

L/lR).

82

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Now let f belong to L p (1R) (l < p < +oo), and suppose We can find a function g E L 2 n Lp such that II f- gil p ~arne trick as before, we write

+ lim sup

< sup t)

0

t-+ o+

0 is given.

•

Using the

ITtg(x)- g(x)l + lg(x)- f(x)l

ITt(f_ g)(x)l + l(f- g)(x)l (since Ttg(x)-+ g(x) a.e. as t-+ o+) I

= (f- g),x) + (f- g)(x) .

So by Lp -boundedness of the maximal function, we have II lim supt-+O+ ITtf()- f( )I lip .S Kllf-gllp .S Kc Letting £ .... 0, we get lim sup ITtf(x)- f(x)l "' 0 a.e., which means t t-+ o+ that T f-+ f a.e. QED.

Section 4. A digression: L 2 theorems. Before we continue our development of Littlewood-Paley theory, we shall pursue a digression. The above maximal theorem and ergodic theorem were posed for semi groups {Tt I which were contractions l2dx :s lJi

c2

~

(1- t:)2n I giJ

~

< +oo ,

n=O

so

Hence l:n":o lpfih 2(x)l 2

< +oo for almost ail x, so Pnh 2 ~ 0 almost

everywhere as n-+ +oo. Finally, then pnf ,... Pnh 1 + Pnh 2 converges almost everywhere as n ... +> for all f in a dense subset of L 2(ffl).

QED.

§4. A

DIGRESSION: L 2 THEOREMS

87

The arguments used in the above theorem originate in the papers of Kolmogoroff-Selivestroff-Plessner in 1928, in which the authors prove the ·almost everywhere convergence of a Fourier series I: =-oo anein(J for which I;~-oo \ ~\

2

logIn\< +oo. Paley [29], Bochner (in his book Fourier

Analysis and Probability 1955) [32] and E. Stein [31] successively widened the scope of these ideas. The following two theorems are proved by essentially the same technique as Theorem 3. THEOREM 4: (L 2 ergodic theorem). Let U be a unitary operator on L 2(»l) such that f > 0 implies Uf ?. 0. Define

~+(f)

n

= -

1-

(n + 1)

_!

Ujf

j= 0

for f

f

L 2 . Then limn -ooo An+ (f) exists almost everywhere.

1 n · Sketch of Proof: Let An(£) = 2 n + 1 Ij = -n uJf. The operators An

are self-adjoint, and satisfy A(n)A(m)f

:5

2[A(2n) + A(2m)]f iff?. 0.

Proceeding as in the proof of Theorem 3, we obtain the maximal inequality l\supn2:_0A(n)f(·)H 2 :5 C\\f\1 2 . Butfor f ?_0, A;£ :5 2Anf, sothemaximal inequality is proved. Almost everywhere convergence follows easily. THEOREMS: (Martingale Convergence Theorem- L 2 variant). Let E 1 , E2 ,

...

be the orthogonal projection operators to sub spaces X1 f X2 f ...

of L 2(»l). If f > 0 implies all E. f > 0, then limn Enf exists almost J -ooo everywhere, for each f £ L 2(m). Sketch of Proof: Use the inequality Em En f .$ Emf+ En f (if f 2:. 0 ). This inequality is vlaid because Em En

= Emin (m,n).

This implies that

1\supnEn£112 .$ 2\lf\12· Some examples of applications of theorems 4 and 5 are in order. The typical operator U of Theorem 4 arises as follows: Let

Ol $

~

\\~11' and \\g\\ 1 $ Kjjf\\ 1 .

En_ 1 (h)jj 1 $ K\\f\\ 1 . In particular l\hil 1 $ K\\fl\ 1 .

96

THE GENERAL LITTLEWOOD-PALEY THEORY

From Gundy's lemma, we can easily prove parts 1. and 2. of Theorem 7. For if f

£

L1

n L 2 , and f

mlxiTi(x)

= g + h + k as in the lemma, then we have

>.\I~ mlxJTag(x) > ~ l+mlxiTah(x) >~I

+ m lxJ T ak(x) > ~I

= I. +II. +III.

Now I.$ mlxl supnlEn(g)(x) ~-01 ~ ~ JJ£11 1 • To estimate II, we note that ITa(h)(x)l ~ In J(En- En_ 1)h(x)J. So II.

~

""

m lxl!n J(En- En_ 1)f(x)l >

~I ~.!}II fJI 1

by (b) of Gundy's lemma. Finally, III. ~ ~ ll fJI 1 • For by (c) of the lemma, we have the inequality II k II

i~

K ,\II f 11 1 • Since T a is a bounded operator

on L 2 ,

Putting together our estimates for I., II., and III., we find that

which is exactly part 2 of Theorem 7. The operator Ta thus has weak-type (1, 1) and strong type (2, 2) (i.e., T a is a bounded operator on L 2 ). By the Marcinkiewicz interpolation theorem, T a is a bounded linear operator on Lp ( 1

· The decomposition of f will be

carried out using the IfnI, and the notion of a stopping time, which we now define.

97

§2. THE INEQUALITIES FOR MARTINGALES

Suppose r(x) is a positive integer-valued function on the measure space (M,m, dx) such that lxl r(x) • nl is measurable not only with respect to

m.

s= n,

but with respect to

for each n > 1.

r(.) is then called a

stopping time.

If r(x) is a stopping time, then

f

(*)

f(x)dx

M

=I

fr(x)(x)dx

M

For the 16ft-hand side of(*) is just

=

i

j=l

f

f(x)dx •

lxjr(x)•jl

Equation(*) generalizes the identity

f

f(x)dx.

M

JM

fn(x)dx •

JM f(x)dx.

We can construct a new martingale from Ifni and r(x). Simply set fn' (x) .. f m1n . ( n,r( x ))(x). lfn' I is called the stopped martingale defined by lfn I and r. (For each x, If~ (x)l looks just like I fn(x)l until time r(x), when If~} "stops".) The proof that If~ I form a martingale, i.e., f~ .. En(f~ +1)

resembles the proof of(*), and is left to the reader.

As a slight extension of the definition of stopping time, we allow r(x) to take on the value +oo. Equation(*) still holds if we define f00 (x) - f(x), and If~ I is still a martingale. (In fact f~

= En(fr(x)(x)) .)

Recall that we are trying to prove Gundy's Lemma. The easiest of the three parts of the decomposition is g, which will be defined by g(x) .. f(x)- ft(x)(x) where t(x) is a particular stopping time. To define t, we let r(x)

= inflnl fn(x) > AI. lx\ r(x) = n}

Next write fn(x)

r(x) is a stopping time, since

= (xI f 1(x) •••• , fn_ 1(x) .$A but fn(x) >A}£ S: n.

= Ik= 1

cpk(x) where cpk = fk- fk_ 1 and f0

= 0;

98

THE GENERAL UTTLEWOOD-PALEY THEORY

and set En(x)

= ¢n(x)

Xfy\ r (y)-= nl (x) •

Obviously En ~ 0. (Think

about it for a moment.) Define a new stopping time s by s(x) = infln\ I~ .. 0 E(Ek+ 1 \ j=k)(x) >..\I. (The reader may check that s is, in fact, a stopping time.) Now set t(x) = min(r(x), s(x)). t is a stopping time, since it is the minimum of two stopping times. I claim that mlxlt(x) < +ool ~ ~ \lf\1 1 , where K isauniversalconstant. First of all, I xI r (x) ~ + oo I =- I xI supn fn(x) > AI, so that mIx I r(x) < +ool ~ ~

1\ fl\ 1

by the martingale maximal theorem. Similarly,

lx\I;,.. 0 E(Ek+l\j=k)(x) >..\I and

"' i !,

(fk+l(x)- fk(x))dx

k=O lxlr(x)=k+ll


0, the left-hand side of this inequality is dominated by J -

I fe:jX{yls(y)~ jldx+~ j

M

J

'" 2I

f

f

E(e:jiS:j-1). X{yls(y):;:: jldx

M

Elx>xt Yl s(y):;:: jjdx (by definition of

~onditional expectation)

M

j

00

~ (fj(x)- fj_ 1(x)) · X{ylr(y)=jl(x)dx (by definition of e:j) j= 1

S2

f

fr(x)(x)dx = 211£11 1 by(*).

M

Therefore (**) is verified with K '"' 2.

§2. THE INEQUALITIES FOR MARTINGALES

101

So far we haven't used any information on s(x) except that it is a

stopping time. We now finish off the proof of the lemma by exploiting the properties of s(x) to prove c. By the properties of g and h already demonstrated, k = f- g- h has

L 1 norm

II kll 1

K II fll 1 ; and we have the representation

~

valid pointwise almost everywhere. Part 3 of the lemma says that 1

k(x)l _$ KA

almost everywhere, so it is surely enough to prove that

""

III

(a)

Yj- X{yj s(y) 2: jl

j= 1

t

.$ KA

and

""

II!

{{3)

E (e) 1 j-t>Xty 1s(y) 2: jl II oo

j= 1

~

KA

(a) follows from the computation 00

00

min (r(x)- 1, s(x}}

=

·

~

J=

cp.{x) = f

J

1

min(r(x}-l,s(x)}

which has absolute value at most A , by definition of r(x). Similarly, ({3) follows from the computation 00

0

~ ~

E(ej11j_ 1 )(x) · X{yis(y)2: jl(x)

j= 1 s(x)

= ~ i= 1

s(x)- 1

E(ej11j_ 1 )(x) ~

~ E(e:e+ 1 11e)(x) ~A,

e .. o

(x)

102

TilE GENERAL LITTLEWOOD-PALEY TIIEORY

by definition of s(x). This completes the proof of Gundy's lemma. QED. Recall that we are trying to prove Theorem 7, and that we already established parts 1 and 2 of the theorem, by using the splitting lemma. To carry on, we need another of the basic tools of Fourier analysis-the family Irk} of Rademacher functions. If k is a non-negative integer, then rk is the function on [0,1] defined by

, =1

1 if j/2k _s t

rk(t)

< O+ 1)/2k, j even

, -1 if j/2k S t < (j+ 1)/2k, j odd.

The Irk} form an orthonormal system on [0,1]. which is, however, very far from being complete. Suppose that F(t) = I.

k

= 0

akrk(t), where I.k = 0 Iaki 2

< + oo

. Then

of course F < L 2 and IIFII 2 = (I.k=O jak1 2) 1h. But we can say much more. For any p (1 S p < +oo), F f Lp[0,1], and

Bpc~. hi')" $I FI p $Ape~. 1·.1

(*)

2 )"

where the constants Ap and Bp depend only on p. For a proof of (*), see Zygmund's Trigonometric Series, Vol. I. Chapter V, [20]. We can now prove part 3. of the theorem, by using part 1. and inequality (*) for the Rademacher functions. Part 1 says that

where T af For t

f

= I.k= 1

fM ITaf(x)jPdx S Ap\lfll/

ak(Ekf- Ek_ 1 f) and a = (ak) is any sequence of norm 1.

[0,1], let ak

= rk(t).

We obtain the inequality

with Ap independent of t. Integrating in t, and changing the order of integration, yields

I [! M

0

1

I

i

k=

(Ekf(x)- Ek_ 1f(x)) · rk(t)jpdtldx $ Ap!/fll: 1

J

103

§3. AN ADDITIONAL "MAX" INEQUALITY

By inequality (*),the expression in brackets is approximately

Ap • (

I

IEkf(x)- Ek_ 1f(x)l 2

k=l

)p/ =

Therefore, fMIG(f)(x)IPdx ~ ApllfiiC for all f

2

£

A;IG(O(x)IP .

Lp(M,dx). This completes

the proof of part 3 of Theorem 7. Part 4 comes from part 3 by the usual duality argument, based on the fact that G is an isometry on L 2 • Thus all parts of Theorem 7 are proved.

Q.E.D. Section 3. An additional "max" inequality. We have proved two big theorems on martingales-the "Paley inequality" and the maximal theorem. There remains one more result, and after we get it out of the way, we can come (finally!) to the link between semigroups and martingales, that will enable us to prove the general semigroup form of the Littlewood-Paley inequality.

1 1 £ 1 2 £ ·· · as before, let Ek denote the conditional expectation operator with respect to 1 k. Suppose that {fkl is any sequence of functions on (M, dx), where fk is not ass111ned to be 1 k-measur able; and let {nk I be any sequence of positive integers. Then THEOREM

8. Given

(l

< p < +oo)

where Ap depends only on p. Proof: The theorem has an easy proof. Let Lp(~q) denote the Banach space of all sequences of functions, {fkl, for which the norm

is finite. (If q • +"" we make the obvious modification

104

THE GENERAL LITTLEWOOD-PALEY THEORY

LP(eq) is really very much like LP. For example, the dual space of LP(eq) is Lp ,(fq,), under the pairing 1/p'+ 1/p

= 1/q' + 1/q = 1,

JM ~k fk(x) gk(x)dx,

=

provided that p

where

f. + oo, q f. + ""'·

We shall use the following generalization of the Riesz convexity theorem: Let T be a linear operator which maps sequences of functions to sequences of functions. Suppose that T is bounded as an operator from LPo(eq 0) to itself, and as an operator from LP 1 (eq 1 ) to itself. Then T is also bounded as an operator from LPt(eqt) to itself, where 1 ph

(1- t)

---

=

Po

+

t Pt

and

= (1-t) + -~

1 qt

qo

ql

(0 ~ t $ 1)

The proof of this theorem is very similar to that of the Riesz convexity theorem. A full proof is found in a paper of Benedeck- Panzone, The Spaces

J.

LP with Mixed Norm (Duke Math.

1961, p. 301- 324), [21]. See also

Calder 6n [39]. Now, consider the operator T, which sends the sequence lfkl of functions, to the sequence lEnkfkl.

f (~ Mk

lEn fk(x)IP) k

p/p dx =

T is a bounded operator on LP(ep), since

~k

f

M

IEn fk(x)IPdx k

~ ~f. lfk(x)jPdx kM

~ ( ~ lfk(x)!P )"'•dx .

•

On the other hand, T is a bounded operator on LP(e 00 ) if l < p $ + oo. This is because

f

M

lsupkEnkfk(x)IPdx $

f

lsupn,k Enfk(x)IPdx $

M

= AP

f

M

(supklfk(x)I)Pdx ,

§3. AN

(where

lOS

ADDI'nONAL "MAX" INEQUALITY

* denotes the maximal function, and

¢ (x) = sup Ifk(x)l ), by the

maximal theorem.

k

We can now apply the generalized Riesz convexity theorem to conclude that T is bounded on Lp~q) if I < p S q S +oo. In particular, if I< p $2, then T is bounded on Lp(f 2 ), which is precisely the statement of the theorem! The case 2 S p < + oo follows by an obvious duality argument involving Lp(fq) -spaces.

Q.E.D.

REMARKS. The result of Theorem 8 does not hold when either p = 1 or p • "", and in fact the true order of growth of the bound Ap is O(p~) or p-+ ""• and O((p-1)-lh), as p-+ 1. This indicates that the theorem cannot be entirely trivial. The fact AP $ Ap~

as p -+ "" follows by an examination of the bounds

arising from the interpolation argument. To show that in fact Ap > Apy,_ in general, let E 1 ,E2 , ... ,Ek ... arise from the "dyadic interval" expecta· tions and set fk(x) = I if rk-t < X $ rk, fk(x) - 0) otherwise. Then

(IIfk(x)i 2 )~

=1, while

COROLLARY: If {fkl is any sequence of functions on (M, dx), then (l

OTtf(·)\\p ~ Ap\lfl\p (recall that Ttf(x) is a continuous function of t f (O,+oo)). This deduction was not necessary at this stage-after all, we already knew a proof of the semigroup maximal theorem (which didn't even require axioms (III) and (IV).) But we can already surmise the power of the martingale theorems when combined with Theorem 9. Proof of Theorem 9:

(0, {3, P) is actually the result of an old construc-

tion from the theory of Markov processes. Imagine a particle located somewhere inside M, (say, at p0 ) at time t

= 0.

At time t = 1 the particle jumps to some other point p 1 of M, ac-

cording to some fixed probability distribution for p 1 . Having reached p1 , the particle forgets that it was ever at p0 • So at time t "'2, the particle jumps to a point p 2

M, and the probability distribution of p 2 depends only on p1 , not on Po • The process continues-at time t = n + 1 the particle jumps from Pn to Pn + 1 , having completely forgotten where it was at times f

0, 1, •.. ,n-1. There is a natural probability space (U, f3, P) for this random process. Suffice it here to define

n.

A point

(U E

n

should describe the complete

history of the peripatetic particle. So it is reasonable to set cu equal to the infinite sequence (p 0 , p1 , p 2 , ••• ). Thus, 0 consists of all possible sequences of points of M, i.e.' n = M X M X M X .... Now let us return to the case of an operator Q satisfying (I)- (IV) above, and try to use the above probabilistic ideas to construct an (0, {3, P). First of all, we agreed that

n = MX M X MX

• .. •

For the Borel field

the sigma- field generated by all sets of the form (*)

f3

we take

108

'niE GENERAL LITTLEWOOD-PALEY 'niEORY

where the Ai are measurable subsets of M. Note that the sets of the form (*)(so-called cylinder sets) already form a Boolean algebra. To start we aregoingtodefine the measure P. Let

s ..

A1

X

A2

X ••• X

AN

X

M

Start with the function XAN on M; then form

X

MX MX

Qx"N;

•••

then multiply by

XA

, to obtain XA • Q(xA ); apply Q again, to obtain N-1 N N -1 Q(XA • Q(xA )); multiply this by XA to obtain N-1 • N-2 XA • Q(xA • Q(xA )); apply Q again. Continuing this process, we N-2 N-1 N finally come to the function XA • Q(xA • Q( ••• (XA • Q(XA )) ••• ). 0 1 N-1 N Set

The reader may check that P is well-defined, non-negative, and finitely additive on the cylinder sets ((IV) is required to show that P is welldefined, since (A 1 x A2 x · •· x AN) x M x M x M x · · · and (A 1 x A2 x · · · x AN x M) x M x M x M x M x ·· · are the same cylinder set). It can be shown that P extends to a countably additive measure on fJ.

The proof is rather technical, so we omit it. The demanding reader may look in the paper of Doob, A Ratio Operator Limit Theorem [27]. Probabilistically, this corresponds to the situation explained at the beginning of the proof, where p0 is distributed according to the probability law Pr(p0

E

A) = fA dx,

and where a particle at position Pn "' x jumps to a position Pn +l cording to the pr~ability law Pr (pn+ 1 l A) Now define

Ml, and set

1n

10

"' (A 0

x M x M x Mx

---------

l

M ac-

= Q(XA)(x).

···I A 0

is a measurable subset of

n+l

= (M x M x M x ··· x M x S I S

Obviously · ·· .S:

E

,8, so that S .S: M x M x M x •··1.

1 n +1 .S: 1 n .S: ••• .S: 1 1 .S: 1 o = f3 -

§4. THE LINK BETWEEN MARTINGALES AND SEMIGROUPS

The mapping i:

n .. M defined

109

by i(xo, XJ' x2, ... ) "' Xo sets up an

isomorphism of measure spaces between (0, j'0 , P) and (M, JR,dx). Thus, part (1) of Theorem 9 is verified. In order to prove part (2), we make two claims: (a). If g:

n .. R

is such that g(lxo, XI' x2, .•• !), depends only on xn then

E(gXIx 0 , x1 , ••• })

=

Qng (x 0 ).

(b). If g: 0 .. R is such that g({x 0 , xl' x2 , ... !) depends only on x 0 (i.e., g is ~0 -measurable), then En(g)(lx 0 , xl' .•. }) = Qng (xn) . From these two claims, it is obvious that E En(i

-to = C 1(Q 2no.

which proves (2) of Theorem 9. Therefore, the proof of Theorem 9 is reduced to the task of checking (a) and (b). Proof of (a): We are given a c~ndidate, Qng(x 0 ). for the conditi~nal

expectation of g with respect to j'0 • Since Qng (x 0 ) is obviously j'0 measurable, it is enough to check that (*)

for S E j'0 , i.e., S = A 0 x M x M x M x ···. Both sides of(*) are equal to JA 0Qng(x)dx if g is the characteristic function of a subset of M, as follows from the definition of P. On the other hand, both sides of (*) are linear in g, and well-behaved under limit processes. So (*) is valid for all g. Note that so far we have not used the self-adjointness of Q. Proof of (b): As in (a), the problem reduces to showing that

(**) where S=MxMx ... xMxA xA 1 x .. ·xA xMxMxMx· .. n n+ n '

'--) Then

(!, ~l(t, x) be any measurable function on (O,oo)x(M,Ilt,dx) suchthat

(J0

00

tlll>(t,x)l 2 dt)Y.! $1 for

every point x. The operators T~: Lp ... Lp defined by T!(f)(x) = ..,

depend analytically on a

f

0

t

00

t[ll>(t,x).l._Ma(f)(x,t)]dt

at

C. Inequality (2) implies that II T;(OIIp $ Apallfllp

(1

< P < +oo)

for Rea> 1 and inequality (1) implies that 1IT<E satisfying the defining condition, we obtain at last

Q.E.D.

120

THE GENERAL LITTLEWOOD-PALEY THEORY

Section 6. Dlmouement The following corollaries elaborate Theorem 10 and give an inkling of the applications of the Littlewood-Paley inequality. COROLLARY 1. For each k ~ l, llgk(flllp .$ Ap I flip (l

< p < +oo),

Proof: Proceed by induction on k. The case k .. l is what we have

just proved: We shall illustrate the induction step by considering k = 2; the general case is left to the reader. As we saw above,

a 0), where M(t) is a bounded frmction on

(O,+oo). Then T m is a bounded operator on all the spaces Lp (l

< p < +oo).

Proof: The arguments we used to prove the Lie group version of this

theorem in Chapter II apply to the present case, to prove that for f 1 1(T mf)

s Kg2(f)

E

Lp,

with K independent of f. Corollaries 1 and 2 now show

that IITmfllp S Apllfllp·

Q.E.D.

COROLLARY 4. Let A be the infinitessimal Aenerator of ITt!. (Thus

Tt "" etA.) Then (-A)it is a bounded operator on Lp (l < p < +oo) for each real t. Proof: ,\it is of ~aplace-transform type.

Q.E.D.

122

THE GENERAL LITTLEWOOD-PALEY THEORY

BIBLIOGRAPHICAL COMMENTS FOR CHAPTER IV

Section 1. For the theory of martingales see Doob [26], Chapter 7. The Marcinkiewicz interpolation can be found in Zygmund [20], Chapter XII. Section 2. See the remarks after Theorem 7. Section 3. Theorem 8 is new. Section 4. For Theorem 9 see Rota [30] and Doob [27].

CHAPTER V

FURTHER ILLUSTRATIONS In this final chapter we indicate some further illustrations of the theory, but our presentation is more in the spirit of Chapter II than Chapter III and Chapter IV.

Section 1. Lie groups We assume that G is a non-compact, connected, Lie group. We let Xl'X 2 , ... ,Xn be a basis for the (left-invariant) Lie algebra, considered as first-order differential operators on G. We set ~+ =

I. a lj.. x.x. 1 J

where laijl is any real symmetric positive definite matrix. (More specific choices of the {a. 1.1 will be made later.) Our first object is to consider the 1

heat-diffusion semigroup

T1 = et~

+

•

THEOREM. There exists a semigroup

I T~l,

which satisfies properties

(1), (II), (III) and (IV) of Chapter[[[, and such that

T~ = e~+ in the follow-

ing two related senses: (a)

(T! -l)f

as

for all sufficiently smooth f (b)

If f f Lp(G), 1 ~ p ~

oo,

then u(x, t)

dU (X, t) dt

= (T~ f)(x)

= A+ u (x, t ) Ll

f

C00 (G x R+), and

.

It should be noted that the operators T~ are left-invariant, that is

123

124

FURTHER ILLUSTRATIONS

La T~

= T~La,

where (LaO(x)

= f(a- 1x),

at G.

According to Hunt's paper (see [12, p. 279]), we can construct a probability semigroup T~ which satisfies (a) and (Ill) (T~ f ~ 0, if f ~ 0) and (IV) (T~ (1) = 1). By that same construction, symmetry (our property (II)) is then also guaranteed by the symmetry of ll+ (see [12, Section 7.3]). (I) is a consequence of (II), (III), and (IV), and the Riesz convexity theo-

rem. The fact that T + also satisfies conclusion (b) follows from the fact that

(T~f)(x) =

1

kt(y)f(xy)dy

G

where k!

t

kt

is a fundamental solution, and

L 1 (G)

Jet

l

C00 (G

X

R+), also

n L (G). (For these facts see Nelson [14].) 00

As a simple corollary of the above we also obtain the existence of the "Poisson semigroup" corresponding to the equation

a2u (x, t) at 2

+ fl+u(x, t)

=0

•

In fact, define P~ by pt 1 + = V"

~~ 0

e-u Tt 2 /4udu VTT +

We then claim that IP~l is a semigroup which satisfies (1), (II), (Ill), (IV), and instead of (b) of the above theorem, we have

a2 u at 2

+ ll +u

= 0,

where u(x, t)

= (P~ f)(x)

•

The details of this passage from T~ to P~ can be carried out as in the analogous case of compact groups treated in Section 2 of Chapter We now come to the g-function. For f ing two expressions (1)

t

n.

Lp (G) we consider the follow-

125

§1. LIE GROUPS

(2)

THEOREM 11. Let f

l

Lp{G),

1 < p < oo, and let g 1{f)(x) denote

either of the two expressions above. Then

This theorem is a direct consequence of Theorem 10 and its second corollary (Section 6 of the previous chapter), as soon as we verify that the projection E 0 is zero in this case. But suppose f l E 0(L 2(G)). Then T~ f = f for all t ~ 0, and so f E C 00 (G), and A+f = 0. Moreover since

kt

l

L 2 (G), for t

> 0 and f l L 2 (G),

it follows that f (x) = f0 ktCy) f (xy) dy,

vanishes at infinity. It may be assumed that f is real-valued. The above then shows that f attains its maximum and minimum values and thus must be constant and therefore zero in view of the local maximum principle of Hopf for the operator A+. Notice also-that T~f = f for all t > 0, if and only if P~ f

=f

all t, and therefore the result applies also the the semi·

t

group P+. The result for (2) can be extended when 1 < p

~

2 by taking into ac·

count the Xj derivatives. To do this define (.:\u)(x,t)

=

G at

+ A+u, and 1Vul 2

=

(*-)

2 +I aij(Xiu)(Xju) .

Let

Observe that g 1(f)(x)

~

g(f)(x).

THEOREM 12.

In proving this we follow closely the argument of Section 2 and 3 of

126

FURTHER ILLUSTRATIONS

Chapter 11. First, it suffices to consider the case where f ? 0, and f is

> 0,

C 00 and has compact support. Note that kt(x)

(see Nelson [14]). and

thus CT! f)(x) > 0, all (x, t) ( G x (0, oo) and hence (Pl f)(x) all (x, t)

f

= u (x, t) >

0,

G x (0, oo). Now it is immediate that

(A)

(See Lemma ~· Section 2 of Chapter II.) Also

by the general maximal theorem of Chapter III, or by the argument for the proof of Lemma 1 in Section 2, Chapter II. Finally,

f1

(C)

0

t(LiF)(x, t)sxdt=

G

1

F(x, O)dx

G

for appropriate F defined in G x [O,oo), and this class of F includes F(x, t)

=

(u (x, t))P. The proof of (C) requires a little bit of care.

We have to observe first that if f ( Lp(G), then [

ju(x,t)jPdx .... 0, as t

-+oo,

when l < p < "?,

G

This assertion (not valid when p

= l)

will be an immediate consequence

of the Lebesgue dominated convergence theorem, the maximal theorem, and the fact that lim u (x, t) = 0, for almost every x ,

t .... OC)

In proving the second assertion when p = 2, it suffices to consider a dense subset of f in L 2 (G). Now if P~ == f 0 00 e-AtdE(A), then we can write f = If- E(f) £1 + E(t )f and E (f )f -+ 0, as f -+ 0, since we already saw that E 0 (f) = 0. But when f 1 is of the form f- E(f) f, then II f 1 11 ~ e -it. Moreover

P!

§1. LIE GROUPS

127

Thus in view of the maximal theorem

This shows that lim u (x, t) = 0 almost everywhere, for f in a dense subt ->oo

set of L 2 (G), and therefore all f in L 2 (G). Finally, L 2 (G) n LP(G) is a dense subset of Lp(G) and therefore lim u(x, t) = 0, almost everywhere t-+oo for all f l LP(G), and finally J0 \u(x,y)\Pdx .... 0. Let us return to the proof of (C) for all F of the form (u (x, t))P. To establish this it will suffice, in view of what has just been done, to prove

1f

(C ')

N

0

t(Ll\F)(x, t)dt

=

G

i

G

F(x,O)dx-

f

F(x,N)dx .

G

Now if F, in addition to the smoothness it already has, also had as a function of x support in a fixed compact set of G, there would be no difficulty in verifying (C ') by the argument of integration by parts given in the proof of Lemma 3 (Section 2, Chapter II). To bring about this situation we construct a sequence { \P(X)¢k(x)\ < k X l G differential operators;

(i) sup

(ii) cpk(x) = 1, for x that Uk

~

l

for any polynomial P(X) of left-invariant

oo,

Uk, where Uk are open sets with the property

G;

(iii) ¢k+ 1 (x) ~ ¢k(x) ~ 0. An example of such a sequence can be obtained as follows. Let 17 ( t), 0

~

t < oo be a monotone Coo function in ( 0, oo ), such that 17 (t) = 1 for t

near zero and 17 vanishes outside a compact subset of t. Let d (x) denote the square of the distance from x

l

G to the group identity, measured by

any fixed smooth left-invariant Riemannian metric. Set cpk(x) = 77((d 2 (x))/k).

128

FURTHER ILLUSTRATIONS

Now with F(x, t)

= ¢k(x)(u(x, t))P

just indicated. We let k -+

co,

the identity (C ') holds, as we have

then the right side of (C ') clearly converges

to fa(u(x,O))Pdx- fa(u(x,N))Pdx. The left-hand side of (C') can be written as the sum of two integrals, whose integrands are respectively -t(&¢k)(u(x,t))P, and t¢k&(u)P. The first integral converges to zero since the 11 ¢k are zero inside Uk, Uk

-+

G; and 111 ¢kl

< A,

everywhere;

also uP(x, t) ·is integrable on G x [0, N]. The second integral converges monotonically to f0 N fa t ,& uPdx dt, since .\uP 2 0, and the ¢k converges monotonically to 1. This proves (C ') and therefore (C). Now that (A), (B), and (C) are established the rest of the proof of Theorem 12 is then the same as the corresponding argument given in the compact case (for 1

< p S 2)

in Chapter II, (see Section 3 of that chapter).

It is important to point out that the argument for p 2: 2 given in the

compact case cannot be extended in the present situation. This is because at that stage we would need to use the assertion that the Xj commute with

P~, which is the same as requiring that the Xj commute with 11+ =

I aij Xi Xj. For similar reaons some of the further applications given in Chapter II for compact groups do not have evident analogies in the case of general non-compact G, but there seem to be interesting possibilities if we make the assumption that G is semi-simple as we shall now see.

Section 2. Semi-simple

case

We now assume that G is a unimodular Lie group, K is a compact subgroup, and we consider the homogeneous space S = G/K. As usual LP(S, ds), (where ds is G invariant measure on G/K) is identifiable with the class of functions If Iff Lp(G), and f (gk) = f (g), k ( KJ. We also make a more specific choice of the left-invariant Laplacian

11+ "" I aij xi xj I by requiring that 11+ is also right-invariant under the action of K. More particularly if we write pk(f)(x) that 11+pkf

= p~+f,

= f (xk),

then we require

k f K, for all sufficiently smooth functions f on G.

We can obtain such a positive definite symmetric matrix laijl, by starting with any positive definite symmetric matrix la~j) I and performing the

129

§2. SEMI-8IMPLE CASE

appropriate integration with respect to the compact group K; (see the argument in Section 7 of Chapter I). When we have such a ~+ which is rightinvariant under K, then we denote by

~

its induced action on functions on

s. Let us denote by 0 the origin in S, that is, the point corresponding to the coset K. Then our non-unique choice of laijl corresponds to a choice of a positive definite quadratic form in the tangent space at 0, invariant under the action of K. For every such quadratic form we get a Riemannian metric on S, invariant under the action of G, and

~

then is the Laplace-

Beltrami operator for this metric on S (see the related problem at the end of Section 7, Chapter I). By the construction given in Section 1 above, the operator

~

leads to

semi groups which we now write as Tt and pt (instead of T~ and P~ ); since the latter semigroups are right-invariant under K, the former semigroups act on Lp(S, ds). It also follows that pt and Tt satisfy properties (1), (II), (III), and (IV), our fundamental properties for symmetric diffusion

semi-groups. We can write symbolically Tt = et~. and pt = e-t(-6.)¥.1 . In addition if U (x, t) ... (Ttf)(x), and u (x, t) = pt(f )(x), then au/at

~ u (x, t), and (a 2 u/at 2 ) + ~ u (x, t)

=

= 0.

There are now certain theorems for Lp(S, ds) which ate immediate consequences of the corresponding results for G (Theorems 11 and 12) in the previous section. We need not reformulate these theorems separately. We now make the assumption that S .. G/K is a symmetric space. If g is the Lie algebra, of G, and have that

!

= ~ + £. where

~

f. is the

respect to the Killing form. Let i~varia~t

Lie alg~bra

!

the sub-algebra corresponding to K, we

n~w

~rtho:onal c~mplement of ~ in

!

with

X1 , _x 2 , ••• , Xn be a basis of the right-

so that xl, ... , xr is a basis for ~ and

Xr+t'Xr+'2''"''Xr is a basis for£.. Then there exists two positive definite symmetric matrices {cijl. l

s i,

j ~ r, and lbkfl· r+ 1:$ k,

e~ n, so that

130

FURTHER ILLUSTRATIONS

c .. x.x. 1J

1 J

l~i, j~r

is not only right-invariant, but also left-invariant.

(~-

in effect is the

Casimir operatot: see Helgason [2, p. 451]). Let us say that a function f is zonal if f (k 1 xk 2 ) = f (x}, where k 1 , k2

f

K. The~e are exactly the functions on S

= G/K, which

are also invari-

ant under the a"ction of K on S. It is to be noted that if f is any smooth zonal function, then

_!

c .. x.x.

l~i, j~r

and therefore

~-f

1J

1 J

f:O,

is a Laplace-Beltrami operator of f, which is identical

with M for appropriate aij. We fix this choice of aij in what follows. Notice also that if

!

aij (Xif)(Xjf),

1_$i, j_$n

then if f is zonal we have .. I v £1 2 =~b . . 1J

o{.ocx.o 1 J

We can now state the version of Theorem 2 in Section 2, Chapter II, valid for all p, l


0 and f f L n L2 . ..

P

For these f 0 we define the Riesz transforms by Ri (f 0 ) .. - fi

=-

XiPt 0 (f)

in accordance with the inequality (*) or (**). Now purely formally since pt

= e-t(-1'\)'1-l

-'1-l = Xi(-&} fo

we have f 0 = -(-A)'n pt 0 (f), and therefore - fi

= XiPt 0 (f)

.

In order to show that these Ri are in fact well-defined or a dense subset of LP, we need to observe the following two simple facts: (i) The set of f 0 of the form

133

§2. SEMI-SIMPLE CASE

f 0 .. apt (f)l (for some to > 0, with f at t•to is dense in LP

E

Lp

n L2 ),

1 < p < oo. To see this, recall that for any

=h

lim P to h - pt 'h -oO t .... 00

t0

in both Lp and L 2 norm (when 1 < p < oc), as is shown in the proof of Theorem 12 of the previous section. Thus the set of f0 of the form f 0 =

n LP

with t' > t 0 , is dense in LP. Each such f 0 can be represented in the form

(Pto- pt ')h, h

l

L2

t

f 0 ..

aP (f) at

It =to

where f = -

j

t

~to

pt(h)dt

I

0

as an easy calculation verifies (ii) To see that the resulting Ri(f 0 ) is well-defined remark the following.

Suppose f 1 and f 2

E

Lp, and

fo .. apt -(fl)

at

Then Pt 1(f 1)

= Pt 2 (f 2).

I t=t 1

• -aPt (f2) j

at

t~t 2

,

This is because

ptj (f)

=-

lim t, .... ""

f

t,

t. J

apt (f)dt at

in the Lp norm, since pt '(f) ... 0, as t ' ... ""· However by the semigroup t t· t+t· property P (P J f) = P J (f) and therefore

from which our desired conclusion follows. The way the Ri(f) have been defined shows that by (*) (or (**)) we have

134

FURTHER ILLUSTRATIONS

for a dense linear subset in Lp' 1 < p < oo and hence the Ri have a unique bounded extension to all of Lp. We summarize this result and elaborate it somewhat as follows.

Suppose 1 < p < oo, and f is zonal f

THEOREM 14. (a)

BpJIIIIp .$ Ii IIR/OIIp $ ApJifiJp ·

(b)

When p = 2,

l

Lp. Then:

The proof of the inequality IJIRi(f)Jip _$ Apllfllp has been given above, and this shows in particular that it suffices to prove (b) for a class of f which are dense in L 2 • Start with f 0 which is C00 and has compact support and set f 1 = Pt 1(f 0 ), t 1 > 0, u(x,t) • Pt+t 1(f 0 ) = Pt(f 1). Apply the identity (C) (of the proof of Theorem 12), wtih F

= u2•

Then since

A(u 2 )

=

21 Wuj 2 , and if

f 0 is

zonal, so is f 1 and u (x, t), we have &u

2

a at

= 2[[__!!]

2

~ ~ + I b .. x.(u)X.(u)] -

..

1 ,J

1J 1

J

Now the identity

is valid for any one of our general semigroups, and follows easily by the spectral representation of p+, which can be written as

§2. SEMI-5IMPLE CASE

135

for appropriate E(.A). We have, in fact, already pointed out that in the present case E 0 ;;; 0. We therefore have

Hence

for all t > 0. This identity is (b) for f "' (apt;at)(f0 ), t > 0 and since this class of f is easily seen to be dense in L 2 the identity (b) is then fully proved. By polarization this identity yields, for f, g

f

fgds "'

s

f" s

~

L2

f

b lJ.. R.(f)R.(g)ds 1 J

and hence by Holder's inequality

where 1/p + 1/q

= 1,

wherever f ( L2 llf!lp =

n Lp,

sup gtL 2 nLq

I gil q:S 1

and g

1js

fgdsl

f

L2 n Lq. However,

136

§3. STURM•LIOUVILLE

Thus

II fliP

s sp- 1 :£ IIRiO,

l