Nilpotent Lie Groups: Structure and Applications to Analysis

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

562 Roe W. Goodman

Nilpotent Lie Groups: Structure and Applications to Analysis

Springer-Verlag Berlin-Heidelberg • New York 1976

Author Roe William Goodman Department of Mathematics Rutgers The State University New Brunswick, N. J. 0 8 9 0 3 / U S A

Library of Congress Cataloging in Publication Data

Goodman, Roe. Nilpotent lie groups. (Lecture notes in mathematics ; 562) Bibliography: p. Includes index. i. Lie groups, Nilpotent. 2o Representetions of groups. 3° Differential equations~ Hypoelliptic. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 562. QA3. L28 no. 562 [QA387 ] 512'.55 76-30271

AMS Subject Classifications (1970): 44A25, 17B30, 22E25, 22E30, 22E45, 35H05, 32M15 ISBN 3-540-08055-4 Springer-Verlag Berlin • Heidelberg ' New York ISBN 0-38?-08055-4 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

Table o f Contents

Chapter I ,

Structure of nilpotent

L i e algebras and L i e groups . . . . . . . . . . . . . .

§ 1. D e r i v a t i o n s and automorphisms o f f i l t e r e d I.I

1

polynomial r i n g s

D i l a t i o n s and g r a d a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

1,2 Homogeneous norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3 Vector f i e l d s

4

w i t h polynomial c o e f f i c i e n t s . . . . . . . . . . . . . . . . . . . . . . . .

1.4 L o c a l l y u n i p o t e n t automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5 Transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i0

1.6 F i n i t e dimensional r e p r e s e n t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I0

1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II

§ 2. B i r k h o f f embedding theorem 2,1 F i l t r a t i o n s

on n i l p o t e n t

Lie a l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2 A l g e b r a i c comparison o f a d d i t i v e and n i l p o t e n t group s t r u c t u r e s . . 13 2.3 F a i t h f u l

unipotent representations ...............................

16

§ 3. Comparison o f group s t r u c t u r e s 3.1 Norm comparison o f a d d i t i v e and n i l p o t e n t 3.2 A l g e b r a i c comparison o f f i l t e r e d 3.3 Norm comparison o f f i l t e r e d

structures .............

and graded s t r u c t u r e s . . . . . . . . . . .

and graded s t r u c t u r e s . . . . . . . . . . . . . . . .

17 19 27

Comments and references f o r Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Chapter i i .

33

N i l p o t e n t L i e algebras as tangent spaces . . . . . . . . . . . . . . . . . . . . . . .

§ 1. T r a n s i t i v e L i e algebras o f v e c t o r i.I

fields

Geometric background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

1.2 P a r t i a l homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.3 L i f t i n g

38

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§ 2. Proof o f the L i f t i n g

Theorem

2.1 Basic L i e formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.2 L e f t - i n v a r i a n t

42

vector fields

....................................

2.3 Formal s o l u t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.4 C~ s o l u t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

iV

§3, Group germs generated by p a r t i a l isomorphisms 3.1 Exponential c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

312 Comparison o f group germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

§4. Examples from complex a n a l y s i s 4.1 Real hypersurfaces i n 4.2 Points o f type

~n+l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m ...............................................

53 55

4,3 Geometric c h a r a c t e r i s a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.4 Siegel domains and the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Comments and references f o r Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Chapter I I I .

67

S i n g u l a r i n t e g r a l s on spaces o f homogeneous type . . . . . . . . . . . . . .

§ 1. Analysis on v e c t o r spaces w i t h d i l a t i o n s 1.1 Homogeneous f u n c t i o n s and d i s t r i b u t i o n s

..........................

1.2 I n t e g r a l formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 70

§ 2. Spaces o f homogeneous type 2,1 Distance f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

2.2 Homogeneous measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

2.3 L i p s c h i t z spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

§ 3. S i n g u l a r i n t e g r a l o p e r a t o r s 3.1 S i n g u l a r kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3,2 Operators d e f i n e d by s i n g u l a r kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

§ 4, Boundedness o f s i n g u l a r i n t e g r a l o p e r a t o r s 4.1 Almost orthogonal decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.2 Decompositions o f s i n g u l a r i n t e g r a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.3 Lp

95

boundedness

( 1 < p < ~ ) ..................................

§ 5. Examples 5.1 Graded n i l p o t e n t groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5,2 F i l t e r e d n i l p o t e n t groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96 99

5.3 Group germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

5.4 Boundedness on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

v

Comments and references f o r Chapter I I I

Chapter IV.

......................................

Applications ...................................................

114

117

§ 1. I n t e r t w i n i n g Operators 1 . 1 B r u h a t decomposition and i n t e g r a l formulas . . . . . . . . . . . . . . . . . . . . . . .

118

1.2 P r i n c i p a l

121

series .................................................

1.3 I n t e r t w i n i n g o p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

1.4 Boundedness o f i n t e r t w i n i n g o p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

§ 2. Boundary values o f

H2

functions

2,1 Harmonic a n a l y s i s on the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . .

138

2.2 Tangential Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

2.3 P r o j e c t i o n onto

146

2.4 Szeg~ kernel f o r

H~(G)

as a s i n g u l a r i n t e g r a l o p e r a t o r . . . . . . . . . .

H2(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

§ 3. H y p o e l l i p t i c d i f f e r e n t i a l

operators

3.1 Fundamental s o l u t i o n s f o r homogeneous h y p o e l l i p t i c 3.2 P r i n c i p a l

151

parts of differential

operators .....

operators ........................

158 163

3.3 C o n s t r u c t i o n o f a p a r a m e t r i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166

3.4 Local r e g u l a r i t y

167

.................................................

Comments and references f o r Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

Appendix:

175

Generalized Jonqui6res Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1

Root space decomposition o f

A.2

Maximal f i n i t e - d i m e n s i o n a l

A.3

Structure of

A.4

Birational

Der(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176 180

m ......................................................

185

transformations ...........................................

192

Comments and references f o r Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

Bibliography .................................................................

202

Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

210

Preface

These notes are based on lectures given by the author

during the Winter

semester 1975/76 at the University of B i e l e f e l d . The goal of the lectures was to present some of the recent uses of n i l p o t e n t Lie groups in the representation theory of semi-simple Lie groups, complex analysis, and p a r t i a l d i f f e r e n t i a l equations. A complementary objective was to describe certain structural aspects of simply-connected n i l p o t e n t Lie groups from a " g l o b a l " point of view (as opposed to the convenient but often unenlightening induction-on-dimension treatment).

The unifying algebraic theme running through the notes is the use of filtrations;

indeed, n i l p o t e n t Lie algebras are characterized by the property of

admitting a p o s i t i v e , decreasing f i l t r a t i o n .

The basic a n a l y t i c tool is a

homogeneous norm, which replaces the usual Euclidean norm and gives a "noni s o t r o p i c " measurement of distances. One obtains a f i l t r a t i o n

on the algebra of

germs of C~ functions at a point by measuring the order of vanishing in terms of the homogeneous norm. This in turn induces a f i l t r a t i o n

on the Lie algebra of

vector f i e l d s and on the associative algebra of d i f f e r e n t i a l operators. To construct (approximate) inverses f o r certain d i f f e r e n t i a l operators, one uses integral operators whose order of s i n g u l a r i t y along the diagonal is measured via the homogeneous norm.

A recurring aspect of our constructions is the approximation of one algebraic structure by a simpler structure; e.g. a f i l t e r e d Lie algebra is approximated by the associated graded Lie algebra; a group germ generated by a Lie algebra of vector f i e l d s is approximated by the Lie group generated by a " p a r t i a l l y isomorphic" n i l p o t e n t Lie algebra. The "order of approximation" is s u f f i c i e n t l y good that results in analysis on the simpler structure can be transfered to corresponding results on the o r i g i n a l structure; e.g. convolution operators on a f i l t e r e d n i l p o t e n t group which are " s i n g u l a r at i n f i n i t y "

have the

Vlll

same LP-boundedness properties as operators on the corresponding graded group.

The notes are organized as follows: Chapter I studies n i l p o t e n t Lie algebras and groups viewed as l o c a l l y n i l p o t e n t derivations and l o c a l l y unipotent automorphisms o f f i l t e r e d polynomial rings. Comparisons, both algebraic and a n a l y t i c , are made between various n i l p o t e n t group structures. These constructions are continued in the Appendix, in the context of groups of b i r a t i o n a l transformations.

In Chapter I I we explore the p o s s i b i l i t y of approximating a f i n i t e l y generated ( i n f i n i t e - d i m e n s i o n a l ) Lie algebra of vector f i e l d s by a ( f i n i t e dimensional) graded n i l p o t e n t Lie algebra. This leads to the notion of " p a r t i a l homomorphism" of graded Lie algebras, and the problem of " l i f t i n g "

a partial

homomorphism. The prototype f o r t h i s s i t u a t i o n is the case of a homogeneous space f o r a group, where the " l i f t i n g "

is obtained by i d e n t i f y i n g functions on the

homogeneous space with functions on the group which are l e f t - i n v a r i a n t under the s t a b i l i t y subgroup of a f i x e d point. The main r e s u l t of t h i s chapter is that a s i m i l a r construction can be carried out r e l a t i v e to a p a r t i a l homomorphismwhich is " i n f i n i t e s i m a l l y t r a n s i t i v e " . In concrete terms, this means that i f one wants to study a set of vector f i e l d s on a manifold which have the property that t h e i r i t e r a t e d commutators span the tangent space at each point of the manifold, then f o r local questions i t suffices to consider the case in which the manifold is a n i l p o t e n t Lie group, and the vector f i e l d s are "approximately" l e f t i n v a r i a n t . We describe how vector f i e l d s of t h i s type arise in connection with real submanifolds of complex manifolds.

Chapter I I I is devoted to constructing a theory of " s i n g u l a r i n t e g r a l operators" which is s u f f i c i e n t l y general to include the "approximate convolution" operators associated with the " a p p r o x i m a t e l y - i n v a r i a n t " vector f i e l d s of the previous chapter. We prove the boundedness of these operators on

~,

I < p <~ ,

and study the i n t e r a c t i o n between the integral operators and the vector f i e l d s .

IX Chapter IV contains the applications to the representation theory of r e a l rank one semi-simple Lie groups, harmonic analysis on generalized upper h a l f planes, and r e g u l a r i t y properties of h y p o - e l l i p t i c d i f f e r e n t i a l operators associated with t r a n s i t i v e Lie algebras of vector f i e l d s .

I t is obvious from the comments and references at the end of each chapter that a large part of these notes is based on the work of E.M. Stein and his collaborators. We have t r i e d to make the notes as self-contained as possible, by g i v i n g complete proofs of almost a l l r e s u l t s .

In the comments and references an

attempt has been made to assign c r e d i t to the various results and proofs; we apologize in advance to authors whose work we have overlooked or miscredited.

The

author would l i k e to thank Prof. Horst Leptin f o r the i n v i t a t i o n to

v i s i t the U n i v e r s i t y of B i e l e f e l d and f o r his gracious h o s p i t a l i t y . To Prof. Leptin and his collegues in the mathematics department go many thanks f o r making the v i s i t s t i m u l a t i n g and enjoyable. Dr. Peter MUller-R~mer provided invaluable assistance in organizing the manuscript, and the splendid job of typing was done by Frau M~nkem~ller.

Much of the w r i t i n g of the manuscript was done while the author

was v i s i t i n g

at the I n s t i t u t des Hautes Etudes, Bures-sur-Yvette, France. The author thanks the I n s t i t u t f o r i t s h o s p i t a l i t y , and Rutgers U n i v e r s i t y f o r providing f i n a n c i a l support through i t s sabbatical leave program. Several sections of the manuscript are based on reseach supported by grants to the author from the National Science Foundation.

Bures-sur-Yvette May, 1976.

Notational Conventions

An attempt has been made to keep the notations

consistent from one chapter to the next; the reader's indulgence is requested f o r the lapses from uniformity which crop up. Each section is numbered s e r i a l l y , and generally contains at most one Lemma, Proposition, or Theorem. In subsequent sections the Theorem of usual, IN, ~ ,

JR, $

§ 3.3

of Chapter I , say, is called Theorem 1.3.3. As

denote the non-negative integers, integers, real numbers,

and complex numbers, respectively.

Chapter I

Structure of nilpotent

L i e alg.ebras and L i e flrou~&

§ 1. D e r i v a t i o n s and Automorphisms o f f i l t e r e d

In t h i s s e c t i o n we w i l l

polynomial r i n g s

c o n s t r u c t a class o f n i l p o t e n t

Lie groups. Every n i l p o t e n t

Lie algebra

Lie algebras and

and simply-connected Lie group can be

embedded in t h i s c l a s s . These "model" a l g e b r a s and groups can be presented e i t h e r as d e r i v a t i o n s and automorphisms o f a polynomial r i n g , o r d u a l l y as v e c t o r f i e l d s and n o n - l i n e a r t r a n s f o r m a t i o n s on a v e c t o r space. This d u a l i t y w i l l

be used in

passing from a l g e b r a i c t o a n a l y t i c p r o p e r t i e s o f the embedded algebras and groups.

1.1 D i l a t i o n s and Gradations. space, and l e t V .

P = P(V)

Let

V

be a f i n i t e - d i m e n s i o n a l

real v e c t o r

be the algebra o f r e a l - v a l u e d polynomial f u n c t i o n s on

We take as our basic datum a d i r e c t - s u m decomposition

V =

(We a l l o w the p o s s i b i l i t y p o s i t i o n , we d e f i n e a

r ~ ~ Vn n:l

that

Vn = 0

f o r some n .)

In terms o f t h i s decom-

one-parameter group of d i l a t i o n s

{6 t : t > o}

on

V

by

setting

6 t ( z Xn) : E tnxn

We then d e f i n e the space

Hn

(x n e Vn) .

o f homogeneous .polynomials o f w e i g h t

Hn = { f e P : f - ~ t = tn f }

F i x a basis

{x i : 1 < i < d}

be the dual basis f o r

Vm ~

If ~i = ~i

for

~ elN d

""~d

~d

V

by

'

such t h a t

x i e Vni ,

is a multi-exponent, write

'

n

w(~) = Z ni~ i

and l e t

{~i }

Since

~i e Hni ,

we have

~

P =

O b v i o u s l y HmHn~Hm~ n Note t h a t when

e Hw(~) .

Thus

z (~ Hn n>o

so the spaces

,

I < n < r ,

Hm g i v e a g r a d a t i o n on the a l g e b r a P-.

t h e r e are both l i n e a r

and n o n - l i n e a r

p o l y n o m i a l s in

Hn .

We w i l l

a l s o need the f i l t r a t i o n

{Pn }

on

P

o b t a i n e d by s e t t i n g

E Hk Pn = k
Evidently

= PO c__.P1 ~ P2 ~ ' " '

IR

~_] P n = P n>O

Pm " Pn ~Pm+n "

We s h a l l

call

Remark. filtration

Pn

Let

the space

o f polynom.i.a..Is.., o f w e i g h t

z Vk . F n = k>n

o f the v e c t o r space

q u e l y d e t e r m i n e d by

{F n} .

F

The subspaces

V .

= F±

CV:':

n+l

we d e f i n e

'

then F:,:1 ~F:,~ 2 ~ . . .

~ F , ~ r = V~':

P

F

and n

w i t h the sum taken o v e r a l l

= S F i

I

indices

are then a d e c r e a s i n g

We c l a i m t h a t the f i l t r a t i o n

Indeed, i f

mn

{F n}

~ n .

...

:'~ik

il+...+i

k ~ n .

{Pn }

is uni-

1.2 Homogeneous norms

The passage from a n a l y s i s on IRn

n i l p o t e n t Lie groups begins by using the d i l a t i o n group multiplication

by

t .

6t

to analysis on in place of s c a l a r

The next step is to replace the Euclidean norms by the

following functions: Definition x ,

A

dilation-homogeneous norm on

~ Ixl

such t h a t

(i)

Ixl ~ o

(ii)

[xl = 0 < ~ x = o

(iii)

l a t x i = t Ixl

The norm is smooth i f

it

for all

is

C~

t

>

0

away from

V

is a continuous f u n c t i o n

.

0 ,

and i t

is symmetric i f

I×I : E-×i. Example

Let

o < p < ~ ,

Ixlp = (zI~i(x)l Ixl~ =max

and l e t

{Ci }

be as in

1.1.

Then

p/n i I / p

)

lq(x)i

1/n i

i are symmetric homogeneous norms. I f a polynomial, so t h a t

Ixl

P

p

is d i v i s i b l e

by

2r!

,

then

(IXlp) p

is

is r e a l - a n a l y t i c away from zero in t h i s case.

In t h i s chapter we shall be using homogeneous norms to measure the rates o f growth or vanishing of f u n c t i o n s . The f o l l o w i n g simple facts w i l l I. If

Ixl

is any homogeneous norm, then there exists.

C> 0

be needed: such t h a t

C-1 lxl~ ~ Ixl ~ c t~1~ •

Hence a l l homogeneous norms are .equivalent. (The set hence

Ixl II.

f e Hn ,

IxI~ = 1

is compact and

is bounded above and below on t h i s s e t . ) If

f

is a polynomial, then

f e Hn if_~_f I f ( x ) l

then If(x)I

~ (maxlf(u)I)Ixl

luI=1

n

~ C'IxI n

(Indeed, i f

Conversely, write assumption letting

f = ~ fk

f o 6t = O(tn)

t ÷ 0

III.

If

then f o r e v e r y

we g e t

m

'

with

'

so l e t t i n g

fk = 0

is a

n > 0

f k e Hk .

for

t + ~

f o 6t = ~ tkfk

we g e t

fk = 0

for

.

But by

k > n ,

and

k < n o)

function

C~

Then

d e f i n e d on a n e i £ h b o r h o o d . . £ ~

t he.r..e.i ~ a u n i q u e p o l y n o m i a l

Pn e Pn

0

in

V ,

such t h a t

m(x) - Pn(X) J S C Ixl n+l for n

x

near

0 .

in t h e T a y l o r

(The p o l y n o m i a l series

1.3 V e c t o r f i e l d s

of

Pn

m .)

o f the a l g e b r a

the l i n e a r

functions,

any d e r i v a t i o n

I f we s e t

Pi = T ( ( i )

'

P . T

polynomial

may be i d e n t i f i e d

coefficients

on

Using the f i l t r a t i o n

Since

P

be t h e L i e

i s g e n e r a t e d by

I

restriction

and to

V

"

w i t h the L i e a l g e b r a o f v e c t o r

fields

with

V .

{Pn } ,

To d e t e r m i n e t h e s t r u c t u r e

of

we d e f i n e

n ,

the following

we w r i t e

Dx

for directional

. 4

Dn = {D x : x e Vn}

: ~

.

t:o

Then i t

Dn Hk ~ Hk_ n

subalgebra of

Der(P):

: XPnC_Pn_ 1 , Vn} .

x e V:

Dx f ( Y )

Denote by

Der(P)

i s d e t e r m i n e d by i t s

= {X e Der(P)

the d i r e c t i o n

Let

then

T = s Pi s / ~ ( i Der(P)

t h e sum o f t h e terms o f w e i g h t

with polynomial coefficients

algebra of derivations

Hence

is j u s t

f(y+tx)

.

i s immediate t h a t

derivative

in

From t h i s

it

is clear that

=

s Hk D n . o
Set r

~k = mZ=kHm-k Dm "

Then one e a s i l y

verifies

that

nj Hk GHk_ j

~-J' ~ Thus we have a g r a d a t i o n o f

~n-J +k • n : r

n =

s nk k=l

-

and

n

,

is obviously a finite-dimensional

Suppose now t h a t

~

nilpotent

Lie algebra.

i s an open neighborhood o f

be the a l g e b r a o f r e a l - v a l u e d

smooth f u n c t i o n s

of vanishing of functions

0

at

on

0

Q .

in

V .

We w i l l

Let

measure the r a t e

i n terms o f the homogeneous norm

Ix I

set Cm = { f e C : f ( x ) (For

m < o ,

set

C,m = Co~..

It

m

.C

n

~C -

=H n

near O} .

QC n

on

C:

,

m+n

f o l l o w s by the graded from o f T a y l o r ' s C

O(lxi m)

This d e f i n e s a f i l t r a t i o n

C = Co~C 1)... C

=

n+l

and

Dk Cn _~ Cn_ k

series that

if

co

C = CR (Q)

n > o ,

then

If

m ~ o,

Let we s h a l l

L : L(~) say t h a t

be the L i e a l g e b r a o f T

is of order

< k

T Cn_~Cn_ k

C~

at

0

vector

fields

on

.

If

TeL

if

a function

if

for all

n .

Set L

For e x a m p l e ,

Dk C Z L k

vanishes to order order

n-k

at

(b)

i s o f o r d e r < k a t O} .

This d e f i n i t i o n

" at

0 ,

formalizes

then i t s

the notion

derivatives

that

of "weight"

k

vanish to

0 .

Proposition (a)

n

= {T e L : T

k

The f o l l o w i n g

properties

hold:

L = L r D _ L r _ I ~_ " . . ~j

(c)

, Lk-]cLj+ k

For e v e r y

k > o ,

nk~L

k

and

Lk = ~-k 0 Lk_ 1

Proof (a)

.

Part

We can w r i t e (b)

L = CoPlg...OCoO r .

follows

from t h e d e f i n i t i o n

Hm Cn

Hence of

L Cn ~ Cn_ r .

Lk .

For part

This g i v e s

(c)

,

we have

Cn-k _~ Cm+n_ k

Taking

k-m = j

,

we g e t

nj By c o n s i d e r i n g

the action

Cn ~ Cn_ j of vector

. fields

on l i n e a r

functions,

we f i n d

that

r

Lk =

The d e c o m p o s i t i o n

~ 0 Cm_k Dm m=l

Cn = Hn 0 Cn+ I

then g i v e s

(c)

,

finishing

the p r o o f .

Associated to the f i l t e r e d

algebra

L = Lr~

Lr_ 1 ~ . . .

is the graded

algebra gr(L) =

r ~ (D (L k / Lk_z) k=l

The Lie algebra s t r u c t u r e on and

(T)j,

(S)k

gr(L)

is defined as f o l l o w s :

denote t h e i r cosets mod Lj_ I

E(T)j,

(S)~

=

and

From part

Corollary

gr(L)

respectively,

finite-dimensional (r = i)

,

value at

L

(c) =

of the Proposition we have

n 0 ,

the i n f i n i t e -

of a l l vector f i e l d s can be "approximated" by the

n i l p o t e n t Lie algebra

n .

When the gradation on

V

is t r i v i a l

t h i s approximation merely consists in replacing a vector f i e l d 0 .

For

r > 1 ,

question in chapter I I ,

Remark

V .

We w i l l

r e t u r n to t h i s approximation

i n connection w i t h subalgebras of

The gradation

by i t s

t h i s approximation is more s u b t l e , as i t d i s t i n g u i s h e s

among the various d i r e c t i o n s in

tions

then

c o n d i t i o n , and obviously s a t i s f i e s

This c o r o l l a r y can be viewed as the statement t h a t near dimensional Lie algebra

T e Lj , S e L k ,

(~,S])j+ k

This is w e l l - d e f i n e d by v i r t u e of the f i l t r a t i o n the Jacobi i d e n t i t y .

Lk_ 1

If

n = nl~.-.@nr

L .

can also be described using the d i l a -

6t : n_k = {T e n_ : T ( f o a t )

= t k (Tf)o6t}

This f o l l o w s immediately from the formula f o r "homogeneous of degree k"

as vector f i e l d s .

~k "

Thus the elements of

~k

are

1.4 L o c a l l y

unipotent

automorphisms

automorphisms o f t h e a l g e b r a

P

is clear

that

N

from

n

onto

If

X e n ,

Leibnitz'

this

conclude that

into

eX

Conversely,

X Pc-

eX : m . First

enX

n = m

v a n i s h e s on identically.

ex e N ,

formula asserts

X e n , f e P ,

define

of

X .

and the map

X~--~ ex

is a bijection

definition

i s a homomorphism. Since given

Fn-1 ' It

that

: Z (~) x k f x n - k g . k

the s e r i e s

m e N , Xf =

have

nilpotence

then

xn(fg)

Clearly

If

N .

Proof.

Substituting

Aut P

Xn f

sum, by the l o c a l

Theorem.

be the group o f a l l

P n G P n _ I , ¥n} .

i s a subgroup o f eXf= z ~

This i s a f i n i t e

Aut (P)

Set

N = { m e Aut P : ( r - I ) It

Let

~ n>l

( - 1 ) n+z

we p r o v e t h a t

e tX e Aut (P)

Hence f o r any

f,g

Z .

But t h i s

and r e a r r a n g i n g ,

eXe -X = I ,

it

follows

transformation

X

we

that on

eX e N . P by

(~ - l ) n f

n

power s e r i e s

o n l y remains t o p r o v e t h a t

> etX(fg)

eX(fg)

define a linear

and by t h e f o r m a l

t,

of

e P ,

X

for all

identity

t = el°g t

t e ~R .

Indeed, for

in

so i t

any

the function

a polynomial

we

is a derivation.

- (etXf)(etXg)

is clearly

,

t

,

must v a n i s h

n e2Z,

Thus X(fg) = ~

e tX ( f g ) t:o d

= dt

t=o(e

tX

f)(e

tX

g)

= (Xf)g + f (Xg) ,

The map for

N .

X~--~ e X

f u r n i s h e s g l o b a l " c a n o n i c a l c o o r d i n a t e s o f the f i r s t

Using t h i s map, we t r a n s f e r

v e c t o r space

n

to the group

N .

X,Y ~

restriction

kind"

the a n a l y t i c m a n i f o l d s t r u c t u r e o f the

If

X,Y e n

and

f e P ,

then the map

e X eY f

i s o b v i o u s l y a p o l y n o m i a l mapping on mined by t h e i r

Q.E.D.

to

n x n .

V* ~ P r

'

it

Since elements o f

N

are d e t e r -

f o l l o w s t h a t group m u l t i p l i c a t i o n

is

a p o l y n o m i a l map when expressed i n c a n o n i c a l c o o r d i n a t e s . Indeed, as i n the p r o o f o f the theorem, i f

Hence i f

we w r i t e

eX eY = eZ

then

Z = log (e X eY)

e

Z = X * Y ,

(~)

X ~ Y =

To d e t e r m i n e the e x p r e s s i o n f o r {X * Y ( ~ i ) } that

,

using

{X * Y ( ~ i ) }

(*)

.

n .

then Z (-l)n+l, n>L n X mY

Since t h i s

i~eX e Y - l ) n

as a v e c t o r f i e l d , s e r i e s is l o c a l l y

are p o l y n o m i a l f u n c t i o n s o f

s h a l l o b t a i n more e x p l i c i t

{X(~i)

we o n l y need c a l c u l a t e finite

, Y(~i)}

on .

P , (In

we f i n d § 2

i n f o r m a t i o n about these f u n c t i o n s using the

Campbell-Hausdorff formula to rewrite

(m)

i n terms o f L i e p o l y n o m i a l s . )

we

10

1.5 T r a n s f o r m a t i o n

groups

a group o f ( n o n - l i n e a r ) {Ci }

be a b a s i s f o r

Theorem

If

The group

analytic V~

a l s o has a dual p r e s e n t a t i o n

transformations

with

m e N ,

N

~i

of weight

of the vector

space

V .

as Let

ni

then t h e r e i s a t r a n s f o r m a t i o n

T : V ÷ V

of the

form (~)

~i(Tx)

with

qi e P n . - 1 '

=

such t h a t

~i(x)

~(f)(x)

+ qi(x)

,

= f(Tx)

.

C o n v e r s e l y , f o r any c h o i c e o f

3

qi e Pn.-i exists

'

formula

m e N

(~)

defines

isomorphism o f

V ,

and t h e r e

such t h a t ~(f)

Proof

an a n a l y t i c

Since

= f o T

m(~i ) = ~i

mod

,

for

Pn -1 '

all

f e P .

there exist

qi

so t h a t

1

m(~i ) = Ei + qi is clear

that

"

Define

is invertible, if

g i v e n any

such t h a t

same p r o o f as i n

that

with

qi e P n i _ l

shows t h a t -1

= e -X .

o f the m a n i f o l d

Aut (P) V ,

subgroup o f a l i n e a r

Xn(~)

,

,

P

there exists One has

m = e

X

Hence

V

i s g e n e r a t e d by

,

it

for

a u n i q u e homomorphism

(~ - I ) some

m e N .

Pn ~- Pn-1 '

X e n . Clearly

corresponding to

representations

so the

In p a r t i c u l a r , m(f) = f o T ,

-I m

,

so

we have

We have p r e s e n t e d the group

and as a subgroup o f t h e group o f a n a l y t i c

two i n f i n i t e - d i m e n s i o n a l

N

isomorphisms

g r o u p s . We may a l s o embed

N

as a

group.

Xn : N + GL (Pn)

o b t a i n e d by r e s t r i c t i o n for

Then s i n c e

Q.E.D.

1.6 F i n i t e - d i m e n s i o n a l as a subgroup o f

"

is the transformation

S o T = T o S = I ,

matrix

(m) .

m(~i ) = ~i + qi

§ 1.4

S : V + V

Let

by

m(f) = f o T .

Conversely, : P ÷ P

T

to relative

be t h e f i n i t e - d i m e n s i o n a l Pn

Since

,n(m) f = f

t o the d e c o m p o s i t i o n

representation mod P k - i

for

Pn = Ho ~ HI ~ ' " ~

of

N

f e Pk ' Hn '

the is

(il : ii

Xn(m) <-->

.

,

n

where

I k = identity

Theorem Proof P

If If

n > r ,

this

1.7 ExamPleS

and

n > r ,

implies that

1)

~{k "

If

Xn then

is faithful. R[V ~ = I .

V = V1 ,

2)

vector fields, If

V = V1 ' V2 ,

d e f i n e d as in

§ 1.3 ,

dim V I = dim V2 = i by

X = ~/ax

and

,

then

N

~t

maps

N

H : V~

(x,y)

Vm g e n e r a t e s

is scalar multiplication

translations and

~1 = Pl 8 V 1 P2

Y = x~/~y ,

and

~2 = P2 "

~-2

i s spanned by

by

~ = all

by elements o f

are c o o r d i n a t e f u n c t i o n s .

and

n ,

H2 = (V~)2 + V~

Heidenberg a l g e b r a in t h i s

t

,

constant-

V . If

~k

is

Suppose Then

@/ay = ~ , Y ]

hi

i s spanned .

Thus

case. As a t r a n s f o r m a t i o n

~

group

acts by

Ii The space

N = all

then

then and

the t h r e e - d i m e n s i o n a l IR2,

and

Since

m= I .

i s the usual space o f homogeneous p o l y n o m i a l s o f degree

coefficient

on

on

the r e p r e s e n t a t i o n

m e Ker (Rn)

as an a l g e b r a ,

Hn

transformation

P2

has b a s i s

t i e IR

÷x+tl ÷Y+t2x+t

i

3

, x , y , x

2

,

and the f a i t h f u l

onto matrices

\0

I

t2

0 0

10l 0

2tl) /

t i e IR

representation

~2

is

12 § 2.

2.1 F i l t r a t i o n s Lie algebra over

B i r k h o f f Embedding ' Theorem

on n i l p o t e n t Lie algebras

IR .

A positive filtration

~1 ~ 2 ~ 3

~ "'"

~ = ~1 '

g-n = 0

Let

F

of

for

n

~ g

be a f i n i t e - d i m e n s i o n a l is a chain o f subspaces

such t h a t

i Proposition

Proof

Set

g

i

~j

' g~]

~

is n i l p o t e n t

=~ ,

?+i

~j+k <--~ there e x i s t s a p o s i t i v e f i l t r a t i o n

= ~g

, ?],

central s e r i e s . By d e f i n i t i o n ,

g

large. Thus we must show t h a t

~_g_m , _gn ]

definition,

for all

n .

large

so t h a t

{gg}

~m+n .

For

Assume t h a t f o r some m i t [-gm+l , gn ]

: [ [_~ , ~ m ]

gn = 0

m= 1

is true f o r a l l

by i n d u c t i o n :

~ [~_ , ~m+n ] + ~ g m , g__n+l]

and g

is n i l p o t e n t ,

and i n d u c t i v e l y ,

(2)

Vn

so t h a t

~

F = {g_n}

-~n = Vn ~ g-n+l " ~ = V1 ~...@ Vr

where

r

ixl

and the spaces H n , Pn

,

Then

, [_g , g_n] ]

n

c_~

.

then

1

= ~1 '

Hence a n = 0

for

so n

large,

Q.E.D.

Fix a p o s i t i v e f i l t r a t i o n subspace

, [~_m , ? ] ] + [ ~

{g_~} is any p o s i t i v e f i l t r a t i o n ,

' ~1 ] ~g-~ '

n .

m+n+1 g

c

= [~1

n

, gg]

~ [g

2

for

t h i s is t r u e , by

by Jacobi:

Conversely, i f

~ .

is the descending

is n i l p o t e n t i f an o n l y i f

_c

of

is the length o f the f i l t r a t i o n .

on

~ .

For each

n ,

choose a l i n e a r

Then as a vector space, , Define d i l a t i o n s

o f polynomials on

g_ as in

~t ' § 1 ,

homogeneous norm r e l a t i v e to

13 t h i s decomposition. Denote by By the remark in

§ 1.1 ,

(Set

Vn = 0

the n i l p o t e n t group c o n s t r u c t e d in

the spaces

depend on the f i l t r a t i o n ,

Definition

N(F)

and hence the group

NIF) ,

only

and n o t on the choice o f complementary subspaces

The decomposition

for

Pn '

§ 1 .

n > r).

(~)

is a g r a d a t i o n o f

E q u i v a l e n t l y , the d i l a t i o n s

decomposition are automor~hisms o f the Lie algebra

~

6t

~

if

[Vj,V~

Lie algebra

~ ,

~Vj+ k

associated w i t h the

in t h i s case.

2.2 A l g e b r a i c comparison o f a.d.ditive and n i l p o t e n t . g r o u p s t r u c t u r e s the n i l p o t e n t

Vn .

we d e f i n e a L i e group s t r u c t u r e on

~

Given

by the

Campbell-Hausdorff f o r m u l a . Recall t h a t the formula asserts t h a t , e.g. i f one considers the algebra o f formal power s e r i e s in two non-commuting i n d e t e r m i n a n t s X, Y,

then

(~)

e X eY

=

eF(X, Y)

,

where

= x +

+ ½ :x,,:

+ 1 Ex,Ex,

i s a ( u n i v e r s a l ) formal L i e s e r i e s in

:

+1

E,,:,,x::+...

X,Y .

Write F(X,Y) = X + Y + T(X,Y) . By the i d e n t i t y

(~)

Now f o r series

x,y e g ,

~(X,Y) .

T : ~ x ~ g

one sees t h a t

,

Since with

define ~

T

has no terms in

$(x,y)

is nilpotent,

by s u b s t i t u t i o n

~(O,y) = T(x,O) = 0 .

structure to

~ .

group generated by

(~)

it

The s t r a i g h t x .

alone or in

Y

alone.

in the formal Lie

t h i s d e f i n e s a polynomial map

xy : x + y + ~ ( x , y ) By the formal i d e n t i t y

X

Set .

f o l l o w s t h a t t h i s composition gives a group line

Denote by

t~--+ tx , t e R , G the space

~

is the one-parameter

w i t h t h i s L i e group s t r u c t u r e .

14

To s t u d y t h e a l g e b r a i c comultiplication canonical

structure

on the a l g e b r a

identification

of

P

o f t h e map

T ,

we d u a l i z e

of polynomial functions

P @P

with the polynomial

on

to obtain

g .

functions

a

We make the

on

]@ ~ .

Then we can d e f i n e an a l g e b r a homomorphism

:P÷P@P u(f)(x,y) Similarly,

using the additive

= f(xy)

structure

. of

, we d e f i n e

4 :P ÷P@P A(f)(x,y)

= f(x+y)

We want t o compare t h e homomorphisms

The f i l t r a t i o n

{Pn }

of

P

Qn = (P @ P)n '

and

4 .

(If

induces c a n o n i c a l l y

(P ~ P)n : Write

~

.

• z

l+j
Proof

If

It

is clear

a linear

P®P

Since

u(f)

4

: O} .

+ mod Qn "

= A(f)

+ + Qm Qn-~Qm+n "

If

~ e~:~ ,

i s a homomorphism, t h i s

we c o n c l u d e t h a t

is a s u b a l g e b r a .

¢ e ~

any l i n e a r

Lemma

If

.

Then

the subspace o f

Thus we o n l y need v e r i f y

then

implies P

that

A(Pn)~Q

f o r which the

the theorem when

f

~(¢) = 4 ( ¢ ) + ~ o T •

But

T

is a Lie polynomial,

terms.

c(x,y)

by

,

function.

Let without

then

that

From t h e s e two p r o p e r t i e s theorem i s t r u e

on

~ = A.)

and l e t

f e Pn '

A(¢) = ~ @ i + i @ ~ .

a filtration

then

Pi @Pj

+ Qn = {h e Qn : h ( x , O ) = h ( O , y )

Theorem

~,g_~ = 0 ,

i s any L i e p o l y n o m i a l w i t h o u t

linear

t e r m s , then t h e r e

is

n •

15 +

exist polynomials

qk e Qk

and elements

c(x,y)

Proof o f Lemma Write are dual bases f o r

=

z k>2

qk ( x , y ) z k

x = z ~i(x)x i

g , g~ ,

with

such t h a t

z k e ~k

,

y = z ~i(Y)X i

x i e Vn. '

,

where

{x i }

, {~i }

~i e Vni

1

Then Ix,y] where

zij

= ~i,xj~

: s ~i(x)~j(y)

e g~i+n j

and

zij

since

~(~k) = 0

when

Remarks 1.

Let

T ,

k > n .

{~i }

The Lemma f o l l o w s by

~i @ Cj e Q~i+n j

i n d u c t i o n on the l e n g t h o f the commutators in A p p l y i n g the lemma to

,

c .

we see t h a t

if

~ e V~ ,

+

then

C o .T e Qn '

This proves the theorem.

be a basis f o r

o f the theorem is e q u i v a l e n t to the f o l l o w i n g

~:~ w i t h

~i e Vn. The statement i statement about the formula f o r xy

in canonical c o o r d i n a t e s :

Set

ai = ~i(x)

,

bi = ~i(y ) , ci = ~i(xY)

.

Then c i = a i + b i + q i ( a l . . . . . ai_ I , b I . . . . . bi_ 1) where it

qi e Q+ (Since qi ni depends o n l y on the a j , b j From t h i s formula i t

is i n v a r i a n t Jacobian

contains no terms in with

d e t e r m i n a n t is

on

~ ~ ~ ,

so t h a t relative

alone or

{bj}

alone,

f o r example, t h a t Euclidean measure on

translation

by elements o f

G ,

g

since the

i .

2. Suppose ~ = V1 ~---@ Vr (i + j = n) ,

{aj}

nj < n i .)

is c l e a r ,

under r i g h t and l e f t

,

Kn

is a g r a d a t i o n o f

g .

Set

Kn = s Hi ~ Hj

c o n s i s t s of the homogeneous p o l y n o m i a l s o f weight

to the d i l a t i o n s

(~t x

, ~ty ) .

Let

K~

n

be the same sum,

I@ but w i t h

i ~ 1 , j > 1

and

i + j = n .

u(f) Indeed,

at

= A(f)

Then

mod

K+ n '

VfeHn

is an automorphism in t h i s case, so

t h i s p r o p e r t y holds f o r l i n e a r f u n c t i o n s the same argument as before, with

f

,

and hence f o r a l l

r e g u l a r r e p r e s e n t a t i o n s of R(x) f ( y )

Thus

p o l y n o m i a l s , by

R, L : G ÷ Aut (P)

be the

P :

, L(x) f ( y )

= f(x'ly)

since the f u n c t i o n s in

. P

separate the

G .

Theorem filtration

R and

F of

Proof right at

Let

G on

= f(yx)

Both these r e p r e s e n t a t i o n s are f a i t h f u l , points of

.

+ replaced by Kn , K ÷n Qn ' Qn

2.3 Fa..ithful unipoten.t r e p r e s e n t a t i o n s r i g h t and l e f t

T ( a t x , 6 t y ) = 6t T ( x , y )

Let

L

map

G i n t o the group

N(F)

associated w i t h the

~ .

Px : P Q P ÷ P

be the homomorphism d e f i n e d by " e v a l u a t i o n on the

x": Px(F)(y) : F(y,x)

Then

RX

PX o ~

a d d i t i v e group o f + Now Qn w(~) ~ 1 ,

Also

.

~

on

x-~ Px o A P ,

is simply the r e g u l a r r e p r e s e n t a t i o n o f the

which c l e a r l y

is spanned by monomials

w(B) > 1 .

is in

C ~ ~

In p a r t i c u l a r ,

,with

w(m) <_ n-1 ,

Combining these f a c t s , we see t h a t when

feP

n ,

Rxf = Px u ( f ) = Px A ( f ) = f

mod

mod Pn-i

N(F)

Px (Q~)

.

w(~) + W(B) ~ n so t h a t

and

Px (Q~) ~ P n - 1

"

17 (we have used the theorem o f This proves t h a t be d e f i n e d by

Jf(x)

the inverse to

§ 2.2

t o pass from

Rx e N(F) .

= f(-x)

x

in

G).

Corollary 1

If

dR

.

To pass to l e f t

Then

Hence

JPn = Pn

L x e N(F) ,

= ~t

translation,

and

JRxJ = L x

let

J e Aut (P)

(Note t h a t

-x

is

Q.E.D.

i s the d i f f e r e n t i a l

dR(x) f ( y )

~ t o A) .

of

R:

f(y(tx)) t=o

then

dR

is a Lie a l g e b r a homomorphism from

Corollary 2

If

Un(X ) = R ( x ) I p n ,

unipotent representation of Un

is f a i t h f u l

when

Suppose

n = n.l Q...@ ~ r

to

Un

I f the f i l t r a t i o n

n .

is a finite-dimensional F

is o f l e n g t h

r ,

then

n ~ r .

Both these c o r o l l a r i e s Remark

G .

then

~

f o l l o w i m m e d i a t e l y from the theorem and

~ = V1 Q . . . Q Vr

be the g r a d a t i o n o f x e Vk ~

This is e a s i l y proved d i r e c t l y

is a ~ r a d a t i o n o f ~

defined in

g .

§ 1.3 .

§ 1.4 - 1.6 .

Let Then

dR(x) e . n k .

from the f a c t t h a t

case, using the remark a t the end o f

6t

is an automorphism in t h i s

§ 1.3 .

§ 3. Comparison o f group s t r u c t u r e s 3.1 Norm compar.iso.n o f a d d i t i v e and n i l p o t e . n t s t r u c t u r e s

In t h i s s e c t i o n

we want to convert the a l g e b r a i c i n f o r m a t i o n in Theorem 2.2 i n t o an e s t i m a t e f o r the d i f f e r e n c e between the a d d i t i v e and n i l p o t e n t group s t r u c t u r e s on the n i l p o t e n t L i e algebra on

~

~ .

Let F

be a f i l t r a t i o n

compatible w i t h the decomposition

on (~)

~ , in

and choose a homogeneous norm § 2.1 .

18 Theorem (t)

There i s a constant

C > 0

such t h a t

I x y - x - y I ~ C { I x l a l y l z-a + I x l a l y l a + i x l l - a l y l

r = length of filtration Proof

Since a l l

{~i }

with

Pie

i s a basis f o r

Q+ ni ,

by

w(~) + w(~) ~ n i

a = 1/r ,

homogeneous norms are e q u i v a l e n t , we may assume t h a t

V~

l~i(x) 1 with

~i(xy-x-y) where

where

F .

txl : max where

a} ,

~i e V~i

= pi(x,y)

Theorem 2.2 . and

1/n i

But

Then

, Pi

is a sum o f monomials

w(~) ~ 1, w(B) ~ 1 .

Since

i~(x)l

~(x)~(y)

~ Ixl w(~)

,

, we

thus have ipi(x,y)I where

1 ~ j,

k

and

~ c max { I x l J l y l k}

j + k ~ ni .

From t h i s we o b t a i n the estimate

I x y - x - y I < C max { [ x l J / n l y [ where the max is taken over a l l and

,

integers

j,k,n

k/n}

with

, j ~ 1, k ~ 1, 2 ~ n < r ,

j+k
IYl Z Ixl > 0

and

write IxlJ/nlylk/n

We may assume and f o r

j,k,n

r > 2 ,

= I~-~) j / n

since o t h e r w i s e

lyl(j+k)/n

~,_g_~ = 0

and

xy = x+y .

in the i n d i c a t e d range we have j/n

~ a ,

2a < ( j + k ) / n

~ i

Hence i x l J / n l y l k/n < (

max { l y l 2a

lyl}

Thus

a < 1/2,

19 Interchanging

x

Corollary

Ixl

y ,

we g e t e s t i m a t e

Suppose the f i l t r a t i o n

i s a homogeneous norm r e l a t i v e

(t,) where

and

Ixy-x-yl a = l/r,

F

(t)

comes from a g r a d a t i o n of

g__ ,

and

to the g r a d a t i o n . Then

~ C { I x l a l y l l - a + I x z Z - a l y [ a} ,

r = length o f

Proof of C o r o l l a r y

Let

F .

Ci ' Pi

be as in the p r o o f above. Since

at

is

an automorphism o f the L i e algebra in the graded case, we have ni Pi ( a t x ' ~t y) = t Hence

Pi

is a sum o f monomials

w(B) ~ i .

C(x)~B(y)

occur in the f i n a l

j + k = n .

w(~) + w(~) = n i ,

w(~) > I ,

Hence the term

The only d i f f e r e n c e between the f i l t e r e d

or

Ixlalyl a

does not

estimate.

these estimates is the behaviour near Ixl ~ ~ > 0

with

By the p r o o f j u s t g i v e n , t h i s leads to the same estimates as b e f o r e ,

but now w i t h the c o n s t r a i n t

Remark

Pi ( x , y )

lyl ~ E > 0 ,

x = O, y = 0 .

estimates

3.2 A l g e b r a i c comparison o f f i l t e r e d

(t)

and

o f the n i l p o t e n t algebra

p o t e n t Lie algebra

using

gr(g) :

F ,

As long as e i t h e r (tt)

are e q u i v a l e n t .

and graded s t r u c t u r e s

be a decreasing f i l t r a t i o n gr(~)

and the graded case in

as f o l l o w s :

~ .

Let

F = {g_~}

We c o n s t r u c t a graded n

Set

z @ (~n / ~n+l ) n>l

and d e f i n e + g-m+1 ' Y + g-~+l~ = ~ ' Y ] when

X e ~m' Y e ~n •

+ g-m+n+l '

The r i g h t - h a n d side o f t h i s formula o n l y depends on the

20 equivalence classes o f filtration into

X, Y mod g-m+1 '

g-~+l '

r e s p e c t i v e l y , by v i r t u e of the

c o n d i t i o n . Extending t h i s bracket o p e r a t i o n to a b i l i n e a r map o f gr(g)

gr(~) ,

we obtain a Lie algebra s t r u c t u r e

(skew-symmetry and the Jacobi

i d e n t i t y f o l l o w immediately from the corresponding i d e n t i t i e s

in

9) .

In t h i s section we want to make an a l g e b r a i c comparison between the Lie algebras

~

and

gr(~) . Pick a l i n e a r map m : g ÷ ~(X) : X + g-n+1

Then

,

if

gr g

such t h a t f o r a l l

X e g_n •

is a l i n e a r isomorphism, and we t r a n s f e r the Lie m u l t i p l i c a t i o n

to

gr(~)

from

by d e f i n i n g ~(x,y) : ~ ([~-Ix, -ly])

I f we denote property

g-n / -~n+l = Vn' gr(g_) = V ,

[~q, ~nl] ~g-m+n

then we see from the f i l t r a t i o n

t h a t the b i l i n e a r map u

can be w r i t t e n as a f i n i t e

sum of b i l i n e a r maps

(1)

P = ~o + P l + ' " + P r - 1

'

where Pk : Vm x Vn ÷ Vm+n+k (r = length of the f i l t r a t i o n ) . p l i c a t i o n on

In p a r t i c u l a r ,

St

on

x e Vn

is a b i l i n e a r map, d e f i n e

~t b ( x , y ) = a l / t The maps Pk

is skew symmetric.

V by

6t x = tnx,

b:VxV+V

is the Lie algebra m u l t i -

V defined above. Each o f the maps Uk

Define d i l a t i o n s

If

Po

b(~ t x, ~t y) "

are thus homogeneous of degree

k :

~ Pk = tk ~k " Thus

at u = Uo + t u l +" . "+ t r - 1 Ur_ 1

n ,

21 In p a r t i c u l a r , lim t+o

6t ~ = ~o "

Note t h a t f o r every

t # o ,

and the Lie a l g e b r a

(V, 6~ ~)

Thus

gr(g) When i s

6t u

is in the c l o s u r e gr(g)

defines a L i e algebra m u l t i p l i c a t i o n

is isomorphic to

choices are o f the form

above so t h a t ms ,

mIvn

where

map v

on

V

to

g ?

Uk = 0

Identity

(mod

from

g

V ,

map 6 t o m.

~ .

This w i l l

for

~ : V÷ V

I f we t r a n s f e r the Lie m u l t i p l i c a t i o n bilinear

v i a the l i n e a r

o f the isomorphism class o f

a c t u a l l y isomorphic

we can choose the map ~

~ ,

on

occur e x a c t l y when

k ~ 1 .

But the p o s s i b l e

i s l i n e a r and

k>nE Vk) . to

V

using

~m ,

we o b t a i n a

such t h a t

m ~ ( x , y ) = ~(mx, mY) • As b e f o r e , we decompose v

(2)

i n t o i t s homogeneous p a r t s :

~ = Vo + ~ I + ' " '+ U r - i

where

'

6~ ~k = tkuk "

We have

~o = ~o '

To compare

since t h i s gives the L i e m u l t i p l i c a t i o n (1)

and

(2) ,

we note t h a t

= I + ml + ' ' ' +

of

gr(~!

.

m can be w r i t t e n as

mr-1 '

where ~k : Vn ~ Vn+k Hence equating terms o f the same degree o f homogeneity ( r e l a t i v e the r e l a t i o n s (3)

Z m+n=p

~m~n(X,y) =

z ~k(~ix,~jy) i+j+k=p

,

to

6t)

gives

22 for

0 ~ p ~ r-1

can pick

~k

(mo = I d e n t i t y )

so t h a t

Vk = 0

.

for

In p a r t i c u l a r ,

= gr(~)

i f and o n l y i f we

k ~ 1 .

To express these equations in a more i n f o r m a t i v e way, we i n t r o d u c e the coboundary o p e r a t o r a s s o c i a t e d w i t h the L i e a l g e b r a space o f a l t e r n a t i n g ,

n-linear

maps from

V

to

gr(#)

V .

.

Let

cn(v,v)

be the

Define

: cn(v,v) ÷ cn+I(v,v) by the formula

a f ( x l . . . . . Xn+l) = i # j

(-1)i+J

f(~°(xi'xj)'

x l . . . . . xi . . . . . x j . . . . . Xn+l)

z ( - I ) i A ( x i ) f ( x I . . . . . x i . . . . . Xn+l) Here

xi

means to omit

i.e.

A ( x ) y = ~o(X,y)

condition

62 = 0 .

xi ,

.

A

is the a d j o i n t

The Jacobi i d e n t i t y

For

6f(xl,x2)

and

n = 1

representation of

gr(~)

for

Uo

is e q u i v a l e n t to the

the formula f o r

~f

becomes

= Uo(f(xl),

x 2) + U o ( X l , f ( x 2 ) )

- f(uo(Xl,X2)) Using t h i s ,

we o b t a i n from

Proposition

the f o l l o w i n g

The L i e algebras

e x i s t l i n e a r maps

mp on

(4) where

(3)

V ,

~

P

e C2(V,V)

gr g

mp : Vn ÷ Vn+ p ,

6rap = Up + Fp F

and

criterion:

,

are isomorphic

<--> There

such t h a t

1 <_ p <_ r-1

is d e f i n e d by

Fp(x,y) =

p-I z mk ~p-k ( x ' y ) k=l

- ~o(mkX'mp-k y)

(F z = O)

Remarks 1.

The set of equations

(4)

seems q u i t e i n t r a c t a b l e .

There is

,

23

additional satisfies

information a v a i l a b l e , h o w e v e r , which we have not used; namely, t h a t the Jacobi i d e n t i t y .

the i n t e r i o r ' ~roduct of

This can be expressed most n e a t l y by i n t r o d u c i n g

f , g e C2(V,V):

(f.g)(xl,x2,x3)

and is defined by

: f(g(xl,x2),x3)

+ f(g(x2,x3),Xl) The Jacobi i d e n t i t y

f - g e C3(V,V)

is then

~'u = 0 .

+ f(g(x3,xl),X2)

.

Using the decomposition

(1)

of

~

this

gives the equations

I

(5)

u o • uo = 0 Uo " ~I + ~i " Uo = 0 p-i -

,

u o • Up + Up • u o = kZ=l ~k

We have already noted the f i r s t

~p-k

of these equations. As f o r the o t h e r s , we c a l c u l a t e

that

u o " Up + Up " uo = -~Up Hence

uI

is a

2-cocycle

(~I

= O) ,

and the

Up

for

p > I

satisfy

p-1 aUp = kZ=l u k - Up_ k

We conclude t h a t necessary c o n d i t i o n s f o r s o l v i n g starting with

p = 1 ,

2. The f i l t r a t i o n ~F~ e H2(gr ~, A)

F

determines an i n t r i n s i c

(A = a d j o i n t representation of

remark

i

,

and

~1

Up + Fp

,?p ,

be zero.

cohomology class gr g__) .

Indeed by

(3) ,

= Uo(~lx,Y) + Uo(X,~ly) - ~lUo(X,y) = ~l(x,y)

uI

recursively for

are t h a t the cohomology classes of

Ul(x,y ) - ~l(x,y)

so t h a t

(4)

,

are representatives of the same cohomology class. By

t h i s class is the " f i r s t

o b s t r u c t i o n " to c o n s t r u c t i n g an isomorphism

24 between

g

and

gr(~)

Examples 1. isomorphic to to

~

in

I f the f i l t r a t i o n

g .

g ,

.

is of length

Indeed, in t h i s case

then

~ (a,b,c,d)

Xl,...,x 7

[Xl,Xn]

= 0 ,

then so i f

gr(g) VI

is

any complement

~J1,V~ ~g-~2 ' ~Vl'g-2~ = 0 .

2. Consider the f a m i l y algebras w i t h basis

~2'~

< 2 ,

o f seven-dimensional n i l p o t e n t Lie

and commutation r e l a t i o n s

~1'x7]

= Xn+ 1 ,

= 0

[x2,x3] = ax 5 + bx 6 + cx 7 )

[x2,x4] =

xz,x

with all

ax 6 + bx 7

:

7

Fx3,x4] :

d x7

o t h e r commutators obtained by skew-symmetry from t h i s t a b l e ( o r equal zero

i f they do not appear in the t a b l e ) . A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t Ad x i

is a d e r i v a t i o n o f t h i s algebra s t r u c t u r e ,

1 ~ i < 7 ,

and hence these

equations do d e f i n e a f o u r - p a r a m e t e r f a m i l y o f Lie a l g e b r a s . The descending c e n t r a l s e r i e s

g_n is given by

n ~_ = span {Xn+ 1 . . . . . x 7} when

2 < n ~ 6 .

Thus

~

is s i x - s t e p n i l p o t e n t .

o f the commutation r e l a t i o n s t h a t { n} .

But c l e a r l y

~ (0,0,0,0)

parameters are non-zero . when

~

W,

with

I t is obvious from the form

gr g ~ g (0,0,0,0) i s not isomorphic t o

,

r e ] a t i v e to the f i l t r a t i o n g (a,b,c,d)

when any o f the

Hence the descending c e n t r a l s e r i e s is graded only

is the s e m i - d i r e c t product o f

Lie algebra

,

ad x I

(Xl)

being the s h i f t

A more i n t e r e s t i n g f i l t r a t i o n

on

~

-~n = span {Xn,Xn+ 1 . . . . . x 7} ,

and a s i x - d i m e n s i o n a l commutative o p e r a t o r on

W.

is obtained by s e t t i n g

1 < n < 7

25

This f i l t r a t i o n

is o f l e n g t h seven, and i t

is obvious t h a t r e l a t i v e

to t h i s

filtration gr(g) = ~ (a,O,O,d) w i t h the parameters writing

Ix,y]

a

and

= ~(x,y),

d

,

arbitrary.

Letting

Vn = span (Xn)

we see t h a t the decomposition

(I)

of

~

,

and

i n t o a sum

o f homogeneous terms is given by

~o(Xl,Xn) = Xn+ 1

,

~o(Xl,X7) = 0

~o(X2,X3) = ax 5 ~o(X2,X4) = ax 6 ~o(X2,X5) = ( a - d ) x 7 ~o(X3,X4) =

dx 7

~1(x2,x3) = bx 6

,

~1(x2,x4) = bx 7

~ 2 ( x 2 ' x 3 ) = cx7

'

u3 = ' " =

u6 = 0

( A l l e n t r i e s not o b t a i n a b l e by skew-symmetry are z e r o ) . Let us examine e q u a t i o n homogeneous o f degree

p ,

(4)

o f the p r o p o s i t i o n in t h i s case. A map

is d e f i n e d by

~p Xn = a n Xn+ p where

an = 0

when

algebra structure

n > 7-p . ~o '

,

The coboundary o f

- mp(~o(Xi,Xj))

p = 1

relative

to the Lie

is thus the skew-symmetric mapping d e f i n e d by

a@p(Xi,Xj) : a i ~ o ( X i + p , x j )

When

~p ,

we c a l c u l a t e t h a t

+ ajuo(Xi,Xj+p)

~p

26

6@l(Xl,X2) = (a2-a3) x 4 aml(Xl,X3) = (aal+a3-a4) x 5 6ml(Xl,X4) = (aal+a4-a5) x 6 ~ml(Xl,X5) = ( ( a - d ) a l + a 5 - a 6 ) x 7 ~l(X2,X3)

= a(a3-as) x 6

6~1(x2,x4) = (da2+(a-d)a4-aa6) x 7

,

w i t h a l l o t h e r e n t r i e s not o b t a i n a b l e by skew symmetry equal zero. Hence the condition

6ml = ~I

i s e q u i v a l e n t to the s e t o f 6 l i n e a r equations

a2-a 3 = 0 a3-a 4 = -aa 1 a4-a 5 = -aa 1 a5-a 6 = (d-a) a 1 a(a3-a5) = b d(a2-a4) + a(a4-a6) = b

One c a l c u l a t e s t h a t t h i s inhomogeneous system has rank 5 , and o n l y i f e i t h e r if

a = 0

and

filtration

b = 0

b # 0 ,

(i.e.

Ul = O) ,

or else

then we cannot solve

Assume

a # 0 .

isomorphism between require solving

and

~mp = Up + Fp ,

when

to those j u s t made, we f i n d t h a t f o r while for

p = 2

the e x p l i c i t

~(a,O,O,d,)

(4)

a ~ 0

and

d # 0 ,

then

.

and the s o l u t i o n

~(a,b,c,d)

The h i g h e r - o r d e r .

d # 0 .

consistency c o n d i t i o n i s expressed by the equation that if

b # 0 .

obstructions

By c a l c u l a t i o n s s i m i l a r

these equations can always be s o l v e d ,

they can always be solved i f

form o f equation

hence the

o b s t r u c t i o n to d e f i n i n g an

p = 2,3,...,6 p > 2

In p a r t i c u l a r ,

~ and

~(O,b,c,d),

Then t h e r e i s no f i r s t - o r d e r ~(a,b,c,d)

a # 0 .

~ml = Ul

i s not e q u i v a l e n t to a g r a d a t i o n on

and is c o n s i s t e n t i f

If

ml '

d = 0 ,

we f i n d t h a t the

5b2 = 4ac .

~ g(a,O,O,d).

then using

Hence we conclude

27

Furthermore,

~ ( a , b , c , O ) ~ ~(a,O,O,O)

same a n a l y s i s shows t h a t

i f and o n l y i f

~(O,O,c,O) ~ ~ ( 0 , 0 , 0 , 0 )

F i n a l l y , the

i f and o n l y i f

is also obvious by i n s p e c t i o n o f the m u l t i p l i c a t i o n

3.3 Norm comparison o f f i l t e r e d

5b2 = 4ac .

c = 0 .

(This

t a b l e i n t h i s case).

and graded structures

Continuing the

n o t a t i o n o f the previous s e c t i o n , l e t us turn now to the question of a metric comparison between the group structures defined by l i n e a r map m : ~ + w i t h the subspace

xy

and

x,y

~

xy

V , xmy ,

group laws on

such t h a t

m(~n) .

the vector space write

gr(~)

V . and

gr(m) = I

g

as in

gr(~) .

§ 3.2 ,

corresponding to the Lie brackets

~

and i d e n t i f y

and

~0 "

We shall

r e s p e c t i v e l y , f o r the corresponding Campbell-Hausdorff

~ ~ xmy ,

[x I

on

is o f length

V .

are "asymptotic at i n f i n i t y "

the homogeneous norm, in the f o l l o w i n g sense (Recall t h a t filtration

Fix a

Thus we have two n i l p o t e n t Lie algebra structures on

Fix a dilation-homogeneous norm x,y

and

Then the maps

when measured by

~ = gr ~

i f the

~ 2) :

Theorem Assume the f i l t r a t i o n Then there is a constant

F

is o f length

r > 3 ,

and set

a = 1/r .

M so t h a t

Ixy-x*y[ ~ M ( I x l Z - 2 a l y l a + I x l a l y l a + I x l a l y [ 1-2a)

In p a r t i c u l a r ,

Proof

Ixy-xmy I ~ M ( I x l + l y I # -a

lim

Ixy-x*Yl

IxI+IYl ->~

Ixl+lyl

x,y e V

~i e Vni .

Thus

0

is given by a universal

s u f f i c e to compare the r e s u l t of e v a l u a t i n g a formal Lie p o l y -

Pick a basis and

Ixl + IYl ~ 1 .

Since the Campbell-Hausdorff m u l t i p l i c a t i o n

formula, i t w i l l nomial at

=

if

,

using the two Lie algebra structures on

{x i }

for

V and dual basis

We can w r i t e , by equation

(i)

{~i } of

for

§ 3.2 ,

V .

V* ,

with

xie

Vni

28

~ =

where

~0

+

B

,

B(Vm'Vn) ~g-m+n+l

Hence f o r the formal Lie element

c(x,y) = ~,~

(~)

~ ( x , y ) = ~o(X,y) + z ~ i ( x ) ~ j ( y )

where

zij

c , ,

= B(xi,xj)

l e t us w r i t e and w r i t e

c(x,y)

cm(x,y)

Then by equation

(m)

e_~ni+nj+ 1

we have

zij

More g e n e r a l l y , f o r any Lie polynomial

f o r the r e s u l t of s u b s t i t u t i o n using the Lie bracket f o r the r e s u l t of s u b s t i t u t i o n using the Lie bracket

Uo

and induction one sees t h a t

c ( x , y ) = cm(x,y) +

s qk ( x , y ) z k k>2 =

÷

where

qk e Qk and

the maps x , y

z k e g-k+1 ( n o t a t i o n of

~ ~ qk(x,y) zk .

As in the proof of Theorem lqk(x,Y)I where

11 i,j

§ 2.2). Thus i t s u f f i c e s to estimate

and

3.1 ,

! C max { I x l i l y l i+j < k .

we have the estimate j}

Since

,

z k e g~+ 1 ,

l q k ( x , y ) Zkl ! C max ( I x l i / n where the max is taken over

k+l < n < r ,

dominant term in t h i s estimate, we assume ixli/n

lylj/n

= (~)i/n

In the i n d i c a t e d range we have

~_)a

lyl j / n }

,

with

i,j

as before. To f i n d the

lyl ~ Ixl > 0 ,

and w r i t e

lyi(i+j)/n

i/n ! a

l q k ( x , y ) Zkl <_ C (

it follows that

and 2a ! ( i + j ) / n

(IYl 2a + lyl

l-a)

! 1-a .

Hence

29 Symmetrising t h i s estimate with respect to

x and y ,

we o b t a i n the f i r s t

estimate o f the theorem. To obtain the second estimate, we note t h a t

( I x l l y l ) a ~ (ixl + lyl) 2a IXl 1-2a l y l a

Since

a ~ I/3 ,

one has

_<

(Ixl + lyl) l-a

2a ~ 1-a ,

( I x l + l y l ) 2a < _ ( I x l + l y l ) 1-a

so t h a t f o r

Ixl + IYl ~ 1 ,

This gives the second estimate and completes

the proof of the theorem.

Corollary

Let

{a t }

group s t r u c t u r e

x*y

of

be the d i l a t i o n s associated with gr(~)

gr(~)

is obtained from the group s t r u c t u r e

Then the xy

of

by

x*y : lim t-~o

Proof we have

Set

al/t

((atx)(6ty))

x t = at x ' Yt = atY "

x*Y = ~ i / t

(xtmYt) "

IxmY - a l / t

Since

at

is an automorphism o f

gr(~)

,

Hence by the theorem

(xtYt)i

= t-1

IxtmYt - x t Y t l

M t -a ( I x l Z - 2 a l y l a + i x i a l y l l - 2 a ) + M t 2a-I I x l j y i . But

xmy = xy

if

a ~ 1/2 ,

so we may assume

Ixmy - a l / t when

t ÷ ~ ,

a ~ 1/3 .

Then t h i s estimate gives

( x t Y t ) I : 0 ( t -a)

proving the c o r o l l a r y .

Remarks 1.

This C o r o l l a r y makes more precise the statement above t h a t the

group structures defined by

~_ and

gr g_ are "asymptotic at i n f i n i t y " .

Namely,

30

to c a l c u l a t e

xmy ,

product

xt Yt

to

w i t h an e r r o r

xmy 2.

length

All r

'

we d i l a t e

and then s h r i n k O(t -a)

and

y

by

back w i t h

6t 51/t

,

t "

v e r y l a r g e , form

the

T h i s g i v e s an a p p r o x i m a t i o n

.

the estimates of this

of the filtration

x = O, y = 0 .

x

s e c t i o n are monotone i n . c r e a s i n g f u n c t i o n s

(a = l / r )

,

as long as we s t a y away from

of the

31 Comments and references f o r Chapter i

§ 1,1-1.2

The notion of "homogeneous norm" (=gauge) and d i l a t i o n s was

systematically developed in Knapp-Stein [ ~ ,

and then in Kor~nyi-V~gi [1]. We

have followed Kor~nyi-V~gi in requiring that the norm be homogeneous of degree one r e l a t i v e to d i l a t i o n s . For many applications the smoothness and symmetry of the norm are i r r e l e v a n t , so we have not included these conditions as part of the d e f i n i t i o n . (In many estimates, norms l i k e

§ 1.3

Ixl~

are the most convenient to use.)

The Lie algebra of vector f i e l d s with polynomial coefficients has

been studied, e.g. in Auslander-Brezin-Sacksteder [ I ] , Arnol'd [ 1 ] , and Goodman [6]. (cf. Appendix). The use of the homogeneous norm to define the order of vanishing of functions at a point is taken from Folland-Stein ~ ]

and Rothschild-Stein [ I ] ,

The Corollary appears to be new.

§ 2.1

One of the e a r l i e s t uses of the f i l t r a t i o n

given by the descending

central series of a n i l p o t e n t Lie algebra is in Birkhoff [1], where i t is extended to a f i l t r a t i o n

of the universal enveloping algebra by ideals

Jn

of f i n i t e

codimension. The process of using a f i l t r a t i o n

on ~

algebraic objects f u n c t o r i a l l y attached to

has been systematically developed

by Vergne [1] and Rauch [1], [2]. integral f i l t r a t i o n s ,

~

to define f i l t r a t i o n s on

In these notes we r e s t r i c t our attention to

because these suffice f o r the study of i t e r a t e d commutators

of vector f i e l d s . For the study of a r b i t r a r y positive f i l t r a t i o n s ,

cf. Goodman [6].

We also r e s t r i c t our attention to one-parameter groups of diagonalizable automorphism

{6t }.

For the case of an n-parameter commutative group of diagonalizable

automorphisms of a n i l p o t e n t Lie algebra, cf. Favre [1].

§ 2.2

The theory of free Lie algebras and the Campbell-Hausdorff formula

can be found, e.g. in Bourbaki [1], Hochschild [I~, and Jacobson [1].

32 § 2.3

The construction given here is the dual to the construction of

B i r k h o f f [ I ] . The l i n e a r dual space to the enveloping algebra of isomorphic to the formal power series functions on ~ a n n i h i l a t o r of the ideal degree

< n .

Jn+l

~

is canonically

(cf. Dixmier [23]),and the

of Birkhoff is the space

Pn of polynomials of

For the analogue of the B i r k h o f f construction in the case of a

solvable Lie algebra, cf. Reed [ I ] .

§ 3.1

The comparison theorem was proved in Goodman [6]. I t implies, in

p a r t i c u l a r , that

I xy[ ~ a ( I x l + IYl + b) ,

f o r some p o s i t i v e constants a and b .

Similar estimates were proved by Guivarc'h [ 1 ] , where i t is shown that there exists a norm of the type

IxI=

f o r which the constant

i f there is a smooth norm which s a t i s f i e s

a = 1 .

I xYl ~ Ixl + lyl + b .

We do not know See also

Jenkins [1].

§ 3.2

The main source f o r the results of t h i s section is the thesis of

Vergne [1]. (cf. Rauch [ ~ ,

[2] f o r f u r t h e r developments). For the general theory

of deformations of algebraic s t r u c t u r e s , c f . Nijenhuis-Richardson [ i ] ;

f o r Lie

algebra cohomology, cf. Jacobson [ I ] . An example of a n i l p o t e n t Lie algebra with every automorphism unipotent was given by Dyer [ I ] . This phenomenon was investigated f u r t h e r by M~ller-R~mer [ ~ ,

who showed that the 7 dimensional algebra

~ (0,I,0,i)

of Example 2 has t h i s property, but that every n i l p o t e n t Lie algebra of dimension < 6

admits d i l a t i n g automorphisms. The f a c t that the cohomology class of

almost always t r i v i a l nilpotent

~ ,

~1

is

in Example 2 contrasts with Dixmier's results [11: For any

one has

dim Hn(~,~) ~ 2

for

i ~ n < dim g - i . R.W. Johnson

[I] has studied the problem of constructing a gradation f o r the descending central series. § 3.3

The comparison theorem between the graded and f i l t e r e d group

structures was proved in

Goodman [6]. The Corollary is due to Auslander-Brezin-

Sacksteder E l i , in a s l i g h t l y d i f f e r e n t formulation.

Chapter I I

N i l p o t e n t Lie algebras as tangent spaces

§ 1.

T r a n s i t i v e Lie a l g e b r a # o f vector f i e l d s

1.1 Geometric background We denote by

~(M)

M w i l l be a real

the algebra of real-valued smooth functions on

TM the real tangent bundle to C=

In t h i s chapter

vector f i e l d s on

M .

If

M .

We l e t

x e M , TMx

L(M)

C~

manifold.

M , and by

be the Lie algebra of a l l real

w i l l denote the tangent space at

x .

In this chapter we want to construct a theory of manifolds "modeled on n i l p o t e n t Lie groups". The motivation f o r such a theory comesfrom several sources (see Chapter IV).

The basic geometric idea, however, is very simple. We would l i k e

to carry out in each vector space in a way that varies smoothly with

TMx

the constructions of Chapter I , and do t h i s

x .

This would mean putting the structure o f a

graded n i l p o t e n t Lie algebra on each tangent space algebra homomorphism from

T~x

to

L(M)

.

TMx , together with a Lie

(Here we are thinking of the group

manifold, on which each tangent space is isomorphic via t r a n s l a t i o n s to

g , and

acts as vector f i e l d s by the regular representation.) Thus t h i s construction would involve s p l i t t i n g each tangent space into a d i r e c t sum of subspaces, assigning a degree of homogeneity to each subspace and commutation r e l a t i o n s among the subspaces, respecting these degrees. We would then l i k e to use analysis on the n i l p o t e n t groups at each tangent space (convolution operators, e t c . ) to carry out local analysis on tangent spaces is t r i v i a l , groups of the vector spaces

M . When the s p l i t t i n g of the

so that the n i l p o t e n t groups are j u s t the a d d i t i v e TMx , t h i s is the essence of the "parametrix method",

which has been enormously developed and refined in recent years (the theory o f p s e u d o - d i f f e r e n t i a l operators and t h e i r symbolic calculus). Despite the f a c t that graded n i l p o t e n t Lie algebras e x i s t in great profusion, the problem of constructing a "bundle o f n i l p o t e n t structures" as posed above seems hopelessly overdetermined, at f i r s t

examination. To recast the problem into more

34 reasonable form, l e t us note the f o l l o w i n g : 1) on M

We want to avoid topological obstructions, so we are only working l o c a l l y

.

2)

For purposes of local analysis, i t is enough to model M on a

homoqeneous space f o r a n i l p o t e n t group, since functions on a homogeneous space can always be l i f t e d to the group

(This avoids the obvious problem of f i n d i n g a graded

n i l p o t e n t Lie algebra of the same dimension as 3)

Since we want a bundle s'~ructure, we should f i r s t

a f i x e d , graded n i l p o t e n t Lie algebra g . together with a map ~ : ~ ÷ 4)

M)

L(M)

~

choose a f i b e r , namely

The data should then be the algebra

Denote by

I t is unreasonable to require that

G the corresponding n i l p o t e n t group. ~

be an exact Lie algebra

homomorphism, since this is unstablle under small perturbations (see the example to f o l l o w ) . Instead, we should only require that 5) point

The condition

x e M,

~(~)x

(2)

~

be a " p a r t i a l homomorphism".

above, stated in terms of

~ ,

should be the f u l l tangent space at

is simply that at each x .

Now that we have relaxed the requirements on the data, there e x i s t many such pairs

(~,~) .

In f a c t , given any f i n i t e l y generated Lie subalgebra h

which is t r a n s i t i v e in the sense that (~,~)

h

--X

= TMx

for

x eM

L~)

we can construct

with the a d d i t i o n a l condition t h a t a set of generators f o r

a set of generators f o r

of

~

maps onto

h . (This follows from the existence of free n i l p o t e n t

algebras). Suppose now that we have a p a i r

(~,x)

satisfying

3), 4), 5) (the precise

d e f i n i t i o n o f " p a r t i a l homomorphism" w i l l be given in the next section). The next step, in viewing M as an "approximate" homogeneous space f o r functions from

G , is to l i f t

M to G . This is no problem, at least l o c a l l y . I d e n t i f y i n g

as manifolds as always, we can l i f t neighborhood ~ of 0

in

g

a function

by setting

f(u) = f(e~(U)x)

f on M to a function

G with f

on a

35

Here x

e ~(u)

is the l o c a l flow on

is a f i x e d p o i n t o f

(the map u + e~(U)x If

M generated by the vector f i e l d

~(u) , and

M . This defines a map W : f ÷ f , which is i n j e c t i v e is s u r j e c t i v e , by

~ were a f u l l

(5)).

Lie algebra homomorphism, then

and the r e g u l a r r e p r e s e n t a t i o n of

~ .

W would i n t e r t w i n e

Since we only assume t h a t

homomorphism, t h i s is not e x a c t l y true. The l i f t i n g

i

is a p a r t i a l

problem, whose s o l u t i o n forms

the core o f t h i s chapter, is to f i n d a p a r t i a l homomorphism A : g ÷ L(2) s a t i s f y i n g the t r a n s i t i v i t y

condition

(i)

A(Y)W : W~(Y)

(ii)

A(Y)

(5) , such t h a t

is " w e l l - a p p r o x i m a t e d " by the l e f t - i n v a r i a n t

vector f i e l d on Once the l i f t i n g

G determined by

Y .

problem is solved, we can use the i n t e r t w i n i n g map W to

t r a n s f e r problems concerning the a c t i o n of the vector f i e l d s about the "almost i n v a r i a n t " vector f i e l d s

theorem", we must make precise the

homomorphism" and the approximation involved in

1.2 P a r t i a l homomorphisms

to problems

^(Y) .

Before s t a t i n g and proving the " l i f t i n g notion of "partial

~(Y)

To motivate the d e f i n i t i o n ,

(ii)

above.

consider the f o l l o w i n g

example: Let line

X = @/@x ,

Y : x ~/@y ,

acting on IR2 . Away from the

x = 0 , t h i s p a i r of vector f i e l d s spans the tangent space. On the l i n e

the vector f i e l d generated by i f we set

Y

X and Y

Z = ~,Y~

vanishes. However, acts t r a n s i t i v e l y , , then

X,Y,Z

[X,Y~ = ~/~y , so the Lie algebra in the i n f i n i t e s i m a l

sense. Furthermore,

e x a c t l y span a n i l p o t e n t Lie algebra. I f

the Heisenberg Lie a l g e b r a , with basis

x = O,

~,n,~

satisfying

~,~]

= ~ ,

~

then the

is

36 map

x : ~ -~ X , n ÷ Y , ~ ÷ Z

vector fields

on

R2 , and

x

extends to an ( e x a c t ) homomorphism from

is a smooth f u n c t i o n w i t h still

algebra generated by

X and Y'

m(n)(o) # 0

Y ' - m(x) a/~y , where

Then the v e c t o r f i e l d s

is g e n e r i c a l l y n e i t h e r n i l p o t e n t nor f i n i t e for infinitely X,Y' and Z'

: ~ ÷ X , n ~ Y' , ~ ÷ Z'

L(IR 2) such t h a t

by

span the tangent space a t each p o i n t . However, the L i e

o n l y the subspace spanned by ~'

x ~/~y

m(o) = 0 , m ' ( o ) = 1 .

X,Y' and Z' -= EX,Y~

let

to

is transitive.

Suppose we p e r t u r b t h i s example by r e p l a c i n g

dimensional ( i f

~

, then

many

n) .

On the o t h e r hand, i t

is

which is g e o m e t r i c a l l y i m p o r t a n t . I f we extends to a l i n e a r map from

~'

~,'([~,~]) = ~(~),~'(n)]

(but in g e n e r a l ,

g

into

~'(~,~)

# [x'(~),x'(~) Let us t u r n to the general s i t u a t i o n .

Suppose

r

_g =

S ~ Vk k=1

,

~j,Vk~ GVj+ k

i s a graded n i l p o t e n t L i e a l g e b r a , and

Definition

A l i n e a r map

graded s t r u c t u r e o f

:

h =

the l i n e a r map

Example

L(M) ~

is a p a r t i a l

homomorphism ( r e l a t i v e

is surjective at

×1 . . . . . Xn e

generated by the

x e M if

L(M) g

on

M , we w i l l

say t h a t

~(~)x = TMx "

are g i v e n , and l e t

{×k } .

e x i s t s a canonical graded Lie a l g e b r a

~

be the Lie sub-

Then f o r every i n t e g e r of length

r

r ~ I , there

and a p a r t i a l

homomorphism

: g ÷ h , as f o l l o w s : Let

F

to the

j + k 2 r .

is the L i e algebra of v e c t o r f i e l d s

Suppose

L(M)

algebra o f

~ : g ÷ h

EX(x) , X(y~

x e Vj , y e Vk , and

When

is any L i e a l g e b r a .

g ) , if

x(Ex,y~) whenever

h

be the free Lie algebra on

n

generators

Y l . . . . . Yn ' and l e t

37

= F/F r + l nilpotent

, with

Yk = Yk + F r + l

L i e a l g e b r a on

Proposition Vk = span

of

n

generators

~ = V1 ~ . . . ~

k-fold

" Then by d e f i n i t i o n ,

Vr

is the free r-step

Y1 . . . . . Yn "

is graded, with

commutators o f

g

and

V 1 = span { Y I . . . . . Yn }

{ Y i } . There e x i s t s

a unique partial

homomorphism : ~ ÷ F

such t h a t

~(Yi ) = Yi

(Thus we can s p l i t

" 0 ,* F r + l ÷ F + g ÷ 0

t h e e x a c t sequence

by a p a r t i a l

homomor-

phism).

Proof

Let

{H a}

be a H a l l

basis for the algebra

L i e monomial whose l e n g t h we d e n o t e by that

Fn

i s t h e subspace o f

F = s ~ Fn , and

Set

n-fold

l~I

Let

H

Fn = span {H

Then

= n}

{yi } .

,

so

Then

.

{Ym : Iml _< r }

Vk = span {Y

is a formal

: Is[

commutators o f t h e g e n e r a t o r s

F r+1 = s{F n : n ~ r + i }

Ym = H (Y 1 . . . . . Yn) .

.

F . Each

is a basis for

_g , and

: I~I = k} .

Define p(Y)

Then Since

~(Vk) = F k {Yi }

for

generate

L e t us r e t u r n property

:

Ha

1 ! k < r , and i t g ,

p is clearly

now t o t h e v e c t o r

o f the f r e e L i e a l g e b r a

F ,

phism : F + h such t h a t

~(yi)

= Xi

,

follows unique.

fields

that

p

is a partial

homomorphism.

Q.E.D.

×1 . . . . . Xn e L(M)

there exists

.

By t h e u n i v e r s a l

a u n i q u e L i e a l g e b r a homomor-

38 Setting

~ = ~ o ~ , we thus o b t a i n a p a r t i a l

homomorphism

: g÷h such that

x(yi)

In terms o f the Hall b a s i s , in

H

C~

= Xi x

i s simply the r e s u l t o f " s u b s t i t u t i n g

Xi f o r Y'"l

: ;~(Y ) = H (X 1 . . . . . Xn) •

Hence

~

has the a d d i t i o n a l p r o p e r t y t h a t ~(Vk) = span o f k - f o l d commutators o f {X i }

In p a r t i c u l a r , < r

suppose t h a t the

Xi

t o g e t h e r w i t h commutators o f the

s u f f i c e to span the tangent space a t

1.3

Lifting

Lie algebra

~

Theorem

into

x

in

Then

homomorphism

, and assume t h a t

Lemma There e x i s t s an open set around

x e M.

Fix a p a r t i a l

L(M)

.

~

~

around

~

~

Xi

of length

is s u r j e c t i v e a t

from a graded n i l p o t e n t

is s u r j e c t i v e a t some p o i n t

0

in

x .

g , and an open set

x e M.

M'

M , such t h a t the map

u ÷ eX(U)x i s a submersion from

Notation

If

be denoted by

~

X

onto

M' .

is a vector field,

t + e tX .

By d e f i n i t i o n ,

then the l o c a l f l o w generated by

X will

the induced a c t i o n on f u n c t i o n s is given

by

(~)

d

TC

f(etXy)

=

(Xf)

Proof o f Lemma The f l o w

(etXy)

u ÷ eX(U)x

exists for

fundamental e x i s t e n c e theorem f o r o r d i n a r y d i f f e r e n t i a l

u

near

0

in

~

by the

equations. The d i f f e r e n t i a l

39

o f t h i s mapping at

u = 0

is the map u ÷ ~(U)x. The i m p l i c i t

theorem thus gives the existence of such an oper set Now f i x the p o i n t

x

,

H by

and replace

function

~ .

H'

(i.e.

drop the prime). We

d e f i n e the i n t e r t w i n i n g o p e r a t o r

w (~)

c~(~l)÷ c~(~)

:

Wf(u) = f(eX(U)x) .

Let

dR : ~ ÷

L(~)

(we give

be the r i g h t r e g u l a r r e p r e s e n t a t i o n

g

the group

s t r u c t u r e determined by the Campbell-Hausdorff f o r m u l a ) :

dR(y) The l i f t i n g dR(y) .

:

It:o

theorem asserts t h a t

(u.ty)

W "approximately"

intertwines

~(y)

(The choice o f r i g h t instead of l e f t actions is d i c t a t e d by d e f i n i t i o n

and (m)).

To make t h i s p r e c i s e , l e t

= V1 ~ - . . ~ Vr be the decomposition o f of

~

~

as a graded Lie algebra. Viewing t h i s as a decomposition

as a vector space, we apply the constructions o f Chapter I , § 1.3:

Let

c = c~(~)

C = { f e C : f ( u ) = O([u[ m) m where

lul is a homogeneous norm on

f i e l d s o f order

g .

Let L

near

O}

= L(~) , and l e t

Lk

be the vector

< k at 0 .

With these p r e l i m i n a r y notions e s t a b l i s h e d , we can now s t a t e the central r e s u l t o f t h i s chapter: L i f t i n g Theorem

If

~

is a p a r t i a l homomorphism from the graded Lie algebra

i n t o the Lie algebra o f vector f i e l d s on a m a n i f o l d , which is s u r j e c t i v e at some point

x

,

then there e x i s t s a neighborhood

~ of 0

in

~

and a

40 l i n e a r map A : g ÷

L(~)

,

such t h a t (i)

W~(Y)

(ii)

A(Y)

(iii)

A

Furthermore, i f

~

=

A(Y)W

=

dR(Y)

r-step

chosen to be a p a r t i a l

homomorphism.

Remark

A

t h a t the map Y ÷ A(Y)o

§ 2. 2.1 Let

(II)

lifting

of

0

n i l p o t e n t Lie algebra, then

~ .

Note t h a t c o n d i t i o n

Proof of the L i f t i n g Let

e X E(X)Y

=

eX V

= ~d

t=O t-O _

eX+tB(X)Y

Here E(X) = I_~_D ~-D

~!-1 k

=

z

Dk

k>O

and B(X) = 0 z ~ 1-e -D = k>O "

(b k = B e r n o u l l i numbers).

(iii)

implies

to the tangent space at O.

× and Y be elements o f an a s s o c i a t i v e algebra.

D(T) = XT-TX .

e X+tY

g

A can be

Theorem

o f formal s e r i e s :

~

(Y e Vk)

is a l i n e a r isomorphism from

Basic Lie formulae

d

(I)

a

D = ad X be the d e r i v a t i o n

identities

mod Lk_ 1

is s u r j e c t i v e at

is the free

We s h a l l c a l l

(Y e ~)

b k Dk

Then one has the f o l l o w i n g

41 The meaning of

( I ) and ( I I )

is that the corresponding graded i d e n t i t i e s ,

obtained by equating terms of the same degree in (I)'

I

(II)'

n'i xny •

Here

~

~(Xny) =

=

(-1)k

z 1 k+m=n k!m!

Since

xmDk(y)

bk

o(xmDk(y))

(I) ,

i (xny + x n - l y x + . . . + y x n ) n+~

B(X) = E(X) -1 note t h a t

i t is clear that

R = L-D ,

is l e f t m u l t i p l i c a t i o n by ~(xny) = ~ ,1

X .

where

Also

(I)

is equivalent to

LD = DL .

Hence

, Xi (L-D)J(Y) ?~ l+j=n

=~

>: ~ j<_n k<_j

= n!

S (-1)k k+m:n T ~ -

z j=k

=

(-l)kxn-kDk(y)

xmDk(y)

~k+l j

We shall also need the i d e n t i t y d

d-T t=O

eX+tY+tZ

d

:~-C

I

eX+tY + d_~ I t=O

In graded terms t h i s is simply the i d e n t i t y (III)'

~(xn(y+z)) = ~(xny) + ~(xnz)

(II)

R is r i g h t m u l t i p l i c a t i o n by

where we have used the basic binomial i d e n t i t y

(III)

n :

is the symmetrisation operator:

~(xny) =

prove

z k+m=n

X , are v a l i d f o r a l l

eX+tZ t=O

To

X and L

42

2.2

Left-invariant

j u s t proved w i l l Let

n

vector fields

Our f i r s t

be to c a l c u l a t e an e x p l i c i t

a p p l i c a t i o n of the formulae

formula f o r

dR(Y)

,

Y e

be the Lie algebra o f § 1.1.3 associated w i t h the graded v e c t o r space

Let

u e g

Then the

~ .

"Bernoulli operator"

B(u)

adu 1_e-a--~-d-~

d e f i n e s a l i n e a r t r a n s f o r m a t i o n on

g

(which is a polynomial f u n c t i o n o f

u ,

by

the n i l p o t e n c e o f adu) . Lemma I f

(~)

v e ~ ,

dR(v)f(u)

Remark

Formula

i s the d i r e c t i o n a l

f e Ca(Q) ,

u e Q ,

then

d d--~t=O f ( u + t B ( u l v )

=

.

(=~=) asserts t h a t a t the p o i n t

d e r i v a t i v e in the d i r e c t i o n

u

Since

dR

Set

X = dR(u)

,

Y = dR(v) .

X,Y e n ,

f

is a polynomial

by Theorem 1.2.3 .

is a L i e algebra homomorphism, we have dR(B(u)v)

Now view

Then

the v e c t o r f i e l d

B(u)v .

Proof o f Lemma I t s u f f i c e s to prove the Lemma when function.

,

G as embedded in

=

B(X)Y

N = exp ~ ,

v i a the r i g h t r e g u l a r r e p r e s e n t a t i o n .

Then f(uv)

=

(eYf)(u)

=

eXeYf(o)

Hence dR(v)f(u)

= ~t

t=O eXetYf(o)

: eXyf(0) d =-aT

t=O eX+tB(X)Y f(O)

dR(v)

43

= y t - d It=O f ( u + t B ( u ) v )

(Note t h a t by the l o c a l n i l p o t e n c e of n

,

a l l series here are f i n i t e

when f

is

a polynomial.)

Remark The formula

(~)

l e a s t , and gives the e x p l i c i t

is v a l i d in any Lie group, when formula f o r l e f t - i n v a r i a n t

u

is near

0

at

vector f i e l d s in exponential

coordinates. The proof is the same as above, but now working in the pseudogroup of germs of a n a l y t i c t r a n s f o r m a t i o n s , instead o f the group

2.3 the f i r s t around

Formal s o l u t i o n step w i l l

u = 0

Let

Y e ~ , and set

Zmn '

where

C~(~)

will

vanishes to order

An e q u a l i t y

simply mean t h a t f o r every

-

By d e f i n i t i o n

mn eC n

(~n e Cn)

= s ~n

e

Given

f e C~(H) , WXf(u)

For t h i s purpose, i t s u f f i c e s to carry out a l l c a l c u l a t i b n s in

(as measured by the homogeneous norm) .

~

X = ~(Y) .

be to c a l c u l a t e the asymptotic expansion of the f u n c t i o n

the algebra o f formal series

for

N .

m

z mk e Cm+1 k<m

of the f l o w generated by a vector f i e l d ,

dtd

t=O Wf(tu) = ( ~ ( u ) f ) ( x )

I t e r a t i n g t h i s , we obtain the formal expansion Wf(u) :

s ~ n>O

x(u) n f ( x )

we have

> n at 0

44 Thus we can w r i t e

W = e k(u)

basic Lie formula

(II)

(a)

in the sense of asympotic expansions. Hence by the

,

wx : ~

exp ~(u) + tB(~(u))~ t=O

(This formula means t h a t to c a l c u l a t e the Taylor series of identities

(II)'

Du = ad x(u) .

to express

~(u)nx

Similar "finite"

in terms of

interpretations

WXf we can use the

a(~(u)mD~(X))

,

where

can be given to the formulas which

follow.) Since if

~

Y e Vk ,

is only a p a r t i a l

homomorphism ,

B(~(u))X : X(B(u)Y) +

of the

However,

then we f i n d t h a t

(b)

where each

B(k(u))X # x(B(u)Y) .

Z

E w(~)+k>r

is a vector f i e l d on

~(Yi) ,

and

{~i }

~(u)Z

M corresponding to a commutator of weight >r

is a graded basis f o r

converges in the asymptotic sense, since

B(u)Y

g__='=

Notice t h a t t h i s expansion

is a polynomial f u n c t i o n of

u .

Let us w r i t e t h i s as B(k(u))X : X(B(u)Y) ~ Ty(u)

S u b s t i t u t i n g t h i s in

(a)

By Lemma 11,2.2

and the formula

t h i s and the Lie formula

,

we get

(I)

,

W = e x(u) ,

the f i r s t

term is

dR(Y)W .

Using

we thus have

WX = dR(Y)W + WE(~(u)) Ty(U)

Suppose

Y e Vk ,

and consider the " e r r o r term"

This is a formal sum of terms field

(III)

It=O exp ~,(u+tB(u)Y) + --~-t It= 0 exp ~ ( u ) + t T y ( u ~

WX = T t

(c)

and using the Lie formula

on

M .

~(u)

From the s u r j e c t i v i t y

T

,

with

E(x(u)) Ty(u)

w(~) > r-k

and

T

in

(c) .

a vector

hypothesis, we conclude t h a t there e x i s t s a

45 neighborhood of

x in M on which every vector f i e l d

combination, with c o e f f i c i e n t s in Hence by s h r i n k i n g ~(u)

~ ~(Z) ,

with

H ,

C~(M) ,

we can w r i t e

can be expressed as a l i n e a r

of vector f i e l d s

E(~(u)) Ty(U)

w(~) > r - k , ~ e C~(H) ,

and

{~(Z)

: Z e g} .

as a series of terms Z e~ .

Applying the operator

W to such a term, we o b t a i n a term w ~ (z)

where

,

¢(u) = ~m(u) W~(u) e Cw(m) .

Let

{Yi : 1 ~ i < d}

be a basis f o r

]

,

and the foregoing a n a l y s i s , we can f i n d f u n c t i o n s d

(d)

W ~(Yi) = dR(Yi)W +

s s n>l j = l

with

Yi e Vni

~!~) e Cn 13

By formula

(c)

such t h a t

~!~) W ~(Yj) lJ

This series converges in the asymptotic sense. Furthermore, we know t h a t n <_ r-n i --> @I]) = 0

To w r i t e t h i s formula in more compact form, introduce the column vectors X = (x(Yi))

,

Y = (dR(Yi))

and the matrices

Define (d)'

W × = (W ~(Yi) ) .

Then

(d)

WX = Y W +

To complete the formal

becomes S ~n W X n~l

s o l u t i o n , we introduce the m a t r i x

series converges in the asymptotic sense, and Hence the geometric series T=

~ Sn n>l

S

S = E ~n "

vanishes to order

This

~ 1 a t u=O .

46

converges in the a s y m p t o t i c sense. Since ÷

(d)'

can be w r i t t e n

as

÷

(I-S) W X = Y W , ÷

(e)

solution

÷

C~

solution

in terms o f

To pass from the formal a s y m p t o t i c expansion

C=

vector fields,

Given a formal power s e r i e s , as i t s T a y l o r s e r i e s a t are

C~

f u n c t i o n s on

u = 0 .

equation

0 o ~ ,

C~

function

such t h a t

T~ = T

(e)

to a

theorem o f E. B o r e l :

having the given s e r i e s

Hence t h e r e e x i s t s a m a t r i x

By the d e f i n i t i o n

(e)

we i n v o k e a c l a s s i c a l

there exists a

T~ ,

whose e n t r i e s

in the sense o f formal T a y l o r s e r i e s

of asymptotic equality,

we can conclude from

that

(t)

W X = (I + T) Y W

where

c

at

follows that

W X = (I+T) Y W •

2.4

at

it

co

_-- #'Ic n n

mod C

is the space o f f u n c t i o n s v a n i s h i n g to i n f i t e

order

u=O.

The space .

C

is i n v a r i a n t

under a r b i t r a r y

(This is not t r u e o f the spaces

m a t i o n s . ) By the i m p l i c i t intertwining such t h a t

operator

W C (M )

Cn ,

C~

changes o f c o o r d i n a t e s on

even w i t h r e s p e c t t o l i n e a r

transfor-

f u n c t i o n theorem we can d e s c r i b e the range o f the

W as f o l l o w s :

There are c o o r d i n a t e s

(t I ..... td)

c o n s i s t s o f the f u n c t i o n s depending on the f i r s t

for

m coordinates

(m = dim M). ÷

Set

÷

Zo = ( I + 7 ) Y .

Then e q u a t i o n

(t)

t o g e t h e r w i t h the above d e s c r i p t i o n ÷

of the range of C~

W implies t h a t t h e r e e x i s t column v e c t o r s

such t h a t ÷

÷

m ÷

W X = Z° W + k=lZ Fk ~Tk

W

Fk

of f u n c t i o n s in

4?

Replace

Z°

by

Z = Z° + ~ F k ( B / ~ t k )

(~) If

w x

{Z i }

.

Then

= z w

are the components o f

Z ,

then the assumption t h a t

is a partial

x

homomorphism t o g e t h e r w i t h t h i s i n t e r t w i n i n g p r o p e r t y i m p l i e s t h a t

(:H:~) if

~i, zj-] w = w ~'([Yi,Yj~)

n i + n j _< r .

We s h a l l

construct

the d e s i r e d l i f t i n g

c e r t a i n commutators o f t h e

Zi .

( ~ )

:

Zi

A

Let us f i r s t

dR(Yi)

mod

by using c e r t a i n

o f the

Zi

and

observe that Ln._1 1 -9

Indeed, we o n l y need to v e r i f y t h i s f o r the formal s o l u t i o n t h i s p r o p e r t y o n l y depends on the T a y l o r expansions a t o f the

Zi .

But

T

is a s e r i e s o f terms

(I+T) Y ,

u = 0

Tml .- ~mk

since

of the c o e f f i c i e n t s

k > 1 ,

and one can

write

( ~ m l " ' " ~mk

with

mij e Ck(r_ni+l ) ,

p r o p e r t y o f the at

0 ,

~!~))

as i s e a s i l y

.

shows t h a t

g

i s the f r e e ,

then we d e f i n e Y l . . . . . Yn '

A

(TY)i

~ij

verified

is of order

r-step nilpotent

as f o l l o w s :

as in

= z

i

Since the v e c t o r f i e l d s

this

If

Y)

§ 1.2 .

Let

F

dR(Yj)

'

by i n d u c t i o n

( u s i n g the v a n i s h i n g

dR(Yj)

of order

~ ni - 1

at

are all 0 .

Lie algebra with generators

be the f r e e L i e a l g e b r a on

By the u n i v e r s a l

property of

< r

F ,

n

Y1 . . . . . Yn '

generators

there exists

a

unique Lie a l g e b r a homomorphism r

such t h a t

F(yi)

: F

= Zi

,

+

L(~)

1 ~ i ~ n .

Let

~ : g+

F be the p a r t i a l

homomorphism

48 of Proposition

1.2 ,

and d e f i n e

A = r o u .

Then

A

is obviously a partial

homomorphism. Note t h a t in terms o f a h a l l b a s i s , one has

A(H~(Y I . . . . . Yn )) = H~(Z 1 . . . . . Zn) , I t f o l l o w s from

(~)

and

(~)

that

A

]~I ~ r

satisfies

properties

(i)

and

(ii)

o f the Theorem. When ~ define

A

i s not the f r e e , r - s t e p n i l p o t e n t L i e a l g e b r a , then we cannot

merely by s p e c i f y i n g the v e c t o r f i e l d s

generators f o r

~ .

A

to

by l i n e a r i t y

properties

(i)

In t h i s case, we set ~ .

and

immediate consequence o f p r o p e r t y 1.1.3.

vector fields o f § 1.1.3 for

in

L0

(~)

1 ~ i < d ,

and

(~)

that

and extend A

vanish a t

A

is surjective at

(ii)

We know t h a t a t 0 .

0 .

and the s t r u c t u r e o f 0 ,

dR(Y) 0 = Dy ,

Hence i f

But t h i s is an L k , g i v e n by

by Lemma I I . 2 . 2 .

n = nI 0 . . . 0 ~ r

associated w i t h the d i r e c t sum decomposition

A(Y)o = Dy 0

But by the s t r u c t u r e o f equations

(T)

~j

,

mod

we have

is in t r i a n g u l a r

njlo

~ = V1 0 . . . 0 Vr ,

Z -J1n
Hence the system o f

form, which i m p l i e s t h a t

dim A(g)o = dim

This completes the p r o o f o f the l i f t i n g

theorem.

A l l the

i s the Lie algebra

Y e Vn

(T)

satisfies

of the Theorem.

I t only remains to v e r i f y t h a t

Proposition

A(Yi) : Z i ,

I t then f o l l o w s from

(ii)

corresponding to a set o f

then

49

§ 3..G..roup germs g..e..nerated by p a r t i a l

3.1 E x p o n e n t i a l length

r ,

coordinates.

Let

~

isomor.phisms

be a graded n i l p o t e n t

Lie algebra of

and l e t

: g + L(M) be a p a r t i a l

homomorphism. We s h a l l

assume t h a t

for

x e M ,

t h e map

u ~(U)x is a linear isomorphism from

~

We define the exponential

onto

TM x .

map at each point

x e M

by

eXPx(U ) = e ~ ( U ) x . The map

eXPx

is a diffeomorphism

o n t o an open n e i g h b o r h o o d o f Lipschitz fix

constants

xo e M ,

for

x .

from some open n e i g h b o r h o o d The s i z e o f t h i s

the coefficients

we can f i n d

a neighborhood

of

of zero in

n e i g h b o r h o o d depends on

of the vector M°

~

xo

fields

X(u)

and a f i x e d

. gcg

Thus i f

we

such

that i)

if

x e Ho ,

eXPx(Q) 2)

~

t h e map

then

eXPx

is a diffeomorphism

,

~ ,

and

Mo

x , u ~-+ eXPx(U )

is

C®

from

We now c o n s i d e r t h e " g r o u p germ" c o n s i s t i n g {mu : u e ~}

on

Mo

x ~

to

M

of the transformations

where

~u(X) = eXPx(U ) • Pick an open s u b s e t

no

compactly contained in

U,V e ~o ~ X e M 0 --> e x ( u ) e ~ ( v )

Given

x e Mo ,

it then follows by conditions

~ ,

such t h a t

x e Mo

I) and 2)

above that there exists

50 a

map

C°°

F

: Q x Q ~ o o

x

such t h a t e x(v) Furthermore,

t h e map

In the s p e c i a l

x,u,v case

e ~(u)

x = exp x ( F x ( u , v ) )

--+ Fx(U,V )

H = G , on

G ,

i s smooth.

x = identity,

of

~

as v e c t o r f i e l d s

then

on

g

d e f i n e d by t h e C a m p e l l - H a u s d o r f f

3.2 Comparison o f group germs. nilpotent

group s t r u c t u r e s

on

g

Fx

and

x = regular

is the nilpotent

representation

group m u l t i p l i c a t i o n

formula.

The comparison between t h e a d d i t i v e

in

§ 1.3.1

and

extends to the present situation,

as f o l l o w s : Theorem exists

Let

a constant

a : i/r c

,

and l e t

all

u,v e ~

Proof the s e r i e s

Fix

o

and

F(u,v)

~ C(lu[alvl

x e H

x e Ho ,

There

z-a + I v l a l u l l - a )

o

and w r i t e

F = Fx ,

i ~(F(u,v))n = z n--F.

in t h e a s y m p t o t i c sense, f o r ,

g .

exp = eXPx

Since

F(O,O) = 0 ,

expansion

f(exp F(u,v)) is valid

be a homogeneous norm on

such t h a t

IFx(U,V)-U-V! for

lul

any

f(x)

f e C~(M) .

But by d e f i n i t i o n

of

we a l s o have an a s y m p o t i c e x p a n s i o n f(exp

F(u,v))

= ~ ~ i

~(u) m ~ ( v ) n f ( x )

Hence t h e s e two a s y m p t o t i c e x p a n s i o n s must a g r e e . We c o n c l u d e t h a t

the v e c t o r

51 field

~ ( F ( u , v ) ) has an asymptotic expansion given by the power series f o r log (e x(u) e x ( v ) )

(where

log T =

s n>o

(-l)n n+z

(T-I) n

.

is the formal series inverse to the exponential

series) . By Dynkin's e x p l i c a t i o n o f the Campbell-Hausdorff formula, we know t h a t the formal series log (eXeY) = X + Y + . . . in the non-commuting indeterminants e n t i r e l y in terms o f

X,Y

rearrangement in the case

X,Y

can be rearranged to be e x p r e s s i b l e

and i t e r a t e d commutators of X = ~(u) ,

Y = ~(v)

X and Y .

Applying t h i s

and using the f a c t t h a t

p a r t i a l homomorphism, we conclude t h a t the asymptotic expansion o f

~

is a

X(F(u,v))

has the form ~(F(u,v))

Here

uv

= ~(uv) + R ( u , v )

is the n i l p o t e n t group product on

g ,

and the remainder

R is the sum

o f terms p(u,v)

where

p

{X }

,

is homogeneous o f t o t a l degree

is the image under

surjectivity in the

~ x

of

~ ,

all

~

n > r

in

o f a graded basis f o r

the commutators o f

C~(M) - module spanned by the

{X } . )

F(u,v)

= uv +

S

~ .

Since

~

,

m e C~(M) ,

and

(Because of the

~(u) , ~(v)

map, t h i s implies t h a t the asymptotic expansion of (~)

(u,v)

which occur in

R are

is an i n j e c t i v e l i n e a r

F(u,v)

is of the form

Fn(U,V ) ,

n>r

where

Fn

is a homogeneous polynomial of t o t a l degree

n

in

(u,v)

( r e l a t i v e to

the graded s t r u c t u r e on ~). To o b t a i n the estimate o f the theorem, we r e c a l l from C o r o l l a r y 1.3.1 t h a t

52

this

estimate is satisfied

to estimate

F(u,v)

by t h e d i f f e r e n c e

- uv,

We n o t e t h a t

as

u and v

uv - u - v .

Hence we o n l y need

range o v e r the bounded s e t

Fn(O,V) = Fn(U,O ) = 0 .

o

Hence

n-i

[IFn(U,v)ll <_ C z

[ulklv[ n-k

k=l

A l s o , on compact subsets o f

g

we have

[wl ~ c I~II a where

a = I/r

estimates to

and (X)

,

,

]]']I denotes a E u c l i d e a n norm on we conclude t h a t f o r

lF(u,v)

- uv[ ~ C

g .

A p p l y i n g these two

u , v e £o '

max

(]ul ak Ivl aj)

k+j=r+l l
To m a j o r i z e the r i g h t

side,

]ul ak Ivl aj

assume t h a t

~ Iv]

ilvl;

lul

~ Clul 1-a

on

Then

Ivl1+a

f o r any

u,v e ~

o

,

IF(u,v) - uv[ ~ C(lul a Ivl + lul Since

.

=/lull <-

Hence we g e t the e s t i m a t e ,

lul

n° ,

this

Iv] a)

completes t h e p r o o f o f t h e theorem.

In t h e course o f p r o v i n g t h e theorem, we a l s o e s t a b l i s h e d comparsion between the group germ

Corollary let

a = 1/r

Let

uv

F

and t h e n i l p o t e n t

be the n i l p o t e n t

(r = length of g)

.

group s t r u c t u r e

group m u l t i p l i c a t i o n

There e x i s t s

a constant

IFx(U,V) - uvl ~ C(lul a ivl + lul Iv[ a)

the following

C

on

~ ,

such t h a t

on

and

g :

53 for all

u,v e ~o

and

In p a r t i c u l a r ,

x e Me

IFx(U,V) - uvl ! C(Iul + Ivb l+a

so t h a t

Remark map

lim

[Fx(U'V)-UVl

l u l + l v l ÷o

lul+Ivl

-0

The c o r o l l a r y shows t h a t the map

u,v ÷ uv

to order greater then one at

u,v ÷ Fx(U,V)

(0,0)

.

is tangent to the

In t h i s s i t u a t i o n we have

perturbed the graded n i l p o t e n t s t r u c t u r e by adding " h i g h e r order" terms. Compare this result with

Theorem 1.3.3 ,

which treated the p e r t u r b a t i o n by "lower order"

terms.

§ 4. Examples from complex anal#sis

4.1 Real hypersurfaces in

{n+l

Some i n t e r e s t i n g examples o f t r a n s i t i v e Lie

algebras o f vector f i e l d s a r i s e in connection with "analysis on the boundary" of domains in

¢n+1

(n ~ i )

.

Here one finds a simple geometric r e l a t i o n between

the "holomorphic flatness" o f the boundary and the length o f the n i l p o t e n t Lie algebras which occur. Let

M c ¢ n+1

be a r e a l ,

C~

M is l o c a l l y definable as the set function with e.g. t h a t

df # 0 .

submanifold of real dimension { f = o} ,

where

U

is open

the algebra of complex-valued C~ functions on

T e

is a r e a l - v a l u e d

C~

in

{n+l .

M ,

and l e t

Denote by

C~(M,C)

= Der(C~(M,C))

denote the Lie algebra o f vector f i e l d s on For any

f

Thus

A l l our considerations w i l l be l o c a l , so we may assume,

M = {f = o}~ U ,

L{(M)

where

2n+I .

Lc(M) , ~

M w i t h c o e f f i c i e n t s in

w i l l denote the conjugate vector f i e l d

T ( f ) = T(~) ,

c~(M,¢)

.

54

and

we d e f i n e Let

A

Re T = ( I / 2 ) ( T + T ) ,

Im T = ( 1 / 2 i ) ( T - ~ )

.

be the algebra o f holomorphic functions on the ambient space

The i n c l u s i o n

map M c { n + l

induces the r e s t r i c t i o n

~n+l

map

A ÷ C~(M,¢) . We d e f i n e the hp.tomorphic vector f i e l d s

on

M to be those vector f i e l d s which

a n n i h i l a t e the a n t i - h o l o m o r p h i c f u n c t i o n s : LI'O(M) = {X e L$(M) : X ~ = 0 f o r a l l f e A} .

S i m i l a r l y , the anti-holomQrphic vector f i e l d s a n n i h i l a t e the holomorphic f u n c t i o n s : LO'I(M) = {X e L£(M) : X f = 0 f o r a l l f e A} .

Clearly L I , 0 = LO, 1

The f o l l o w i n g p r o p e r t i e s also hold: Proposition

LI ' 0

and

L0 ' I

L1 ' 0 ~

At each p o i n t

p e M ,

are Lie subalgebras o f

L~(M) ,

with

L0'1 = 0 .

dim~ ( L l ' O ) p

=

n ,

so t h a t

(L I ' 0 + LO'l)p

is a 2n-dimensional complex subspace o f the c o m p l e x i f i e d tangent space

Proof

I t is c l e a r from the d e f i n i t i o n

subalgebras o f

L$(M)

.

If

X e L1'0

n

L0 ' I

that ,

L 1'0

then

and

dim{ ( L l ' O ) p .

L0'1~ are Lie

X a n n i h i l a t e s the real and

imaginary parts o f the complex coordinate f u n c t i o n s , and hence remains to c a l c u l a t e

T~Mp .

X : 0 .

It

55

I f the e q u a t i o n p e r f o r m i n g an a f f i n e

f = 0

defines

transformation

M near on

p ,

{n+l

,

with

df # 0 ,

we can assume

p = o

then a f t e r and

d f p = dZn+ 1 + dZ-n+1

where

{z k}

are the complex c o o r d i n a t e s on

(~n+l .

L e t us c a l l

Zn+ 1 = w .

Then

the v e c t o r f i e l d s

Lk = fw ~-~k - f z k !~W

are t a n g e n t to

M ,

since

1<

'

k < n -

'

Lk f = 0 ,

Here

where

zk = xk + iy k .

Evidently

L k e LI ' 0

(Lk)p

by the above n o r m a l i z a t i o n

of

f .

p

and a t

P ,

'

This shows t h a t

d i m $ ( L l ' O ) p >_ n

We must have e q u a l i t y ,

,

d

however, because

2 d i m ~ ( L l ' O ) p = dim~(L 1 ' 0 + L O ' l ) p

_< dim T{Mp = 2n+l

.

This completes the p r o o f o f the p r o p o s i t i o n .

4.2 P o i n t s o f type m

By P r o p o s i t i o n

p a r t s o f the h o l o m o r p h i c v e c t o r f i e l d s o f the

2n+1

on

d i m e n s i o n a l t a n g e n t space t o

4 . 1 , we see t h a t the r e a l and i m a g i n a r y M M

span a

2n

d i m e n s i o n a l subspace

a t each p o i n t .

The q u e s t i o n o f

56

i n t e r e s t now i s to determine whether the missing d i r e c t i o n t a k i n g i t e r a t e d commutators o f these v e c t o r f i e l d s .

It

can be o b t a i n e d by

is t e c h n i c a l l y s i m p l e r

and more n a t u r a l t o work in the c o m p l e x i f i e d tangent space to

M .

Then i t

becomes a question o f examining i t e r a t e d commutators o f holomorphic and a n t i holomorphic v e c t o r f i e l d s

Definition C~(M,$))

on

M .

For each i n t e g e r

of vector fields

m~ 1 ,

let

generated by Lie brackets of l e n g t h

holomorphic and a n t i - h o l o m o r p h i c v e c t o r f i e l d s type

Lm be the module (over

on

M .

~ m o f the

A point

p e M ~s of

m if

(Lm+l) p =TcM p

and

(Lm) p # T6Mp .

(If

We s h a l l say t h a t s

<

m

.

o f type

Then i f < m .

p

no such

m exists,

p e M is o f type

is o f type

m ,

p

< m if

(L1) p

it

type.)

is o f type

s

f o r some

i t has a neighborhood c o n s i s t i n g o f p o i n t s

Note t h a t L I = L1, 0 + LO, 1

so

is of i n f i n i t e

,

is never the e n t i r e tangent space. Hence

p

is o f type

~ I ,

n e c e s s a r i l y . The n o t i o n o f type is o b v i o u s l y i n v a r i a n t under holomorphic changes o f l o c a l c o o r d i n a t e s in the ambient space

Proposition

Suppose

holomorphic v e c t o r f i e l d s free

m + I - step

YI . . . . . Yn "

such t h a t

p e M

on

~n+l

i s o f type

M which span

m <

Let

Z 1 . . . . . Zn

L 1'0

at

p ,

real n i l p o t e n t Lie algebra on

2n

generators

Then t h e r e e x i s t s a p a r t i a l

~ ( X j ) = Re Z j ,

homomorphism

~ ( Y j ) = Im Zj , x(~)p = T Mp

and

and l e t

g

be be the

X1 . . . . Xn,

57 Proof. assumption,

A l l t h a t needs to be v e r i f i e d {Re Z j }

and

{Im Z j }

span

is the s u r j e c t i v i t y

LI

(as a

C~(M,C) module)

By i n d u c t i o n we see t h a t t h e i r commutators o f l e n g t h Since

p

i s o f type

thus the f u l l

m ,

the span over

R

~ r

span

near

Lr

p .

around

o f commutators o f l e n g t h

~ m+l

p . is

tangent space.

4.3 Geometric c h a r a c t e r i s a t i o n .

The type o f a p o i n t

by the degree o f "holomorphic f l a t n e s s " of

Theorem

c o n d i t i o n . By

A point

p e M

is o f type

p e M is determined

M at

p ,

m< ~

i f and only i f

p l e x submanifold o f codimension one tangent to

M at

in the f o l l o w i n g way:

p

to o r d e r

codimension one complex submanifold tangent to a h i g h e r o r d e r a t

there is a comm ,

but no

p .

As a p r e l i m i n a r y to proving the theorem, we make an a f f i n e change o f c o o r d i n a t e s so t h a t

p = o

and the f u n c t i o n

f

d e f i n i n g the boundary is given

by f=w+w+m where

(z I . . . . . Zn,W )

write

z = (z I . . . . . Zn)

,

are l o c a l holomorphic c o o r d i n a t e s and

d~o = 0 .

and fj

= afl@zj ,

f~ = ~ f l @ ~

fw = ~fl~w

,

fw = ~fl~w

Lk = fw ~

- fk Tff

The v e c t o r f i e l d s

and t h e i r conjugates thus span

L1

T = f~ ~-

around

fw

p .

,

l < k < n

,

The v e c t o r f i e l d

We w i l l

58 together with The type of

L 1 spans the e n t i r e complexified tangent space to p

is the smallest integer

t~ around

m such that

Tp e (Lm+l) p

With the local coordinates around vector f i e l d

(~)

X around

p

chosen as above, we can w r i t e any

as

_• + d ~

X=c

p

+ nz a i ~ + b i i=l az i ~T.I --7

In p a r t i c u l a r ,

f o r the vector f i e l d

a

i

:

6ij

X = [ L j , Lk] ,

we calculate that

( f k fww - fw few )

bi = -6ik ( f j fww - - fw f j ~ ) c

= ( f ~ f j ~ - f~ f j ~ )

d

= -(fw f j ~ -

f j few )

Furthermore, under the action of

ad L. , J

the c o e f f i c i e n t s

of

X transform as

follows:

~ ai

÷ ~

1 c

÷ ~

Using these formulae:

-

6 i j {Cfww + dfw~ +

+ Cfjw + d f j ~ +

i t is a straightforward

of i t e r a t e d commutators of the vector f i e l d s

Lemma i

Suppose

n

z k=l

n ak bk s fwk + fw~ } k=l

ak

fj k

+ bk

fj~

induction to establish the form Lk' Lk :

X is a sum Of Lie monomials of length

m> 2

in the

59

vector fields

1 < k < n} .

{ L k ' Lk ;

1) The c o e f f i c i e n t s

a],bl,c,d

Then

i n the e x p r e s s i o n (~)

are sums o f terms o f

t h e form ± D1(f) D2(f)

where each

Di

d I + d2 + . . . +

to

z,z

form

to

D(f)

,

to o r d e r

di

(1 ~ d i ~ m) ,

and

dm = 2m-i . a i or b i

and a t o t a l

3) Each such term i n relative

Dm(f)

is a d i f f e r e n t i a t i o n

2) Each such term in relative

...

z,7 ,

m-i

, where

D

From Lemma 1, i t

of

involves differentiation

m

c or d

times r e l a t i v e

to

w,w ,

is a d i f f e r e n t i a t i o n

is c l e a r t h a t i f H

at

p

in

a total

of

m times

and c o n t a i n s a f a c t o r o f t h e

z,z

only of order

< m .

t h e r e e x i s t s a complex, codimension one

to o r d e r

(Lm) p #

o f m-1 times

w,w

involves differentiation

times r e l a t i v e

submanifold t a n g e n t t o

to

a total

m ,

then

T6Mp

Indeed, i n t h i s case we can make a p r e l i m i n a r y change o f c o o r d i n a t e s so t h a t the hyperplane

{w = o}

means t h a t

D(f) = o

at

o

By p a r t

3)

o f lemma 1 t h i s

order

< m.

coefficients

o f any

is tangent to when

H D

X e Lm vanish a t

to order

m

at

p = o .

is any d i f f e r e n t i a t i o n i m p l i e s t h a t the

p .

in ~/~w

This s i m p l y z,z and

only of ~/~w

This proves t h a t the t y p e o f

p

is

a t l e a s t as b i g as the maximal o r d e r o f tangency o f complex codimension one submanifolds.

To e s t i m a t e the t y p e o f

p

from above, we f i r s t

use the fomulas above

and an i n d u c t i o n argument t o prove the f o l l o w i n g :

Lemma 2. IJI + IKI = m .

Let

J, K eLNn

be g i v e n , w i t h

Then t h e r e e x i s t s a v e c t o r f i e l d

IJl

>_ I , X

IKI >_ I ,

as in

and

Lemma I ,

such t h a t

60

c =

where

R

some

is a sum o f terms o f the form

Di

is a differentiation

With t h i s type m < ~ {w=o}

K f

(z = (z I . . . . . Zn)

m .

---

,

Dm(f) ,

only of order

and i n each such term

~ m-1 .

the p r o o f o f the theorem. Suppose

p

is o f

By a h o l o m o r p h i c change o f c o o r d i n a t e s we can arrange t h a t H

(~)Jf(p)

Pm

z,~

lemma we can f i n i s h

is tangent to

where

in

D1(f)

+ R

at

p ,

= o

and

,

as above) .

IJl<

m

Indeed, i t

suffices

to r e p l a c e

w by w' = w + pm(Z),

is the h o l o m o r p h i c p a r t o f the T a y l o r p o l y n o m i a l f o r

(Since

Pm(O) = o ,

the f u n c t i o n s

h o l o m o r p h i c c o o r d i n a t e s around

p .)

z I . . . . . z n, w'

still

f(z,o)

of

degree

form l o c a l

We are assuming t h a t i f

k ~ m ,

then

(Lk) p # T6Mp By Lemma 2,

this

implies

inductively

f
for all this

J,K

with

that

= o

IJ I + IKI <_ m .

Together w i t h the above n o r m a l i z a t i o n o f

shows t h a t the complex h y p e r p l a n e

{w=o}

is t a n g e n t to

H

to o r d e r

f

> m .

This completes the p r o o f o f the theorem.

Remarks 1.

Suppose the p o i n t

p e H

shows t h a t t h e r e are h o l o m o r p h i c c o o r d i n a t e s so t h a t the T a y l o r s e r i e s f o r (t)

f(z,w)

f

= W + w+

is o f t y p e

m < =

The p r o o f above

w and z = (z I . . . . . Zn)

is o f the form ~ [aI+l~I=r CaB Za

~

+..-

around

p

,

61

Here

r = m + I

llw [I2

+

,

and the n e g l e c t e d terms are o f t h e o r d e r

Hwll llzll+

Ikll2r+1

This approximation

for

f

Furthermore, fits

naturally

the constants into

CaB

the g e n e r a l

are n o t a l l

zero.

framework o f homogeneous

norms. Namely, we d e f i n e i(z,w)I

llzl12=

where

germs o f

S 7iz i

i~II 2

,

C~ f u n c t i o n s

Then the e r r o r equality

= (llzll 2 r + Ilw!12) 1 / 2 r

at

term i n

modulo

C

r+l

p

and we d e f i n e

as in

(t)

is

§ 1.3,

a filtration

relative

O(1~,w) l r + l )

,

to this

i.e.

{C k}

on t h e

homogeneous norm.

equation

(t)

i s an

"

2. By t h e remarks j u s t approximation"

= ~w ,

,

made, we see t h a t

t o the m a n i f o l d

w + ~ +

M

at a point

i s the r e a l - a l g e b r a i c

s

of type

m

,

the "first

hypersurface

z ~ yB = 0

i l+iBl=r ( r = m + 1) , {Cij}

where

and

is a non-zero Hermitian matrix.

of coordinates

C. = 0 J

4.4

C B = CBm . In t h i s

In p a r t i c u l a r ,

or

Siegel

= w + w

+ z Cj llzjll 2 + . . .

_+ 1 .

Domains and the H e i s e n b e r g group

L e t us c o n s i d e r

i n more d e t a i l

H : {(z,w)

: Im(w) :

the example o f t h e h y p e r s u r f a c e Ilzll 2

z e sn

is the boundary o f t h e domain D = {(z,w):

Im(w)>

llzll 2}

,

when

case we can make a l i n e a r

so t h a t f(z,w)

where

{CAB} # o ,

w e ~}

m = 1 , change

82 which is a

Siegel domain o f type I I

.

By the foregoing a n a l y s i s

H

is the

simplest example o f a "complex-convex" manifold of real codimension one. (Observe t h a t the complex hyperplane D

lies strictly

H

is tangent to

is the zero set of the f u n c t i o n ,

w - w - 2i IIz[I2 ,

1 < j ~ n

span the holomorphic tangent space everywhere on

{Lj

0 ,

and

0 .)

the vector f i e l d s

,

M

,

and mutually commute.

commutation r e l a t i o n s are [Lj

where

to order one at

D and blows up e x a c t l y at the boundary p o i n t

Lj = ~ - ~ + 2i ~j ~ J

The n o n t r i v i a l

H

on one side o f t h i s hyperplane. As a consequence, the f u n c t i o n

i/w is holomorphic in Since

{w = o}

, Lk] = 6jk N

N = -2i ( ~ +

~)

[-j : I ~ j < n}

.

,

This shows t h a t the complex Lie algebra generated by

is isomorphic to the complex

(2n+l) - dimensional

Heisenberg algebra. As a basis f o r the real form o f t h i s algebra, we take

.FXj = Re(Lj) = ~1 ~-~~ + xj ~-~ ~ + yj

Yj

where

=

zj = xj + i y j

Im(Lj) =

2I ~yj ~

, w = s + it

Yj Tt

+

xj

in terms o f real coordinates. Thus

I

{Xj,Yj,Z

; 1 ~ j ~ n}

span the real tangent space to

H at each p o i n t , and

s a t i s f y the Heisenberg commutation r e l a t i o n s [Xj,Y~

: 6jk Z

( a l l o t h e r commutators being zero). Let

g

be the real Heisenberg algebra of dimension

n i l p o t e n t Lie group pbtained from

~

2n+l ,

and

G the

as usual by the Campbell-Hausdorff formula.

63 Since

is t w o - s t e p n i l p o t e n t ,

UV = U +

Pick c o o r d i n a t e s

the group s t r u c t u r e

V + - Zi

[u,v]

is

u,v e

= (C 1 . . . . . ~n) , n = (n I . . . . . nn)

for

and

g

so t h a t

the map n

~ , n , ~ ) = j=lZ (Cj Xj + nj Y j ) + ~ Z

i s a L i e a l g e b r a isomorphism from {Xj,

Y j , Z}.

For

u e g ,

where

~.n = s ~ j n j

•

on

from the p o i n t

0

write g

u = (~,n,~)

= (o,o

in

with initial

¢n+1 .

The i n t e g r a l

!zj

(*)

we f i n d

dyj d~

'

~.y+n.x+~,

conditions

T = 1 ,

of

= (~%',n+n',

- ~"~)

G

,

d-c

-

is

~+~' + ½ (~'~' - ~ ' - n ) ) .

: ~ (~j - inj)

2).

~(~,n,~)

,

starting

0

equations

= _ 1 ~- nj

-

g-x-n.y

xj = yj = s = t = 0 . t h a t the p o i n t

,

curre

the system o f o r d i n a r y d i f f e r e n t i a l

=

Thus in these c o o r d i -

the f l o w g e n e r a t e d by the v e c t o r f i e l d

~dxj _ i I d--~-- - ~- ~j

setting

, ~.n'

Hence the group s t r u c t u r e

• ~-~ e T ~ ( ~ ' n ' ~ ) satisfies

,

are

, (~',n',~'~

(~,n,~) ( ~ ' , n ' , ~ ' ) Let us c a l c u l a t e

onto the L i e a l g e b r a spanned by

we s h a l l

nates the commutation r e l a t i o n s [(~,n,~)

~

S o l v i n g these e q u a t i o n s and

e~ ( ~ ' n ' ~ )

. 0 e H

has c o o r d i n a t e s

64

This shows t h a t the map u~-+ eX(U).o The i n t e r t w i n i n g operator

is a diffeomorphism from

W : C~(M) + C~(G)

defined by

Wf(u) = f ( e ~ ( U ) . o ) can be expressed in these coordinates as Wf(~,n,~) = f ( z I . . . . . Zn,W) , where

z I . . . . . Zn,W are given by

(m)

G onto

M

65 Comments and references f o r Chapter I I

§ 1.1

The i n s p i r a t i o n f o r t h i s section was the work of Folland-Stein B ]

and Rothschild-Stein [ I ] . We have t r i e d here to recast these constructions i n t o a more geometric form, emphasizing the analogies with the case of a homogeneous space. For f u r t h e r d i f f e r e n t i a l - g e o m e t r i c aspects of t h i s s i t u a t i o n , cf. Tanaka

[2]. § 1.2

The notion of " p a r t i a l homomorphism" was introduced by Rothschild-

Stein [1] f o r the case of the free n i l p o t e n t Lie algebras. For information about free Lie algebras and P.Hall bases, cf. Bourbaki [ i ] .

§ 1.3 Stein [ ~ ,

The " L i f t i n g Theorem" was f i r s t

stated and proved by Rothschild-

from a d i f f e r e n t point of view and by somewhat d i f f e r e n t methods (the

problem of "adding variables" to make vector f i e l d s "free up to step r . " ) The present treatment is taken from Goodman [71 .

§ 2.1-2.2

The basic formula f o r the d i f f e r e n t i a l of the exponential

mapping can be found in Helgason [1]. The formal inverse to t h i s formula, i n v o l v i n g the "Bernoulli operator," also appears in Berezin B ] ,

§ 2.3-2.4

Goodman [31 , and Conze [1].

One can reverse the order of construction here, and obtain

the a p r i o r i existence of the vector f i e l d s

{Z i }

by the i m p l i c i t function

theorem. This was pointed out to the author by P. Cartier. The formal series solution then serves to calculate the Taylor expansion of the c o e f f i c i e n t s of the

{Z i }

,

and the l i f t i n g

A

is constructed from the

Zi

as in the t e x t , cf.

Goodman [7]. This procedure has the advantage of being applicable in any category where the i m p l i c i t function theorem applies, e.g. analytic.

Ca ,

real a n a l y t i c , or complex

66

§ 3.1-3.2

Theorem 3.2 is i m p l i c i t in the paper of Rothschild-Stein [1],

The presentation here is taken from Goodman [7].

§ 4

The results and proofs here are taken from Kohn [3] and Bloom-

Graham [ 1 ] , who also obtain s i m i l a r results f o r surfaces of codimension

k > 1

which are in "general p o s i t i o n " . For the c l a s s i f i c a t i o n of the i n v a r i a n t s of real hypersurfaces under holomorphic coordinate transformations, cf. Chern-Moser

[11

Chapter I I I

S i n g u l a r i n t e g r a l s on spaces of homogeneous type In t h i s chapter we shall construct a general theory of " s i n g u l a r i n t e g r a l operators" on a class of l o c a l l y compact spaces of "homogeneous t y p e . " Such a space

X will

have a "distance f u n c t i o n "

c o n d i t i o n t h a t the b a l l s of radius

p

and a measure

~

r e l a t e d by the

R have measure of the order

RQ ,

where

Q

is a p o s i t i v e number (the "homogeneous dimension" of the space). The distance function

p

is required to s a t i s f y a c e r t a i n L i p s c h i t z - c o n t i n u i t y c o n d i t i o n ,

which serves as a replacement f o r the t r i a n g l e i n e q u a l i t y . The operators we shall study are of the form T f ( x ) = PV f K ( x , y ) f ( y ) d~(y) , X where

K is a kernel which is s i n g u l a r along the diagonal

v a l u e ) . The major r e s u l t of t h i s chapter is t h a t when

(PV = p r i n c i p a l

K satisfies certain

homogeneity c o n d i t i o n s , mean-value c o n d i t i o n s , and smoothness c o n d i t i o n s , then T

defines a bounded o p e r a t o r on

L2 (X , dp) .

This is a g e n e r a l i z a t i o n of the

Calderon-Zygmund theory of s i n g u l a r i n t e g r a l o p e r a t o r s , which applies both to n i l p o t e n t Lie groups and to the group germs studied in Chapter I I . To make t h i s connection, we begin, as in Chapter I , with some basic d i f f e r e n t i a l

and i n t e g r a l

calculus on graded vector spaces w i t h d i l a t i o n s and homogeneous norms.

§ 1. Analysis on vector spaces w i t h d i l a t i o n s

1.1. Homogeneous functions and d i s t r i b u t i o n s .

Let

dimensional vector space, with a direct-sum decomposition V =

r z ~ V n=l n

V

be a r e a l , f i n i t e -

88

D e f i n e the d i l a t i o n s

{6 t

Lebesgue measure on

V

: t > o}

by

dx .

on

V

as i n

Q = s n dim (Vn)

dimension of

V .

A function

(If f

.

We s h a l l

V = V1 ,

on

V

will

refer then

g

on

V

will

f e Cc (V)

.

measure

dx

defines

g

on

V

to the i n t e g e r

( t > o)

integrable

the d i s t r i b u t i o n

g ( x ) dx

Q

as the homogeneous

be c a l l e d

homogeneous o f degree

if

f o r m u l a above shows t h a t

which i s homogeneous o f degree

which i s homogeneous o f degree

is homogeneous o f degree

{xR. : 1 < i < d}

u (u e C)



a distribution function

if

~ (~ e ~) ,

.

For e x a m p l e , t h e i n t e g r a t i o n

is a locally

Fix a basis

,

be c a l l e d homogeneous o f degree

t Q = t - u for all

dx

formula

Q = dim V).

f o ~t = t u f A distribution

§ 1 . 1 , and d e n o t e

We then have the i n t e g r a t i o n

LQ I f ( ~ t x) dx = ~ f ( x ) V V where

Ch I ,

for

V

u ,

the

El .

If

then

~ .

such t h a t

xR, e V n . ,

and l e t

{~i }

1

be t h e dual b a s i s f o r

V* .

Write

l[~fll~ : sup { I D i f ( x ) l Here

Ixl

Lemma. of degree

Suppose ~ .

: Ixl

1 < i ~ d}

is a

- f(y)[

~ M Ix-yl

Proof.

f

If

V .

CI

function

a constant

~ M l]vfl~

(Here g

= i,

and d e f i n e

.

Then we have the f o l l o w i n g

version of

homogeneous f u n c t i o n s .

There e x i s t s

If(x) Ixl

'

i s any homogeneous norm on

t h e m e a n - v a l u e theorem f o r

when

Di = ~ / ~ i -

on

M ,

Ixl ~-I

V ~ {o}

which is homogeneous

independent of

ix-yl

f

,

such t h a t

,

~ = Re ~) .

is a function

on

V ~ {o}

which i s homogeneous o f degree

u,

69 then

Ig(x)l ~ 1loll lxl ~ , where

llgIl~ : sup { I g ( x ) l

estimate for y(t)

f(x)

= x+t (y-x)

- ni .

: Ixl

- f(y) ,

: I}

,

and

z = Re u •

o b t a i n e d by i n t e g r a t i n g

0 < t ~ I .

df

We observe t h a t

Dif

Let us a p p l y t h i s along the path

i s homogeneous o f degree

Hence If( x ) " f(Y)I

~

sup s l D i f ( Y ( t ) ) I I ~ i ( x - y ) [ O~t~l M l[vfll® { sup z i y ( t ) [ x - n i O~t~l

p r o v i d e d the path To Itx[

t o the

y

does not pass through

0

0 < t < i

,

where

Ixl ~ K ( l ~ ( t ) [

IY(t)[~ so t h a t i f

Ixl

~ 2K I x - y l (2K) - I

In p a r t i c u l a r ,

the path

¥

]x+Yl S K

Ixl + I Y l )

and

Hence

+ Ix-yl)

+ Ix-yl) ,

K2(ixl ,

K ~ 1 .

,

(M = max ll~ill~ ) .

c o n t i n u e these e s t i m a t e s , we note t h a t

~ K l x I when

Ix-y[ ni}

then Ixl

~ iY(t)l

~ 2Kmlxl

does n o t pass through

i ¥ ( t ) l X-hi

Ix-ylni

• 0 ,

L Co i x l ~ - n i

and

I x - y l ni

! C1 Ixl ^ I x - y l ,

provided

Ixl

L 2K I x - y l

depending o n l y on

Corollary. it

is Lipschitz

,

K , X ,

since and

ni ~ 1 . ni .

CO and

C1

are c o n s t a n t s

This proves the Lemma.

Suppose the homogeneous norm continuous:

Ilxl - l y [ l

Here

~ c Ix-yl

ixl

is

C1

on

V ~ {o}

.

Then

70 1.2 I n t e g [ a l formulas. Suppose

Ix[

is a homogeneous norm on

V .

Then

we have the f o l l o w i n g i n t e g r a t i o n f o r m u l a :

a ~ i xB I |

where

C

Ixl -Q dx

=

C log (B/A)

i s a constant independent o f

dimension o f

A,B ,

,

and

Q

is the homogeneous

V .

To prove t h i s f o r m u l a , set

f(t)

=

I

Ixl -Q dx

( t ~ 1) .

1~Ixi~t Because the measure

Ixl -Q dx

is i n v a r i a n t under d i l a t i o n s ,

we f i n d t h a t f o r

s,t ~ I , f(st) In p a r t i c u l a r , 0 < t < I , R+

= f(s) + f(t)

this implies that then

f

.

f(1) = 0 .

I f we d e f i n e

f(t)

= -f(1/t)

i s a continuous homomorphism from the m u l t i p l i c a t i v e

to the a d d i t i v e group ~ .

Hence

More g e n e r a l l y , suppose t h a t

f(t)

= C log t

,

, group

as a s s e r t e d .

i s any continuous f u n c t i o n on

which i s homogeneous o f degree zero. Then there e x i s t s a constant

V ~ {o} m(m)

such

that I f([xl) V f o r any

f

m ( x ) I x l -Q dx = m(m) I f ( t ) R+

e L 1 (IR+ ; t -1 d t ) .

function of

LA,~

(IR+= ( 0 , ~ ) )

.

t -1 d t

(The case

f = characteristic

i s proved as above, and the general case f o l l o w s by dominated

convergence.) We s h a l l c a l l

m(m)

the mean-value o f

m .

The map ~ -~ m(~)

is a

continuous l i n e a r f u n c t i o n a l on the space o f continuous f u n c t i o n s homogeneous o f degree

0 ,

relative

to the norm

m(~) log R =

I

I~LxI~R

llmll~ •

Indeed, we have, f o r

m(x) Ix[ -q dx ,

R > i ,

71

so t h a t

lm(~)l

log R 2

114L

f

Ixl -q dx

12]xl~R and hence

(We n o r m a l i z e the measure

dx

on

V

by the c o n d i t i o n

m(1) = 1 .)

§ 2. Spaces o f homogeneous t y p e

2.1

Distance f u n c t i o n s

Let

X

be a l o c a l l y

compact H a u s d o r f f space.

Suppose

p : x× We s h a l l

call

p

x+[o,~)

.

a d i s t a n c e f u n c t i o n on

X

p r o v i d e d the f o l l o w i n g c o n d i t i o n s are

satisfied:

I)

p(x,y)

# 0 if

2)

p(x,y)

= p(y,x)

3)

The sets

x # y , and p ( x , x )

{y : p(x,y)

~ r},

for

basis f o r the neighborhoods o f

4)

There e x i s t s c o n s t a n t s [p(x,y)

We s h a l l c a l l

p(x,y)

c a l l e d the exponent o f that

- p(x,z)l

C > 0

r > 0 ,

are compact and are a

x and

~ C p(y,z) a ~(x,y)

the d i s t a n c e between

p .

= 0

0 < a < I

such t h a t

+ p ( y , z ~ 1-a

x and y .

The number a w i l l

Note t h a t a i s r e q u i r e d to be s t r i c t l y

positive,

be so

1 - a < 1 . Examples I .

dilations

6t .

Let Let

V I'I

be a graded r e a l v e c t o r space as in

§ 1.1 ,

be a smooth, symmetric homogeneous norm on

with V ,

and

72

p(x,y) = Ix'YI.

set

Then by C o r o l l a r y

Ip(x,y) Thus

4)

- p(x,z)I

is satisfied

with

2. More g e n e r a l l y , Lie algebra

V

! C [ x - y - x+z I = [Y-Zl

a = 1 .

let

X

e(x,y)

The o t h e r p r o p e r t i e s

I'I

(as a L i e a l g e b r a ) .

on

= log ( x - l y )

V

,

structure

on

map).

is clear that conditions

4) V

It

in this

as n o t e d . Then

Corollary

1.3.1

a = i/r,

function

u = x-ly

e(x,y)

p(x,y)

le(x,y)l

=

(If

,

we d e f i n e the group

,

1) - 3)

are s a t i s f i e d .

v = x-lz

= u, e ( x , z )

,

w = y-lz

= v, e(y,z)

,

= w .

is the i d e n t i t y

To e s t a b l i s h and i d e n t i f y

But

estimate ×

v = uw ,

with

so by

we have

IV-U-Wl

where

Take a smooth,

using the C a m p b e l l - H a u s d o r f f f o r m u l a , then log

case, s e t

L i e group whose

as in example 1, and d e f i n e

where log i s the i n v e r s e to the e x p o n e n t i a l map V

are obviously verified.

be any s i m p l y connected n i l p o t e n t

admits a g r a d a t i o n

symmetric homogeneous norm

1.1 ,

~ C ( l u l a l w l l - a + l u l l - a l w l a) ,

r = length

o f the g r a d a t i o n on

V .

Further,

f o r any norm

one has

lu-vl ~ K(lu-v+wl + I w l ) . By c o n s i d e r i n g the cases

[u I ~ Iw[

and

lul

! lwl

separately,

we see t h a t t h i s

g i v e s the e s t i m a t e

lU-Vl ! C {luil-alwl a + lwl} (Note t h a t we may assume

a ~ 1/2 ,

commutative and

lul

other properties

are o b v i o u s .

3.

Let

X

= Iluli.)

be a p a r t i a l

s i n c e the case

By C o r o l l a r y

a = I

means

X

1.1, this gives condition

homomorphism from a graded n i l p o t e n t

is 4) .

The

Lie a l g e b r a

V

73 i n t o the Lie a l g e b r a o f v e c t o r f i e l d s Assume t h a t f o r from to

V H

x e M ,

the map

u --+ ~(U)x

o n t o t h e t a n g e n t space a t

be d e f i n e d as in

§ 11.3.

on a m a n i f o l d

x .

M ,

as in Chapter I I .

is a linear

isomorphism

L e t the e x p o n e n t i a l map from

The analogue o f the map

V

x , y ÷ log ( x ' l y )

considered in the p r e v i o u s example is the map

e :H defined implicitly

o

×H

÷V

o

by the i d e n t i t y

exp x ( e ( x , y ) ) Here in

Jqo c

g

is a s u f f i c i e n t l y

X = Ho, p ( x , y )

s i n c e t h e maps condition

eXPx ,

4)

= Io(x,y)l x e Ho ,

w i t h the a d d i t i v e

4. Suppose filtration

it

V

Then are a l l

p

obviously satisfies

as

group o f

X

II.3.2

to compare the group germ generated by

is any s i m p l y - c o n n e c t e d n i l p o t e n t

= Ilog(x-ly)l

satisfies

of

4)

~.

V .

on t h e L i e a l g e b r a

p(x,y)

1) - 3) ,

d i f f e o m o r p h i s m s . The v e r i f i c a t i o n

V

of

X ,

L i e group. L e t

,

as in example 2. Then

in e v e r y subset where

F

be a

and put a graded v e c t o r space s t r u c t u r e

by choosing complementary subspaces t o the f i l t r a t i o n ,

C = C(~). lul

xe e g ,

i s made by e x a c t l y t h e same argument as in the preceding example,

but t h i s t i m e using Theorem

Let

small neighborhood o f a g i v e n p o i n t

§ 11.3.1.

Set

on

= y .

p

as in Chapter I .

satisfies

p(x,y) ~ ~ > 0 ,

1) - 3) ,

and

with a constant

Indeed, using the same n o t a t i o n as in example 2 above, we have

= p(x,y)

estimate for

~ e .

Hence by Theorem 1 . 3 . 1

Iv-u-wl

Remarks 1.

Remark 1 . 3 . 1 , we get the same

as in t h e graded case.

Suppose

p

i s a d i s t a n c e f u n c t i o n on

mean-geometric mean i n e q u a l i t y Ip(x,y)

and

- p(x,z)l

we have ! K~(x,y)

+ p(y,z~

.

X .

By the a r i t h m e t i c

74 In p a r t i c u l a r , p ( x , z ) ~ 2K ~ ( x , y ) so t h a t

p

satisfies

2. Condition compared to implies that

z

,

a weak form o f the t r i a n g l e

4)

p(x,y)

+ p(y,z~

on

,

p

i.e.

will y

be used in s i t u a t i o n s where

is close to

is also f a r from

inequality.

x

z

but f a r from

x .

and t h a t the d i f f e r e n c e

i s o f s m a l l e r o r d e r o f magnitude than the d i s t a n c e require

p(y,z)

p(x,y)

is small

In t h i s case 4)

ip(x,y)

(recall

- p(x,z)l

t h a t we

a > o) .

3. In a l l

the examples

2) - 4)

above the number

i n t e g e r equal to the l e n g t h o f the g r a d a t i o n on the weaker e s t i m a t e

4)

V .

r = 1/a

is a p o s i t i v e

The l o n g e r the g r a d a t i o n ,

becomes, in terms o f the comparison o f distances described

in the previous remark.

2.2 Homogeneous measures p o s i t i v e Radon measure on relative

X .

Let

X, p

be as in

We s h a l l say t h a t

to the d i s t a n c e f u n c t i o n

p ,

§ 2.1. ~

Suppose

u

is o f homogeneous t y p e ,

i f t h e r e e x i s t constants

C, Q > 0

that (~)

S p ( x , y ) -Q d~(y) ~ C log (B/A) A~p(x,y)~B

for all

x e X ,

Lemma

(i) (ii)

(iii)

and a l l

Assume

u

0 < A < B .

satisfies

(~) .

Let

m> 0 ,

is a

0 < A < B .

~ ( { y : p ( x , y ) ~ B}) ~ CBQ f p ( x , y ) -Q+~ du(y) <_ (C/m) B~ p(x,y)_
.

Then

such

75

Here the c o n s t a n t

Proof.

C

is independent o f

We e s t i m a t e the i n t e g r a l s

o f the r e g i o n o f i n t e g r a t i o n k e ~

~,A,B .

by the c l a s s i c a l

into concentric shells.

method o f decomposition

Take

R < 1 ,

and f o r

define. S k = {Y : BRk+l < p ( x , y )

Then by

~ BRk} .

(m) ,

I

p ( x , y ) -Q+e d u ( y ) ~ CB~ Rk~ log ( I / R )

.

Sk

S

p ( x , y ) -Q-~ dB(y) _< CB-~ R- ( k + l ) ~

log ( l / R )

.

Sk

Now the spheres r > o

and any

= Q

{x : p ( x , y )

y e X .

= r}

have

In p a r t i c u l a r ,

~-measure z e r o , by

~({x})

= 0

for every

(m), x .

f o r any

Hence t a k i n g

i n the e s t i m a t e s above, we have oo

p({y

: p(x,y)

~ B})

z U(Sk) k=o ClBQ

where

C1

i s independent o f

To prove

(iii)

,

B .

we choose

The same argument g i v e s n

so t h a t

(ii)

BRn+l < A ~ BRn .

e s t i m a t e s above g i v e s R" ( k + l ) ~ n k=o

p ( x , y ) - q - ~ d~(y) ~ CB-~ log ( l / R ) A~p(x y)2B

C log (I/R) ~RA)-~ _ B-~ i - R~

<_ Letting

R~ 1 ,

we o b t a i n

(iii)

,

Q.E.D.

ERA)

-

. Then the

76 Remark

I f we l e t

~ ÷ o

We s h a l l

c a l l the t r i p l e

in

(iii)

(X,p ,~)

,

we regain e s t i m a t e

satisfying

the above c o n d i t i o n s a space

o f hgmogeneous t y p e , and we s h a l l r e f e r to the number dimension o f

( X , p , ~ ) . (In a l l

from the Lemma t h a t

A + 0

or

R has measure

dimension has a geometric j u s t i f i c a t i o n . c o n t e x t , we w i l l

write

Examples 1.

X .

X ,

as in

Identifying

In t h i s case i t (~).)

< CRQ ,

by t r a n s l a t i o n - i n v a r i a n c e

.

Examples

V

for

Hence

u .

V

(X,p,~) satisfies X×X

(X,p,~)

is c l e a r from the

be Haar

by the e x p o n e n t i a l map,

S ivl -Q dv , A
f o l l o w s from the i n t e g r a l formula of in the Lemma become e q u a l i t i e s V

in t h i s case,

is a L i e - a l g e b r a

but not graded, as a L i e a l g e b r a , then

i s o f homogeneous type "near i n f i n i t y " , 4)

~

in

§ 2.1

when

in the sense t h a t

(x,y)

p

only

is away from the diagonal in

. 2.

Let

X,O

be as in

§ 2.1 , Example 3. Take f o r

which in l o c a l c o o r d i n a t e s has smooth, s t r i c t l y Lebesgue measure. For ×

~

is a space o f homogeneous t y p e , in the sense j u s t

i s only f i l t e r e d ,

condition

(i)

Then

by the same i n t e g r a t i o n f o r m u l a . I f the g r a d a t i o n on

V

is c l e a r

so t h i s use o f the term

I , 2 o r 4. Let

X w i t h i t s Lie algebra

Furthermore, the i n e q u a l i t i e s

defined. If

be an

Note t h a t by p a r t

When the choice o f

§ 2.1 ,

S p ( x , y ) -Q d~(y) = A~p(x,y)~B

g r a d a t i o n , then

will

be a simply-connected n i I p o t e n t L i e group,

we can take Lebesgue measure on

§ 1.2 .

(m)

d~(y) = dy .

Let

p(x,y) = flog (x-ly)I measUre on

B-+ =

Q is u n i q u e l y determined by

o f the Lemma a b a l l o f radius

Q as the homogeneous

the a p p l i c a t i o n s the e s t i m a t e

asymptotic e q u a l i t y when e i t h e r

(m) .

x e × ,

onto an open, r e l a t i v e l y

t h i s mapping i s o f the form

the map y

compact subset ~(x,u)du ,

,

any measure on

positive density relative ~ e(x,y)

U of V .

where

u

du

×

to

is a diffeomorphism from The image o f

d~

under

i s Lebesgue measure on

V

77 and

m is a bounded, p o s i t i v e ,

smooth f u n c t i o n on

Hence we have the

X × U .

i n t e g r a l formula

f f(Io(x,y)l) X

d~(y) = f f ( [ u l ) U

v a l i d f o r any Borel f u n c t i o n this implies that Q

~

and

f

on

IR+ .

~(x,u)du ,

By the i n t e g r a l formulas o f

p(x,y) = le(x,y)I

condition

(*)

i s the homogeneous dimension o f the graded n i l p o t e n t a l g e b r a

V .

2.3 L i p s c h i t z spaces complex-valued f u n c t i o n order

~

if

f

on

(X,~,~) X

t h e r e is a constant if(x)

for all

Let

x,y e X .

- f(y)[

Set

A(B) = { f

: I[flI~ < ~} .

Banach algebra r e l a t i v e

a uniform L i p s c h i t z c o n d i t i o n o f

~ C p(x,y) B B >0

we d e f i n e

+ sup { p ( x , y ) -B I f ( x ) x~Y

It

where

such t h a t

In g e n e r a l , f o r any

I l f l l # = sup I f ( x ) l x

,

be a space o f homogeneous type. A

satisfies C

satisfy

§ 1.2

- f(Y)I}

•

is a standard e x e r c i s e to v e r i f y

to the norm

IIfI[#

isa

that

Let

Ac(~ ) = { f e A(B) : Supp(f) is compact} .

Then by axiom

3)

f o r the d i s t a n c e f u n c t i o n , we see t h a t

space o f continuous f u n c t i o n s vanishing a t i n f i n i t y Lemma Co(X )

when

Proof.

Let a be the exponent o f

(X,p,~)

.

on Then

Ac(B ) ~ C o ( X ) ,

the

X . Ac(~ )

i s dense in

o < B < a .

Let

~ e C~ (IR)

and the l o c a l compactness o f

and set X ,

f(y)

= ~(p(x,y))

the f u n c t i o n

f

will

.

By axiom

3)

for

be compactly supported.

78 Since

m i s smooth, we o b t a i n from axiom If(y)

This shows t h a t of

X

verifies

if

- f(z)l

Hence the f u n c t i o n s in

(Note t h a t

Ac(a ) ~ Ac(B))

.

Ac(~ )

separate the p o i n t s

As remarked above, one e a s i l y

that

IIfgII~ <_ [LfEI6 Ilgl[ 6 so t h a t

the e s t i m a t e

~ C ~ ( y , z ) a ~ ( x , y ) 1-a + p ( y , z ~

f e A (a) .

o < 6 ~ a

4)

Ac(£ )

is a self-adjoint

,

subalgebra o f

Co(X ) .

The Stone-Weierstrass

theorem thus gives the asserted d e n s i t y statement.

Corollary

If

o < £ < a ,

the space

Ac(6 )

i s dense in

Lp(X,du)

for

l
§ 3.

Singulariiiintegral

3.1 S i n g u l a r k e r n e l s . homogeneous dimension diagonal in

X × X ,

Let

X,p,~)

Q and exponent

operators

be a space o f homogeneous type, w i t h

a .

Let

D = {(x,x)

: x e X}

be the

and assume t h a t K:XxX~D÷¢

i s a continuous f u n c t i o n .

Define

K="(x,y) = K ( y , x ) c (c = complex conjugate)

Definition. C, M, R > 1

F=K:

K

is a s i n g u l a r kernel on

X

if

there e x i s t constants

such t h a t the f o l l o w i n g e s t i m a t e s are s a t i s f i e d

by

F = K and

79

(I)

IF(x,Y)I

(II)

If

£ C p ( x , y ) -Q

I F (A,x) =

then f o r a l l

I

f F ( x , y ) dy , A£p(x,y)£AR x e X ,

[IF(A,x) I £ C A

(III)

If

p(y,z)

Ir(y,z)

We s h a l l c a l l condition ,

IF(A,x ) = 0

and

~ M p(x,z)

-

(I) (Ill)

,

,

if

A < R

,

if

A > R

then

(y,x)i ' <- c ~ ( y , z ) ]

P'Y'z'-Q~

the ~omio9eneity c o n d i t i o n the smoothness c o n d i t i o n

s i n g u l a r k e r n e l s , which w i l l

, .

(II)

the mean-value

Before g i v i n g examples o f

e x p l a i n the t e r m i n o l o g y , we r e c a s t these estimates

from the p o i n t o f view o f f u n c t i o n a l

analysis.

Namely, we

fix

M, R > i

and

d e f i n e the f o l l o w i n g norms: [[K[[,I

sup

p(x,y) Q [K(x,y)l

=

sup

A- I

IIKIIIII =

sup

[IK[]II

p(y,z) Q iK(y,z)

the suprema being taken over in the case o f

I IK(A,x)I

IIKIIII I , p ( y , z )

x,y,z,A

such t h a t

~ M p(x,z)

- K(y,x)l

x #y,

x # z, 0 < A <_ R ,

and

.

Define the space KM, R ( X , p , u )

= {K : K and Ks: s a t i s f y

I,II,III

We norm t h i s space by s e t t i n g

!IKHM,R = I[K[[ + [[K~[I ,

[[KI[ = [[K[[I + [IKI[ll + IIKI[ll I .

Then i t

where

is s t r a i g h t f o r w a r d

a Banach space in t h i s norm. For example, c o n d i t i o n

f o r some C} .

III

to v e r i f y for

K

that and

K

KM,R

is

implies

80

that

K

i s c o n t i n u o u s on

X x X ~ D .

norm on any compact s u b s e t o f functionals

K i

Examples nilpotent

X x X ~ D ,

> I K (A,x)

mean-value condition

A l s o the norm

[IKII I

and hence a l s o d o m i n a t e s t h e

Thus t h e c o n t i n u i t y

are p r e s e r v e d u n d e r l i m i t s

of

K

and t h e v a n i s h i n g

i n the norm

C o n s i d e r two c l a s s e s o f examples from

group case (example i ) ,

dominates t h e sup

§ 2.2

IIKIM,R

,

namely t h e graded

o r the group germ case (example 2 ) .

cases we have a graded v e c t o r space

V

o f homogeneous dimension

In both

Q

and a

Q

be t h e

mapping 0 : X×X÷V

,

such t h a t p(x,y) where

I'I

~

V .

We o b t a i n

on

V ~ {o}

dilations

on

V .

~

(V)

Give t h i s

sup { I k ( x ) i

Let

k e r n e l s as f o l l o w s :

t o be t h e space o f

which are homogeneous o f degree

x ,

relative

CI

to the

space t h e norm

[[ktt~ + flvklL

ilklL=

V .

a class of singular

be a complex number, and d e f i n e

functions

where

,

i s a smooth, s y m m e t r i c homogeneous norm on

homogeneous d i m e n s i o n o f

Let

ie(x,y)l

=

: ix

,

= 1} ,

as i n

§ 1.1.

Then

K~(V)

i s a Banach

space.

Lemma

Suppose

r = exp ( 2 ~ / i s l ) Then t h e r e e x i s t s continuous this

from

~ = -Q + i s

,

with

a real

number. Set

. an

M > o

K~(V)

into

and an i n t e g e r KM, R

(X,p,~)

map i s c o n t i n u o u s from the subspace

zero i n t o

s # o

KM, R ( X , p , ~ )

for all

n ~ 1 ,

with

K°Q (V)

sufficiently

large

so t h a t

map

R = rn of functions R .

k

When

~--~ k o e ~ = -Q ,

with mean-value

is

81

Proof

Let

k e Kx(V)

,

and s e t

IK(x,y

and hence

<

(II

,

i s homogeneous o f degree z e r o . R = exp ( 2 ~ n / I s l )

,

Then

we o b s e r v e t h a t

the function

[x] -~ k ( x )

formula of

§ 1.2,

= ~(x) if

then

k ( x ) dx

m(m) = o

.

llklI~ # ( x , y ) -Q ,

Hence by the i n t e g r a l

A
= k(e(x,y))

IIklL

ii~I I

To check c o n d i t i o n

K(x,y)

R ,

=

AR m(~) ~

t -1+is

m(~)~ is

(R i s - l ) / ( i s ~

Ris = 1 ,

so t h i s

dt

vanishes.

.

In case

s = o ,

then

by a s s u m p t i o n .

In t h e n i l p o t e n t

group case t h e t r a n s l a t i o n

invariance

gives the

of

formula

I K (A,x)

=

S k ( u ) du = 0 A
In t h e group germ c a s e , we can w r i t e

I K (A,x)

=

By t h e compact s u p p o r t o f sufficiently

large.

k(u)

S

E~(x,u) - ~ ( x , o £

When

m , A < R

this

vanishes for

we can e s t i m a t e

A > R = rn ~(x,u)

bound IIK(A,x)I

} l u l -Q+I du ~ C llkll ~ A
< C A [Ikil~ This g i v e s t h e e s t i m a t e

du

A
for

IIKIIII

.

if

- m(x,o)

n

is

and g e t t h e

82

By Lemma 1.1, we have the e s t i m a t e iK(y,z) - K(y,x)i when

p(y,z) ~ C le(y,z)

from t h i s i n e q u a l i t y , le(y,z) (cf

- e(y,x)I

we r e c a l l

- o(y,x)[

§ 2.1, Examples) .

or graded n i l p o t e n t o f an a r b i t r a r y

~ C Ilvkli ~ I o ( y , z ) - e ( y , x ) I p ( y , z ) -Q-1 , .

In o r d e r to o b t a i n e s t i m a t e

for

K

that

~ C p(x,z) a D(x,z)

+ p(y,z~Z-a

This e s t i m a t e i s v a l i d f o r any

group case.

nilpotent

(III)

(It

group.)

is v a l i d f o r

But now i f

x,y,z

in the group germ

p(x,y) ~ ~ > o

p(y,z) # M p(x,z)

in the case ,

then

le(y,z) - e ( y , x ) l ~ C M-1 [M + ~ Z - a p ( y , z ) . Since when

l-a < 1 ,

we can choose

p(y,z) ~ M p(x,z)

,

M so l a r g e t h a t

M- I

the above e s t i m a t e f o r

[M + ~ l - a

< C-2 .

IK(y,z) - K(y,x)l

Then

applies,

and we get

IIKIIII I ~ c llvkLI~ (Here of

a

k.)

i s the exponent o f Since

p ,

o(y,x) = -e(x,y)

in I I I ,

,

and the choice o f

the same estimates hold f o r

M i s independent K~ .

This

completes the p r o o f o f the Lemma.

3.2 Operators defined by s i n g u l a r kernels singular kernel. use a l i m i t i n g

K ~ KM, R (X,p,t~)

In o r d e r to d e f i n e an i n t e g r a l o p e r a t o r w i t h kernel

process. Given Km,n f ( x )

The i n t e g r a l

Let

=

m < n e2~ ,

K ,

we must

we d e f i n e the t r u n c a t e d o p e r a t o r

f K(x,y) f(y) Rm
dy

converges a b s o l u t e l y f o r any bounded, measurable f u n c t i o n

IIKm,nfll~ <- Cm,n IlflL I1KIII

be a

f

,

and

83

IIflI=

(Here This

= sup

estimate,

If(x)l

A(~)

Theorem

o < ~ <_ a

Kf

.

Letting

is a slight

was d e f i n e d i n

If

d e f i n e d in the p r e v i o u s s e c t i o n . )

use, s i n c e

l o g (R n-m)

The n e x t r e s u l t

(The space

ILK111 was

f p ( x , y ) -Q dy , Rm~p(x,y)2R n

i s o f the o r d e r

information.

and

however, i s o f l i t t l e

Cm, n =

and t h i s

,

=

m + -~

or

n ÷ +~

improvement o v e r t h i s

g i v e s no

crude e s t i m a t e .

§ 2.3):

and

f e Ac (B)

I im

K

m~n

m-~-m

,

then

f

n÷+oo

exists

in

Lp

for

1 < p ~ ~ ,

and

<_ Cp ~(Supp(f)) IIKI[ Ilfll B

IIKflIL P where

Cp

is a constant independent of

Proof

We s h a l l

K and f

.

write

Then

Km = Kin, 1

,

m < 0

Kn = K1, n

,

n >_ 1

Km, n = Km + Kn

,

and we s h a l l

investigate

each summand

separately.

Let

(II)

on

mm = Km I K

,

where

1

is the function

we see t h a t

Ii~mII~ < R IIKIIII and lim

~m(X) : m(x)

,

identically

1 .

By c o n d i t i o n

84 e x i s t s u n i f o r m l y in

X

,

I] [L

with

<-- R

IIKI]zI

.

We may w r i t e Kmf(X ) =

Hence i f

f K(x,y) Rm~p(x,y)~R

m < n < 0 ,

I Kmf(X)"

!f(y)

- f(x)l

dy + ~m(X)f(x)

.

we have

Knf(X)]

<

llKI]i IlfllB

Y

p ( x , y ) "Q+B dy

Rm
+ l lK(Rm-n,x)

I

CR(m-n)a(llKlll

by Lemma 2.2.

m< 0 .

+ [IKIIiz)[IflI B

,

This shows t h a t

e x i s t s u n i f o r m l y in in

Ilfll B

lim

Kmf(X ) : K f ( x )

x .

If

f

Hence t h i s l i m i t

To e s t i m a t e

Knf ,

is compactly supported,

e x i s t s in a l l

Lp

Kmf ,

uniformly

norms, I < p ~

we use the Schwarz i n e q u a l i t y

IKmf(x) - Knf(x)l_
so is

to o b t a i n ,

if

O<m
,

f p ( x , y ) -2Q dy} I / 2 Rm<_p(x ,y)<_Rn

< C R-Qm IIKIII llfllL2 ,

by Lerm~a 2.2.

This shows t h a t l i m Knf(x) = K~f(x) n-~

e x i s t s u n i f o r m l y in

x .

To show t h a t t h i s l i m i t kernel

K(x,y)

,

e x i s t s in

Lp, I < p < ~ ,

t r u n c a t e d in the region

p(x,y)

< R ,

we observe t h a t the is in

Lp

relative

to

X~

85 for fixed

y ,

and v i c e - v e r s a .

Let

g e Lq ,

where

1/p + 1/q = 1 .

Then by

H~Iders i n e q u a l i t y , 1

llgHLq

I K(x,y) g(x) dx I < C IlK~III p(x,y)~R - P

where Cp

=

[ p ( x , y ) -pQ dx p(x,y)~R

by Lemma 2.2, since f

p > 1 .

<

(Here we have used c o n d i t i o n

I

for

Km.)

Thus i f

is compactly supported, then < Knf'g > =

f f n K(x,y)f(y)g(x) y~ Supp(f) R~p(x,y)~R

dx dy .

By the estimate above, we have i < Knf,g > I <_ Cp p(Supp(f))IIK~'~III llfl~ ligllLq

where the constant IIKnfIIL

Cp

is independent of

<_ Cp ~(Supp(f))

K,f,g

.

,

This shows t h a t

[IKII Elf[I= ,

P and completes the proof of the theorem.

§ 4. Boundedness of s i n g u l a r i n t e g r a l operators 4.1 Almost orthogonal decompositions and denote by Tm w i l l

B(H)

Let

H be a complex H i l b e r t space,

the algebra of bounded operators on

H .

If

T e B(H) ,

denote the operator a d j o i n t to T: (Tx,y)

We denote by

lITII

=

(x,T~y)

the operator norm of

r e l a t i o n s between the norm and the

, T ,

x,y e H . and we r e c a l l the fundamental

m-algebra s t r u c t u r e on

B(H):

then

86

[iT[[

=

IIT*TII 1/2

:

lira II(T"T) n [[1/2n . rl-~oo

(The f i r s t

e q u a l i t y Is immediate from the d e f i n i t i o n o f

i n e q u a l i t y . The second e q u a l i t y

T'~ and the Schwarz

f o l l o w s from the s e l f - a d j o i n t n e s s of

T¢=T and

the spectral theorem.) Suppose now t h a t t h a t the

{T k}

such t h a t

j,k e Z .

(Set

For example, i f then

it

(¢:)

Tkl I

~ e Z 1 (E)

when

z re(k)) . keZZ

is a f a m i l y of mutually orthogonal p r o j e c t i o n s on j # k ,

is s a t i s f i e d with

and

m = 26

IIPjP~II = llPjll

=

1

zPj

Furthermore, converges in

because of the o r t h o g o n a l i t y , even though each term has norm 1. The next

theorem generalizes t h i s geometric property to the case of almost orthogonal f a m i l i e s of operators. Theorem (~':).

H ,

In t h i s case

.

(6 = Kronecker d e l t a ) .

is a basic f a c t in H i l b e r t space geometry t h a t the series

B(H) ,

say

<_ re(j-k) 2

IlmtlZ1 =

{Pk }

PjP~ = P~Pk = 0

estimate

We w i l l

and

IIT~ Tkl * ]]Tj

for all

H .

are almost orth,o~enal i f there e x i s t s a f u n c t i o n

m> o

(:'~)

{Tk}ke2Z is a f a m i l y o f operators on

If

Jc7/

Suppose

{T k}

is a f i n i t e Sj

=

is a f a m i l y o f operators on

H which s a t i s f i e s

subset, l e t ~ keJ

T k

Then iISjil <- II~!IZl

and hence the p a r t i a l sums Sj

•

are u n i f o r m l y bounded in

B(H) .

87 Corollary:

If

lim

........

then

Sjx ~ Sx exists for

x

in a dense subspace of

H ,

Iji~

S e B(H)

and

Ilsll ~ i~llZl

Proof of Theorem

We shall estimate

This is bounded by the

"~"H(SjSj)nlE.

sum of terms IIT:~I J1 Ti2 Tj2 where

i k ' Jk e J .

submultiplicative

Tin Tjn

Grouping the factors

'

i k ' Jk

together and using the

property of the norm, we obtain from m(il-J 1

)2

(4)

the estimate

~ (i2-J2)2 . . . ~(in-Jn )2

for such a term. Grouping the factors

Jk' ik+l

together, we s i m i l a r l y obtain the

estimate ~(0) ~ ( J l - i 2 ) 2 for the same term. (Note that

• .

- ~(Jn_l-in )2

~

(0)

IITjll = II~II = IIT~ Tjll 1/2 ~ ~(0).)

Taking the

geometric mean of these two estimates, we see that the term is bounded by the product ~(0) ~ ( i l - J 1) ~ ( j z - i 2 ) We may now sum this over all

""

~(jn_Z-in)

i k ' Jk e J .

~ (in-J n) •

This gives the estimate

tt(ssj)nll (J) (o)tl II Taking the ilsjll

2n-th

root and l e t t i n g

n ~ ~ ,

we obtain the stated inequality for

•

4.2 Decompositions of singular integrals geneous type, and l e t

Let

K be a singular kernel on

(X,p,u)

be a space of homo-

X in the sense of

§ 3.1.

88 We want to use the Theorem of the preceding paragraph to e s t a b l i s h t h a t the operator defined by

K

is bounded on

L 2 (X, du).

For t h i s we shall decompose

K i n t o a sum of almost orthogonal operators. Let the constants

M,R > 1

be such t h a t

K e KM, R (X,p,u)-

For

j e Z ,

define I K ( x , y ) , i f Rj ~ p ( x , y ) ~ Rj + l Kj ( x , y ) =

L o

,

otherwise

and set Tjf(x) for

:

f e L2 (X, d~) .

L2(X, du). {Tj}j~ °

f K j ( x , y ) f ( y ) dy Let

IIAII

,

denote the operator norm of an operator

A on

Then we have the f o l l o w i n g estimate which shows that the two f a m i l i e s and

{Tj}

are both almost orthogonal.

(This separation of

j > o

j~o and

j ~ 0 corresponds to the two s i n g u l a r i t i e s

of

K :

the p o i n t at i n f i n i t y

and the d i a g o n a l . ) Lemma

There is a constant

C ,

independent of

K ,

such t h a t when

j,Z

have the same sign, then

[ETj T~I] + IIT~ TZl] 2 CR-ajj-zl (The norm

HKHM,R was defined in

Proof.

If

A

IIKLI2 R

§ 3.1.

Here

a

is the exponent of

is an i n t e g r a l operator w i t h kernel

A(x,y),

(X,p,~).)

then by the

Schwarz i n e q u a l i t y one has the pointwise estimate I A f ( x ) l 2 <_ ( I I A ( x , z ) I d z )

IIA(x,y)IIf(Y)I

2 dy o

I n t e g r a t i n g t h i s estimate, we f i n d t h a t (*)

llAI1 < [sup f I A ( x , y ) idy~ I / 2 ~ u p f I A ( x , y ) I d x ~ 1/2 x y Let us apply t h i s estimate to

Tj .

By c o n d i t i o n

(I)

on

K ,

we have

89

I

lITjll <_ iIKi]I

p ( x , y ) -Q dy

RJ

<_ (c log R) [[Klit , where

C is independent of

bounded, and i t w i l l N.

(N w i l l

K .

Thus the operators

{Tj}

are a l l uniformly

s u f f i c e to prove the lemma when l j - l l

~ N ,

f o r any fixed

be chosen in the course of the proof and w i l l depend on

the distance function

p

,

but the choice w i l l

M,R ,

and

be independent of

K e KM,R ( X , p , ~ ) . ) The operator

TjT#

is an i n t e g r a l operator with kernel

Gjz(x,y) = ~ K j ( x , z ) K l ( y , z ) c dz.

Since both

K and

K

estimates we obtain f o r

are assumed to s a t i s f y IITj T~I1 w i l l

to e s t i m a t e the r i g h t side of

(~)

(I),

(II),

also apply to

when

(111)

IIT~ T l l I .

of

§ 3.1 ,

the

Thus i t is enough

A(x,y) = GjZ (x,y) .

We shall e s t a b l i s h the i n e q u a l i t i e s

I sup

K .

(x,y)Idy

if

I ~ j

sup IIGjl (x,y)Idx < CRa(z-j) llKII~,R , Y

if

j ~l

provided t h a t of

flGjl

ilKll~,R ,

X

j,l

~

CRa ( j ' l )

are of the same sign and

lj-ll

~ N ,

where

Once t h i s is done, we observe t h a t Gjl ( x , y ) : G/j ( y , x ) c

Hence (m*)

I

implies that i f

jl

~ 0

and

lj-ll

sup flGjz (x,y)tdy < CR-alj-sl IIKIIMZR Syp ./'tGjl

(x,y)ldx < CR-alj-zl

IlKll2 R

~ N ,

then

N is independent

90 Using t h i s in

(m),

To prove

we w i l l

(~m) ,

Suppose f i r s t

that

E = {(y,z)

obtain the estimate of the lemma.

we consider the cases

Z > j .

Fix

x ,

and

j ~ Z

separately.

and set

: Rj ~ p ( x , z ) ~ Rj + l

A l l estimates take place on

I ~ j

,

RZ ~ p ( y , z ) L RZ+l} .

E .

We w i l l make a three-term collapsing sum estimate of the left-hand side of

(mm) ,

of

§ 3.1

corresponding roughly to the three conditions on the kernels

GjZ ( x , y ) = f

K and K~ .

Kj(x,z)

+ ~(y,x) + Kz(y,x

(I),

(II),

(III)

Namely, we w r i t e

~Kz(y,z ) - K ( y , x ~ c dz

- Kz(y,x~C f K j ( x , z ) dz

)c

f K j ( x , z ) dz

Thus we have f I G j z ( x , y ) I d y 5 ~f I K j ( x , z ) I I K ( y , z ) + ff E

- K ( y , x ) I d y dz

IKj(x,z)IIK(y,x)

- Kz(y,x)Idy dz

+ I f K j ( x , z ) dz I f I K z ( y , x ) i dy

(Note t h a t

Kz(y,z ) = K(y,z)

I I , 12 , 13 ,

13

is the simplest to estimate. Indeed, by

I IKz(y,x)I dy <_ IIKIII (II)

Call the terms in t h i s sum

r e s p e c t i v e l y . We shall estimate each term separately.

The term

and by

when (y,z) e E.)

(C log R) ,

,

I f K j ( x , z ) dz I <_ llKllll "

I

CRaj

,

if

J ~Z 0

0

,

if

j>O

(I)

,

91

Thus i f and 1

j > 0 ,

then

11 = 0 ,

while if

j ~ 0 ,

then

1 < 0

also, since

are assumed to have the same s i g n . Hence in any case we g e t the e s t i m a t e

13 <_ C R a ( j - l )

To e s t i m a t e

12 ,

n o t e t h a t the i n t e g r a n d i s zero when

RZ < p ( x , y )

F u r t h e r m o r e , when

IIKII I IIKHII •

< RZ+l

(y,z)

e E ,

then by axiom

4)

on the d i s t a n c e f u n c t i o n we

have Ip(x,y)

- p(z,y)l

~ C p(x,z) a [o(x,z) CRl + a j + ( l - a ) Z

+ p(y,z~l-a

EI+RJ-/]

< CR/+a(J -Z)

since

j-1

< 0 .

Here

C

is a constant independent of

J+ = {y : ( y , z ) then t h i s

e E f o r some z, and p ( x , y )

.

I f we w r i t e

~ RZ+1} ,

l a s t e s t i m a t e shows t h a t J+c

Similarly,

j,l

if

{y : RZ+l < _ p ( x , y )

<_ RZ+l

~

+ CRa ( j - / ) ]

} .

we w r i t e J_ = {y : ( y , z )

e E f o r some z , and p ( x , y )

< RZ} ,

then the same e s t i m a t e shows t h a t J_c{y

We now choose

: RZE1 - C R a ( j - l ~ < p ( x , y ) < RZ} .

N so t h a t CR-aN

and we require t h a t

lj-ll

<_ i / 2 ,

>_ N .

Then we can estimate

12

as f o l l o w s : Write

j

92 12 =

f f

IKj(x,z)l[K(y,x)l

/(y,z)eEL

dy dz .

LyeJ+vJ_] By condition

(1)

on the kernel

12 _

IIKII~

K ,

we thus have

RJ<_P(x,z)
J

By the homogeneity condition f o r the measure u and J_ ,

c : CRa ( j - / )

< I/2

0 <_ c < 1/2 , To e s t i m a t e

kernel

K .

If

.

¢I+ci ~T:T-~J

'

But

¢1+cl L.-i--:TEj < 3c

log if

J+

this gives the estimate

12 <_ c IKII 1 log where

and the above estimates f o r

,

so t h i s gives the d e s i r e d estimate f o r 11 , (y,z)

12 .

we must use the smoothness c o n d i t i o n e E ,

(III)

on the

then

p(X,Z) < Rj-£+1 p(y,z) . We now impose the additional requirement on N that RN'I

>_

M

,

where M is the constant in condition

(Ill)

.

Then i f

l-j

we have a

IK(y,z)-

K(y,x)I

<_ !IK!III I ( ~ )

p(Y,Z) -Q

< [IKIIII I Ra ( j - / ) + a

p ( y , z )-Q .

_

Thus II = U

IKj(x'z)IIK(Y'Z)

- K(y'x)I

dy dz

> N and

(y,z) e E ,

93

<- IIKIIIII Ra(J-/) E SSlKj(x'z)I

C NKIIII I IIKll I

Ra ( j - Z )

This completes the estimates in the case For the case

j ~ I

,

P(Y'z)-Q dy dz

I > j .

we simply interchange the roles of

x

and

y

in the

above argument, i . e . we w r i t e Gjl(x,y ) = S~j(x,z)

- K(x,y~

K l ( y , z ) C dz

+ [K(x,y) - Kj(x,y~

S Kl(Y,z)C dz

+ K j ( x , y ) } K / ( y , z ) c dz

,

and make the analogous estimates. This completes the proof of the lemma.

Remarks 1.

Suppose the kernel

K a c t u a l l y has mean value zero , i n . t h e

sense t h a t S K(x,y) dy : 0 A~p(x,y)~AR f o r some R > 1

12 = 13 = 0

and a l l

A > 0 .

,

Suppose the same also holds f o r

K

Then

in the above argument, and gjl(x,y)

= f~Kj(x,z)

- K(x,y~

Kz(y,z) c dz

The geometric basis f o r the e n t i r e lemma is then t h a t on the set

E where the

i n t e g r a t i o n takes place, the r a t i o of the distances admits the bound

~ The smoothness c o n d i t i o n terms of t h i s r a t i o . estimate f o r 2.

~ Rj-£+1 (III)

for

K ,

on the other hand, is expressed in

This explains the appearance of the q u a n t i t y

j-I

in the

[ITj T~I .

I f we r e s t r i c t e d our a t t e n t i o n to kernels w i t h mean value zero, in the

94

sense of the preceding remark, then i t would be s u f f i c i e n t measure

~

to know t h a t the

satisfied ~({y : p ( x , y ) ~ R}) ~ CRQ

f o r some constant

C and a l l

x,R .

Indeed, in t h i s case we would have the

estimate S p ( x , y ) "Q d~(y) ~ C(B/A) Q A~p(x,y)~B and the estimate of the i n t e g r a l estimate on the i n t e g r a l of

11

t h i s q u a n t i t y as

A -7 B

in the proof above only needed t h i s l a s t

o ( x , y ) -Q .

c o n d i t i o n , however, we must replace

,

In order to perturb the mean-value-zero

(B/A) Q by

Log (B/A) .

The vanishing o f

then compensates f o r the non-zero mean-value.

3. For the case o f kernels with mean-value zero, axiom

4)

on the distance

f u n c t i o n is not needed in the proof o f the lemma. This only entered i n t o the estimation of

12 .

95 4.3

~p

Boundedeness

neous type, and l e t in

§ 3.1 ,

KM,R (X,p,~)

w i t h norm

IK~M,R .

Theorem 3.2 we know t h a t by

K)

(1 <..~ }o...~.)

be a space o f homoge-

If

K e KM,R is a s i n g u l a r k e r n e l , then by

K defines a

"principal-value"

Ac(~ )

into

operator (also denoted

Lp(X,u) ,

for

o < ~ ~ a

and

We can now strenthen t h i s r e s u l t , as f o l l o w s :

Theorem

The operator

K maps Lp

More p r e c i s e l y , there e x i s t s a constant

(~'=)

HKfHLp <_

for

(X,p,~)

be the Banach space of s i n g u l a r kernels defined

mapping the L i p s c h i t z space

I < p < = .

Let

1
continuously into

Cp ,

independent of

Lp , K ,

for

1 < p <

such t h a t

Cp JJKJJM, R nfJJLp ,

. By Lemma 4.2 and C o r o l l a r y 4.1, we know t h a t

(~)

holds when

p = 2 . Define the truncated kernel K~(x'Y) = l J ( x ' Y ) ' ,

otherwise i f ~ <_ p ( x, , y ) . < 1/~

Then we claim t h a t there e x i s t constants

,

M and C ,

K and

such that

(*~)

f p(x,z)~Mp(y,z)

JK ( x , y ) - K ( x , z ) I

dx ~ C IIKIJ .

Indeed, i f we c a l c u l a t e the same i n t e g r a l K

independent of

,

and use the smoothness c o n d i t i o n ( I I I )

on

f o r the kernel K ,

K instead of

we obtain the m a j o r i z a t i o n

IIKjl p ( y , z ) a p ( x , z ) ~ # p ( y , z ) P(x'Y)-Q-a dx .

This in turn is majorized by C IIKJl by part ( i i i )

of Lemma 2.2 .

to estimate the e r r o r involved in replacing

K

because of the i n e q u a l i t y

+ P(Z.Y)]

vanishes when

p(x,y)

and

p(x,y) ~ K~(x,z) p(x,z)

by

K in ,

(*m) .

Thus we only need Note t h a t

the integrand i n

are outside an i n t e r v a l

(C-I~, C~- I )

(~) ,

where

96

C > 1 (I)

i s a c o n s t a n t depending o n l y on

on

K ,

K and M .

we can thus bound t h e e r r o r

2 IIKII

I AuB E

From t h e h o m o g e n e i t y c o n d i t i o n

by

p ( x , y ) -Q dx , E

where A

= { x : C- I

g

B

s _< p ( x , y )

= { x : £-1 ~ p ( x , y )

< ~} S C~ - 1 }

By t h e d e f i n i t i o n

o f a homogeneous measure, t h i s

independently of

£ .

With

(*)

This p r o v e s

for

p = 2

.

integral

i s bounded by

(*m)

and

(*m)

established,

the proof for

f o l l o w s from a " c o v e r i n g lemma", which p r o v e s t h a t t h e o p e r a t o r s type

(1,1)

,

uniformly

in

£ ,

and from t h e M a r c i n k i e w i c z

which p r o v e s t h a t t h e o p e r a t o r s < C IIKIJ . range

KE a r e bounded on

Since t h e same i s t r u e f o r

2 < p < ~

f o l l o w s by d u a l i t y

§ 5

5.1

K~

Graded n i l p o t e n t

(resp.

K°Q)

homogeneous o f degree real

s # o

Lemma 3.1

(resp. and

£

a r e o f weak

interpolation

theorem,

Lp ,

1 < p < 2 ,

K~ ,

t h e boundedness i n t h e

s

w i t h norm

( c f . Comments and r e f e r e n c e s ) .

groups 6t

Let

be a g r a d e d , s i m p l y - c o n n e c t e d

nil-

and smooth, s y m m e t r i c homogeneous norm

Ixj

be t h e space o f ~ = -Q + i s

,

CI with

Theorem 4 . 3

,

a bounded o p e r a t o r

K

G

functions

-Q

G ~ {e}

G ,

From

the kernel

= k(x-ly)

Lp(G)

,

1 < p < ~ ,

when

k

.

which a r e

with mean-value zero).

we c o n c l u d e t h a t

on

in

Q = homogeneous dimension o f

homogeneous o f degree

K(x,y)

defines

the o p e r a t o r s

K

1 < p < 2

Examples

potent Lie group, with dilations Let

C1 IIKII,

is a function

and

97 in

K~ or

K°Q .

Evidently

Under d i l a t i o n s ,

K commutes with l e f t

K transforms by K(~o~t)

t -is

=

In p a r t i c u l a r , when k e K°Q ,

(K~)o~

then

t

•

K commutes w i t h d i l a t i o n s .

For the a p p l i c a t i o n s to group representations in Chapter IV, i t w i l l important to know t h a t f o r homogeneous kernels o f degree mean-value o f the kernel is not o n l y a s u f f i c i e n t , f o r boundedness on kernel

k

G .

t r a n s l a t i o n by elements of

L2(G )

-Q ,

be

the vanishing

but also a necessary c o n d i t i o n ,

of a s i n g u l a r convolution o p e r a t o r . Since any such

can be w r i t t e n as

k(x) = ko(X) + c l x l -Q , where

ko

has mean-value zero, i t s u f f i c e s to consider the p a r t i c u l a r kernel

Ixl -Q

Theorem G ~ {e} .

Let

~

be any d i s t r i b u t i o n on

Then convolution by

bounded o p e r a t o r on

Proof

~

(defined on

G such t h a t C~(G))

~ = Ixl -Q

on

does not extend to a

L2(G ) .

(Sketch)

be " r e g u l a r i z e d " at

e

Since

Ix[ -Q

is not i n t e g r a b l e at

to give a d i s t r i b u t i o n .

x = e ,

i t must

One such r e g u l a r i z a t i o n is

e v i d e n t l y the d i s t r i b u t i o n

(~)

< T'~ > = ~ I (~1 e { ~)( x}) " [ x

If - T

u

is any o t h e r d i s t r i b u t i o n

is a f i n i t e

6t ,

convolution o p e r a t o r on {e}

was bounded on

I

such t h a t

~(x) dx

o

~ = ixl -Q

away from

e ,

l i n e a r combination o f d e r i v a t i v e s o f the d e l t a f u n c t i o n at

Using the d i l a t i o n s

f u n c t i o n at

~+ id^x,

it L2(G )

is simple to show t h a t i f then

p - T

p

Thus i t

e .

defined a bounded

would be a m u l t i p l e of the d e l t a

(no d e r i v a t i v e s could occur). This in turn would imply t h a t L2 .

then

is enough to show t h a t convolution by

T

is not

T

98 bounded on

L2 . T ,

To prove the unboundedness o f o f the f u n c t i o n control

I x l -Q

either

o v e r the s i n g u l a r i t y

Take a f u n c t i o n

f e Ca(G) f(x)

By the i n t e g r a t i o n

e ,

=

the n o n - i n t e g r a b i l i t y

Since we have a l r e a d y gained some

i s e a s i e r t o use the s i n g u l a r i t y

f > o

and

Ixl) -I

[xl ~ 2 .

when

we have

F u r t h e r m o r e , by the mean value theorem, i f

l lxl-

at

dx ! C1 + C2 ~ dt 2 t(log t) 2

But we know from C o r o l l a r i e s

1/a

it

§ 1.2 ,

If( u) - f(v)i

where

or at

such t h a t

formula of

I [f(x)l 2

f e L2(G ) .

at

e

= Ix[ -Q/2 ( l o g

G

Thus

at

we may e x p l o i t

! C(Iul-

Ivl)Ivl

lUl ~ IVl

,

then

-(Q+2)/2

1 . 3 . 1 and 111.1.1 t h a t

IxwLE ~ c [xy-xl < C (Ixy-x-yL

+ IYl)

< C (Ixll-alyl

a + { x l a l y l 1-a + l Y l )

i s the l e n g t h o f the f i l t r a t i o n

on

G .

It follows

from these e s t i m a t e s

that If(xy)

,

- f ( x ) l 2 ! C Ixl -Q-2a j y l 2a

whenever

Ixl ~ A IYl

following

e s t i m a t e f o r the

IYI < i .

and

llRyf - f[Ik2

if

IYI <- i ,

this

modulus o f c o n t i n u i t y

L2

(~)

Integrating

!

where

of right

C lYl a ,

Ryf(X) = f ( x y )

inequality,

.

we g e t the

translations

on

f

:

99 To show t h a t r i g h t convolution by in

(m) .

T

is unbounded, w r i t e

T=o+p

,

as

Then

f ~'~ ~ : ly{
dy

lyl Q so i t is evident from

(mm) that

f * o e L2(G) .

On the other hand,

•

f ~ p(x) = IY ~1 f(xy)

dy

LylQ But by the estimates above, i f

IYl ~ ~Ixl Hence i f

Ixl ~ I/~ ,

~ > o

is small, then

---> IxYI ~ ~ Ixl

then since

f > o ,

f m p(x) 7 l ~ l y l ~ i x l

we have

f(xy)

dy

!yl Q

> C ixl "Q/2 (log I x l ) - I I~ Ixl i > C ixl -Q/2 by the i n t e g r a t i o n formula of

dy lyl e

,

§ 1.2 . This makes i t

clear that

f m p ~ L2(G) ,

and proves the theorem.

Remark

In the proof j u s t o u t l i n e d , we used

f e C~(G)

which does not

have compact support. To make the proof complete, one must show t h a t there is a sequence o f t r u n c a t i o n s { f n ma}

is bounded in

fn of f , L2

but

{fn m p}

done using the same estimates ( c f .

5.2 and l e t

F

smooth and compactly supported, such t h a t is unbounded in

L2 .

This can be

comments and references).

F i l t e r e d n i l p o t e n t 9roups be any p o s i t i v e f i l t r a t i o n

Let on

g_ ,

g

be a r e a l , n i l p o t e n t Lie a l g e b r a , as in

§ 1.2.1 .

Let

{6t }

and

10o Ixl

be d i l a t i o n s

the f i l t r a t i o n

and a homogeneous norm, r e s p e c t i v e l y , which are compatible w i t h

F ,

as in

§ 1.3.1 .

(Thus

dt

is not an exact Lie algebra

automorphism, i n g e n e r a l . )

Denote by

G the v e c t o r space

g

w i t h Lie group

s t r u c t u r e given by the Campbell-Hausdorff f o r m u l a . In

§ 1.3.3

we found t h a t near i n f i n i t y ,

the group s t r u c t u r e o f

G was

asymptotic t o the group s t r u c t u r e associated w i t h the graded L i e algebra Thus we might expect t h a t the r e s u l t s o f § 5.1

should also be v a l i d on

provided we deal w i t h kernels which are o n l y s i n g u l a r infinity. of

g

This i s the content o f the next theorem.

relative

Let

k

be a l o c a l l y

and homogeneous o f degree assume a l s o t h a t

k

G ,

not i n t e g r a b l e ) a t

(Q = homogeneous dimension

Proof

-Q + i s

i n t e g r a b l e f u n c t i o n on

on a neighborhood

f~

f mk

i s bounded on

of infinity.

Lp(G) ,

The o p e r a t o r in question has kernel

boundedness on

(If

p(x,y) = Ix-lyl

.

Since

k

j,l

K(x,y) = k(y-lx)

.

To prove

§ 4.2, relative

i s assumed to be l o c a l l y

l a r g e and p o s i t i v e ( n o t a t i o n as in

Going back to Example 4 and Remark 2 o f Remarks I , 3 of § 4.2 ,

§ 4.2)

§ 2.1 ,

s = o , Then the

i s only necessary in t h i s case to e s t i m a t e the norm o f the o p e r a t o r s for

C1

1 < p < ~ .

L2 , we use the method o f decomposition in

distance function

G which is

has mean-value zero on a neighborhood o f i n f i n i t y . )

convolution operator

T~T/

.

to the given f i l t r a t i o n . )

Theorem

it

(i.e.

gr(g)

to the

integrable, TjT~_

and

. Lemma 3.1 ,

we f i n d t h a t the p r o o f o f Lemma 4.2 is s t i l l

and v a l i d in

t h i s c o n t e x t . The e s s e n t i a l p o i n t i s t h a t the estimates o f Lemma 4.2 o n l y i n v o l v e K(x,z)

,

K(x,y)

and

K(y,z)

for

Rj ~ p ( x , z ) < _ Rj + l When

j and I

i s also l a r g e

are s u f f i c i e n t l y

x,y,z ,

such t h a t

R1 ~ p ( y , z ) < _ RZ+I .

l a r g e , then these i n e q u a l i t i e s

(§ 2 . 1 , Remark 2 ) , and hence

as in the graded n i l p o t e n t case. This proves

p

and L2

K satisfy

imply t h a t

p(x,y)

the same estimates

boundedness. Boundedness on

Lp ,

101

1 < p < ~ , f o l l o w s as in Theorem 4.3 .

5.3

Group germs

potent Lie algebra

V

Let

x e H ,

v

from

isomorphism from

§ II.3.1

that

be a p a r t i a l

homomorphism from a graded n i l -

i n t o the L i e a l g e b r a o f

We assume t h a t f o r a l l

is a linear

~

H

C= v e c t o r f i e l d s

on a m a n i f o l d

M

the map

~

~(v) x

V

o n t o the t a n g e n t space t o

can be covered by open sets

M

X ,

at

x .

We know

f o r which the map

e :XxX÷V defined implicitly

by the i d e n t i t y ex ( e ( x ' y ) )

x = y ,

1)

e

C=

2)

For each

e x i s t s and s a t i s f i e s

is

; x ,

the p a r t i a l

maps

y ~-~ e ( x , y )

are diffeomorphisms o f Let

p(x,y) = lo(x,y)l

,

nates has smooth, s t r i c t l y (X,p,p)

positive

~

u

e

is near

satisfies

density relative

the i d e n t i t i e s

e(x,y) : -e(y,x)

4)

0(x,eX(U)x)

so t h a t

:

X

V .

which in l o c a l c o o r d i -

t o Lebesgue measure. Then

§ 2.1 Example 3, and § 2.2 Example 2 ) .

3)

0 in V ,

o n t o an open subset o f

be any measure on

i s a space o f homogeneous t y p e ( c f .

Note t h a t the map

provided

and l e t

X

;

u ,

e1(U)x e X .

.

102 In o r d e r t o a n a l y s e the i n t e r p l a y fields

on

X ,

m i s a f u n c t i o n on

defined for e C~(X) , a function

on

V

and v e c t o r

we i n t r o d u c e the n o t a t i o n ~(x;u) = ~(e1(U)x)

if

between v e c t o r f i e l d s

x e X then

H .

and

u

~(x;u)

f on V ,

,

We t a k e

X

sufficiently

small so t h a t

l y i n g i n a f i x e d neighborhood is

C~

on

and d e f i n e

X × ~ .

~ o f 0 in V .

In p a r t i c u l a r ,

m(x) = f ( e ( y , x ) )

,

for

m(x;u) If

i f we s t a r t

y

fixed,

is

with

then by 4) we

have

~(y;u) = f(u)

Recall from the l i f t i n g then t h e r e i s a v e c t o r f i e l d

.

theorem o f Chapter I I t h a t i f T

x~w

on ~

( ~ ( w ) ~ ) ( x ; u ) = dR(w) m(x;u) + TX, w m(x;u)

(If)

Tx, w

Here

dR

§ 11.2.2) Tx, w

denotes the r i g h t ,

and

< m-1

vector field dR(w) .

holds, i.e.

x(w)

(IZl)

if

V

The p o i n t o f the l i f t i n g

theorem i n t h i s

i n the e x p o n e n t i a l c o o r d i n a t e s centered a t

~

i s o f t h e form

~(x) = f(e(y,x))

(cf.

the v e c t o r f i e l d

is " w e l l - a p p r o x i m a t e d " by the l e f t - i n v a r i a n t

In p a r t i c u l a r ,

from (4) and ( I )

u=o .

Note t h a t in t h e p r e s e n t s i t u a t i o n

is u n i q u e l y d e f i n e d by p r o p e r t y ( I ) .

case is t h a t ( I I )

at

r e g u l a r r e p r e s e n t a t i o n o f the L i e a l g e b r a

m e Ca(M) .

x e X ,

such t h a t

(1)

is o f o r d e r

w e Vm and

x ,

the

vector field ,

then we o b t a i n

the r e l a t i o n

(~(W)~)(X) : ( d R ( w ) f ) ( o ( y , x ) ) With these p r e l i m i n a r i e s

settled,

+ Ty,wf(O(y,x))

.

we now c o n s i d e r o p e r a t o r s on

X

o f the

form

K~(x) = I m(Y) K ( x , y , X where

K(x,y,u)

is a f u n c t i o n on

8(y,x))

X x X × V ,

du(y)

,

smooth i n

x

and

y ,

and having

103 a prescribed singularity

in

u

at

superposition of multiplication

u = o .

Such an o p e r a t o r can be viewed as a

o p e r a t o r s and " a p p r o x i m a t e " r i g h t - c o n v o l u t i o n e(y,x) = y-lx)

o p e r a t o r s (Recall t h a t in the group case,

.

The classes o f kernels

we s h a l l consider are the f o l l o w i n g : Definition

1.

A function

f o r every p o s i t i v e i n t e g e r

K(x,y) =

K on

X×X

i s a kernel o f type

s 7 o

if

m we can w r i t e N E ai(x) i=I

k i ( e ( Y , X ) ) b i ( Y ) + Em(x,y) ,

where (a)

The f u n c t i o n s a i ,

b i e C~(X) ;

(b)

The f u n c t i o n s k i ( u )

homogeneous o f degree s = 0 (c) (Here m ,

and

~

= -Q)

C~ on V ~ { o } , and

mi ~ s - Q ( w i t h mean-value zero i f ;

The remainder term

Q = homogeneous dimension o f

are

V .

Em e C~(XxX) . The number

N

is allowed to vary w i t h

of course.)

Theorem 1

If

K

is a kernel o f type

A m(x) = f K(x,y) m(y) d~(y) X initially

d e f i n e d on

l
Cc(X ) ,

(The i n t e g r a l

Proof

then the o p e r a t o r

,

extends to a continuous o p e r a t o r on

Lp(X) ,

i s taken in the p r i n c i p a l - v a l u e sense when

s > o ,

~ C p ( x , y ) -Q+s

this inequality

and Lemma 2 . 2 , gives the estimates

,

for

s = o.)

We e v i d e n t l y have IK(x,y)I

In the case

s > o ,

t o g e t h e r w i t h the compact support o f

104 I sup x

f IK(x,y)l X

d~(y) <

sup y

f IK(x,y)l X

du(x) <

i

Using H~Ider's i n e q u a l i t y , 1 5 P < ~

one then shows e a s i l y t h a t

( c f . p r o o f o f Lemma 4 . 2 , f o r the case

A

is bounded on

p = 2).

The case

Lp ,

s = o

follows

from Lemma 3.1 and Theorem 4.3 .

D e f i n i t i o n 2. is o f type

s ,

A linear operator

s > o ,

A

mapping

C~(X)

i n t o f u n c t i o n s on

X

if

A m(x) : f K ( x , y ) m(y) d~(y) , X where

K

is a kernel of type

s .

If

s = o ,

we say

A

is o f type

0

if

A m(x) = PV f K ( x , y ) m(y) du(y) + a ( x ) m(x) , X where

K

i s a kernel o f type

0

and

By Theorem 1, an o p e r a t o r t o r from

Lp(X)

differential

to

o p e r a t o r on

distribution

D(Af)

Remarks 1. into

Lp(X) ,

C~(X) .

for

X with

on

× .

If

A

Indeed, suppose

J ( x , v ) dv

o f type

I < p < ~

J

C~ f u n c t i o n o f

x .

o f a kernel o f type

If

C~ c o e f f i c i e n t s ,

m e C~(X) .

extends to a continuous operaf e Lp(X)

and

D

then

will

denote the

DAf

s > o ,

then

A

is a

maps C~(X)

Then we can w r i t e

m(y) d~(y) = I k(v) m ( x ; - v ) J ( x , v ) dv ,

i s the image o f the measure

By assumption,

s > o

is an o p e r a t o r o f type

i k(0(y,x))

where

A

a e C#(X) .

d~(y)

under the map y ~-~ 0 ( y , x )

.

is a smooth f u n c t i o n , so the r i g h t side o f t h i s formula is a The a s s e r t i o n f o l l o w s from t h i s c a l c u l a t i o n and the d e f i n i t i o n s ,

105

2. The space o f o p e r a t o r s o f type multiplication 3. I f

by

C'(X)

,

f e C~(Xx~)

K(x,y) f ( x , 0 ( y , x ) )

s e r i e s in

i s i n v a r i a n t under l e f t

and r i g h t

and also under t r a n s p o s i t i o n . and vanishes t o o r d e r

by the homogeneous norm on then

s

V ,

cf. § I.I.3)

is a kernel o f type

,

m > o and i f

s+m

at K

u = o

(as measured

is a kernel o f type

(Expand

f(x,u)

in a T a y l o r

u .)

To t r e a t the i n t e r a c t i o n between o p e r a t o r s defined by kernels o f type and d i f f e r e n t i a l

3.

which is spanned over

DO(X)m be the module o f d i f f e r e n t i a l

Let C'(X)

by

x(w I ) wi e V

s

operators on X we i n t r o d u c e the f o l l o w i n g f i l t r a t i o n :

Definition

where

s ,

o p e r a t o r s on

and the o p e r a t o r s o f the form

1

~(w k) ,

...

is homogeneous o f degree

ni ,

and

n I + . . . + nk ~ m .

We s h a l l X-degree ~ m .

r e f e r to elements o f

operators of

These modules are i n v a r i a n t under t r a n s p o s i t i o n ( r e l a t i v e

p a i r i n g given by i n t e g r a t i o n over

Theorem 2 m< s ,

DO(A)m as d i f f e r e n t i a l

If

A

to the

M .)

is an o p e r a t o r o f type

then t h e r e e x i s t o p e r a t o r s

AI and A2

s > o ,

o f type

and

s - m ,

D e DO(X)m

5

with

such t h a t

[AA ~ = A 1 Dm for all

~ e Cc(X ) .

Proof A

A2~

I t is o b v i o u s l y s u f f i c i e n t

an o p e r a t o r w i t h kernel

homogeneous o f degree

m ,

to c o n s i d e r the case

K(x,y) = a(x) k(e(y,x)) and

k

is a

b(y)

Ca f u n c t i o n on

,

D = X(w)

where V ~ {o}

w e V ,

and is

homogeneous

106

o f degree

s - Q .

Denote by the function

K(1 )

the function

x r-+ k ( e ( y , x ) )

(m)

.

K(I ) (x,y)

o b t a i n e d by a p p l y i n g t h e

By f o r m u l a

(III)

= dR(w) k ( e ( y , x ) )

Since t h e v e c t o r f i e l d

D

to

we can w r i t e

+ Ty,w k ( e ( y , x ) )

is of order

Ty,w on

vector field

.

< m - I

,

it

can be

e x p r e s s e d as a sum o f terms o f t h e form

a(y,u) where

a

is a

v e V

i s homogeneous o f degree

Ty,w k(u)

C~ f u n c t i o n

a

,

which vanishes to o r d e r n .

(cf.

> n - m + I -

Chapter I ,

§ 1.3).

at

u = o ,

and

Hence the f u n c t i o n

i s a sum o f terms

a(y,u) with

Dv

as above and

expansion in s - m + i

u ,

kI

kl(U ) ,

homogeneous o f degree

we f i n d

t h a t the f u n c t i o n

s - n - Q .

Ty,w k ( o ( y , x ) )

Taking the T a y l o r i s a kernel o f t y p e

.

To a n a l y s e the l e a d i n g term

in

(m) ,

set

F = dR(w) k

where t h e d e r i v a t i v e s the distribution s - m - Q .

If

a r e taken i n t h e d i s t r i b u t i o n

k(u) du . s > m ,

f(u)

It

follows

Then as a d i s t r i b u t i o n ,

F

is

k

with

homogeneous o f degree

the pointwise derivative

= ~ t t= 0

i s a homogeneous f u n c t i o n

sense by i d e n t i f y i n g

k(u(tw))

o f degree

u ~ 0

s - m - Q ,

that the distribution

F = f(u)

,

du

and hence i s l o c a l l y

integrable.

107 in t h i s

case.

In the case We c l a i m t h a t

(~)

f

s = m ,

the f u n c t i o n

f(u)

is homogeneous o f degree

-Q .

has mean-value z e r o , and t h a t F = PV(f) + Ca ,

where

C

we f i r s t

is a c o n s t a n t , and

a

i s the d e l t a f u n c t i o n a t

choose the c o n s t a n t

0 .

so t h a t the mean-value o f

To prove t h i s ,

g(u) : f(u)

-

~lul -Q

is z e r o . The d i s t r i b u t i o n

G

=

F

-

PV(g)

is then homogeneous o f degree

-Q ,

mlul -Q But t h i s

implies that

and d e r i v a t i v e s

G

linear

o f the d e l t a f u n c t i o n ,

luI 1

however,

T

f

C # 0 .

Hence

G

where

lul

T

Iol]

I

m

lul Q

~(o)

i s n o t homogeneous o f degree

a(x) f(e(y,x))

ax

lul du

lU >1

t r a n s f o r m s by

The f o r e g o i n g a n a l y s i s thus proves t h a t

where

BT

is the d i s t r i b u t i o n

has mean-value z e r o , and by homogeneity we o b t a i n

modulo o p e r a t o r s o f t y p e

c o i n c i d e s w i t h the f u n c t i o n

c o m b i n a t i o n o f the d i s t r i b u t i o n

< T,~oa t > = < T,~ > + C l o g ( t ) with

0

du .

is a finite

< Under d i l a t i o n s ,

and away from

b(y)

s - m+ 1

i s the d e l t a f u n c t i o n a t

DA

-Q

unless

B = 0 .

Thus

(**) is an o p e r a t o r w i t h k e r n e l

, (plus a term

C a ( x ) b(y) a x ( y )

when

x) .

Taking t r a n s p o s e s , we o b t a i n the same c o n c l u s i o n f o r

AD ,

Q.E.D.

s = m ,

108 5.4

Boundedness on Sobolev spaces

In t h i s section we want to e s t a b l i s h

the smoothing p r o p e r t i e s o f the class of i n t e g r a l operators o f type

s

introduced

in the previous s e c t i o n . For t h i s purpose we introduce the f o l l o w i n g class o f "Sobolev spaces." t h a t the set

We continue the assumptions and n o t a t i o n o f

X has compact closure in

neighborhood o f

w

I I I = I~11 + . . . +

If

l~nI

{w }

for

Definition

I

with

S~(X) III

V ,

I = { a l . . . . . an}

is smooth on a

If

~ m .

and denote by

I < p < =

(D1f

~ .) .

the degree of

i s an o r d e r e d c o l l e c t i o n

and

of indices,

set

m is a non-negative i n t e g e r , then the f e LP(x)

is the d i s t r i b u t i o n

Dlf e LP(x)

for all

d e r i v a t i v e , using the dual~ty defined

=

l i~<m

IIDIfI[Lp(X)

As an immediate consequence o f Theorems 1 and 2 o f If

such t h a t

Set

Ilfllp,m Theorem i

la[

~(w n)

consists of a l l functions

by the measure

u

and d e f i n e

DI = ~(w 1) . . .

space

and the measure

and assume

X .

Pick a graded basis homogeneity of

M ,

§ 5.3 ,

A

is an o p e r a t o r of type

§ 5.3 , m, m a

we have non-negative i n t e g e r ,

then A : L P ( x ) + SmP(X) continuously, for

l
We would l i k e to extend t h i s r e s u l t , and show t h a t S~

continuously i n t o

S~+m f o r a l l

k .

For t h i s we w i l l

A

o f type

m maps

need the f o l l o w i n g

commutation formula: Theorem 2

Assume t h a t the graded Lie algebra

elements o f degree one. Let

A

V

is generated by i t s

be an i n t e g r a l o p e r a t o r o f type

s > o

on

X and

109

let

D e DO(~)m .

operators

Then t h e r e e x i s t

Di e DO(X)m

operators

Ai

of type

s

and d i f f e r e n t i a l

such t h a t n

DA =

Theorem 2 w i l l

follow

does n o t r e q u i r e t h a t this

result,

% A i Di i=l

V

we choose a graded b a s i s

Theorem 2'

type

Let

V

A

is

As

of type

{w }

s + l~i

more general

result,

which

elements o f degree one. To s t a t e for

V

as above. Assume t h a t the

r .

be an i n t e g r a l

be homogeneous o f degree s ,

from the f o l l o w i n g

be g e n e r a t e d by i t s

l e n g t h o f t h e g r a d a t i o n on

w e V

easily

m .

operator of type

Then t h e r e e x i s t

- m ,

and

R

integral

of type

s

s > o

on

X

operators

s + r - m + i

and l e t Ao

,

of

such

that

X(w) A = A X(w) + Ao +

E

A

l~i>m

Remark ~(w ) e DO(X)r involving

R

Suppose for all

m = 1 . ~ ,

it

with

Theorem 2 f o r t h e case

As ,

and

x(w)

V ,

is o f t y p e

in t h i s

0

then

case.

x(ws)

the s p e c i a l

case i n which

is the left-invariant

i s the sum o f p r o d u c t s o f

So suppose

k

we can compose

case f o l l o w s

i s homogeneous o f degree

s .

Isl

I~I

- I

of

This gives

by i n d u c t i o n .

idea of the proof,

is exactly

vector field

t h a t the terms

I f we a l s o assume t h a t the

the essential A

Since

§ 5.3

an o p e r a t o r o f t y p e

The g e n e r a l

To c l a r i f y

s + r .

from Theorem 2 o f

and o b t a i n

m = 1 .

P r o o f o f Theorem 2'

V ,

A

x(w ) .

s

o f degree one. By Theorem 2 o f § 5.3 ,

these v e c t o r f i e l d s

consider first

R

follows

can be absorbed in

elements o f degree one g e n e r a t e vector fields

Then

x(ws) + z R s

let

us

a c o n v o l u t i o n o p e r a t o r on

dR(w) .

s - Q ,

and

m e C;(V)

.

Since

110 we are d e f i n i n g the group s t r u c t u r e on the formula f o r the a d j o i n t (~)

V

using the Campbell-Hausdorf f o r m u l a ,

representation of

V

becomes

vwv -1 = ead(v)w

Thus we can w r i t e (m m k)(uw) : ~ m(uwv "1) k(v) dv V = f ~(uv -1 ead(v)w) k(v) dv . V Replacing

w

by

(~)

tw

in t h i s formula and d i f f e r e n t i a t i n g

dR(w)(m mk)(u) = f ~ ( u v - l , v ) V

at

t = o ,

we f i n d t h a t

k(v) dv ,

where ~ ( u , v ) : dR(ead(v)w) ~(u) .

We can expand ead(v)w = w + z pc(v) w

where

p~

are polynomial f u n c t i o n s on

the automorphism

6t

to the i d e n t i t y

p~(~t v) = t l~l-m p~(v) Since

p~

i~I ~ m .

is a polynomial and

,

V

(We f i x

(~),

w

in t h i s argument). Applying

we f i n d t h a t

.

p~(o) = 0 ,

this implies that

p~ = o

for

Thus the f u n c t i o n s ks(v ) = p~(v) k(v)

are homogeneous o f degree

(~)

as

s + [~[ - m - Q .

Using them, we can w r i t e formula

111 dR(w)(m :'= k) : (dR(w)m) :'= k +

~

I~i>m

( d R ( w ) m ) :~, k s

This is the d e s i r e d commutation formula i n t h i s s p e c i a l case. To t r e a t the general case, we need formulas s i m i l a r to Consider f i r s t

(~) v

the analogue o f the a d j o i n t r e p r e s e n t a t i o n . For

near

0 in V ,

X given by

we have a l o c a l one-parameter group o f l o c a l diffeomorphisms o f t

(~)

and

~-+ e ~(v) e t~(w) e - ~ ( v )

Denote the g e n e r a t o r o f t h i s group by

E(v,w) :

E(v,w) ~(x) = T t t=o ~ ( e - ~ ( v ) e t a ( w ) e~(v) x) ,

for field

m e C~(X) . on

It

X which depends smoothly on

Lemma functions

If

f

on

1)

2)

i s c l e a r from the d e f i n i t i o n

w e V X × ~

v ,

when v

is homogeneous of degree

functions

v ,--+ f

E(v,w)

is a

v a r i e s in

Q .

m ,

C~ v e c t o r

then there e x i s t

C~

such t h a t

E(v,w) = x(ead(v)w) + z f ( - , v )

the

that

(x,v)

Proof o f the Lemma:

vanish

to order

x(w ) ;

r

- m + 1

at

v = o

.

S t a r t i n g w i t h the formal i d e n t i t y

e X Y e-X = eadX Y , (adX(y) = XY-YX) ,

one employs the same s o r t o f argument t h a t was used in the

p r o o f o f Theorem 3.2 o f Chapter I f .

The d e t a i l s are l e f t

Completion o f p r o o f o f theorem: degree § 5.3 ,

s-Q

Let

k e C~(V~{o})

(and w i t h vanishing mean-value, in case

given

m e C#(X) ,

we can w r i t e

to the reader.

s=o) .

be homogeneous o f Then by Remark I in

112

(~)'

/ k(e(y,x)) X

~(y) d~(y) = I m ( x ; - v ) k(v) J ( x , v ) dv.

I f we apply the v e c t o r f i e l d will

z(w)

to the r i g h t s i d e , the d i f f e n t i a t i o n s

o n l y c o n t r i b u t e an o p e r a t o r o f type

z(w) ~ ( x ; - v ) =

s .

By d e f i n i t i o n ,

on

one has

~ t t=o ~ ( e - Z ( v ) etZ(W)x)

= (E(v,w)m)(e-~(V)x) .

Using the Lemma above and the formula f o r

~(w) ~ ( x ; - v ) =

exp(ad v) w ,

we f i n d t h a t

(~(w)~)(x;-v) +

z

p~ ( v ) ( x ( w ) ~ ) ( x ; - v )

]~l>m +~f

C~

C~

S u b s t i t u t i n g t h i s in Am o f type

(~)'

s + l~I - m ,

,

(x,v)(~(w)~)(x;-v)

we see t h a t the terms i n v o l v i n g

by

pm give o p e r a t o r s

as in the case o f an exact c o n v o l u t i o n o p e r a t o r . The

a d d i t i o n a l terms i n v o l v i n g the f u n c t i o n s s + r - m+ i ,

.

Remark 3

in

§ 5.3 .

f

contribute operators

R

o f type

This proves Theorem 2' in the general

case.

Corollary

Assume t h a t the Lie algebra

o f degree one. Let

A

V

i s generated by i t s elements

be an i n t e g r a l o p e r a t o r o f type

s > o

on

× ,

with

s

an i n t e g e r . Then

A : s~(x) ~ S~+s(X) continuously, for Proof

1 < p < ~

Let

as a sum o f products

and

D e DO(~)m+s DID 2 ,

with

o f t h i s s e c t i o n and Theorem 2 o f

m = 0,1,2,--. By the g e n e r a t i n g c o n d i t i o n , we can w r i t e D e DO(~)s § 5.3 ,

and

D2 e DO(~)m .

we f i n d t h a t

D

Using Theorem 2

113

DA = z Ai Di ,

with

Ai

operators of type

and t h e i r transposes

map C~(X)

equation is v a l i d not only on then

Af e Lp(X)

0

and

Die DO(~)m .

into itself, C~(X) ,

Ai ( D i f )

This completes the proof.

derivative

•

Ai ,

Di

i t f o l l o w s t h a t t h i s operator

but also on

and the d i s t r i b u t i o n

Since the operators

S~(X) . D(Af)

Thus i f

is the

Lp

f e S~(X) , function

114 Comments and references f o r Chapter I I I

§ 1.1

See Folland ~ ] and Folland-Stein [ ~ for further information about

homogeneous functions and d i s t r i b u t i o n s . The proof of Lemma I . I is adapted from KorAnyi-V~gi [ i ] .

§ 1.2

These integral formulas appear in Knapp-Stein [ I ] . A d i f f e r e n t i a l -

geometric construction of the fibering of Lebesgue measure by the "spheres" { I x l = r}

is given in Cotlar-Sandosky ~ ] .

§ 2.1-2.2

The presentation here is a synthesis of the treatments in

Koranyi-Vagi ~ i ] , Knapp-Stein [ i ] , Folland-Stein [ i ] and Rothschild-Stein [1]. In p a r t i c u l a r , Kor~nyi-V~gi were the f i r s t

to emphasize the role played by the

"Lipschitz-condition" 4) on the distance function. The map e in was introduced by Folland-Stein ~ ] .

§ 2.1, Example 3

The v e r i f i c a t i o n that the associated

distance

function s a t i s f i e s axiom 4 was done by Rothschild-Stein [1]; cf. Goodman [7]. I f (X, p, ~) is a space of homogeneous type, in the sense of § 2.2, then i t also s a t i s f i e s the axioms of Chapter I I I of Coiffman-Weiss [1], by v i r t u e of Lemma 2.2. The additional conditionsthat we have imposed which are not used by Coiffman-Weiss are the Lipschitz condition (4) on the distance function, and the logarithmic estimate (~) r e l a t i n g the measure and the distance function. In return, we are able to prove Lp-boundedness of singular integral operators, while they must assume an a-priori

L2 estimate (or prove

L2

boundedness via harmonic analysis, in

applications).

§ 2.3

This is adapted from Kor~nyi-Wgi

§ 3.1

The same references as in

[1].

§ 2.1-2.2. Our goal in this axiomatic

formulation is to isolate the a p r i o r i information necessary for proving boundedness of singular i n t e g r a l s . For example, Lemma 3.1 can be generalized to include kernels of the form

k(x, e ( x , y ) ) ,

where

k(x,v)

is

C1 on

X x (W{o})

and

115 homogeneous of degree

-Q

in

v ,

with vanishing mean-value. Kernels of this sort

n a t u r a l l y occur in the generalizations of the results of Chapter IV, § 3 concerning h y p o e l l i p t i c operators (cf. Rothschild-Stein [1]).

§ 3.2

This is adapted from Knapp-Stein [ ~ .

§ 4.1

The results of t h i s section go back to Cotlar, in connection with

estimates f o r the classical H i l b e r t transform, cf. Knapp-Stein [11 and CoiffmanWeiss [ ~ ,

Chapter VI.

§ 4.2

These estimates are taken from Knapp-Stein [1], Folland-Stein [ ~

and Rothschild-Stein [ I ] , but adapted to the present axiomatic formulation.

§ 4.3 estimate

(~)

§ 5.1

For the proof of and

L2

Lp

boundedness,

1 < p < 2 as a consequence of

boundedness, cf. Coiffman-Weiss [ i I , Chapter I I I .

These results are due to Knapp-Stein [11 . The proof of the "un-

boundedness" theorem given here is taken from Goodman [5]. Strichartz [~ has studied singular integrals via the (additive) Fourier transform on certain n i l potent groups.

§ 5.2

These results are new. I t would be interesting to extend the

comparsion theorem in with kernels

k(y - I x)

§ 3.3 and

to the graded structure, and

§ 5.3

of Chapter I to a comparison between the operators k(y - I m x) , k

where

~ means m u l t i p l i c a t i o n r e l a t i v e

s a t i s f i e s the conditions of Theorem 5.2.

The results here are taken from Folland-Stein [1] and Rothschild-

Stein [ I I , reformulated in the context of Chapter I I .

§ 5.4

The d e f i n i t i o n of the chain of Sobolev spaces is adapted from

Folland-Stein [ ~ ,

Folland [21, and Rothschild-Stein ~1]. Theorem 2' is stated by

116 Rothschild-Stein. The proof here, based on the a d j o i n t representation,is new, as is the Lemma. The Corollary was proved by Folland [2] in the context of a " s t r a t i f i e d " n i l p o t e n t group (a graded group generated by i t s elements of degree one). For comparisons between these Sobolev spaces and the usual Sobolev spaces, and f o r the corresponding L i p s c h i t z spaces, cf. Folland-Stein ~1], Folland [2], and RothschildStein [ 9 . In t h i s chapter we have r e s t r i c t e d a t t e n t i o n to operators on scalar-valued f u n c t i o n s , to minimize the notational burden. Everything works equally well f o r functions with values in a H i l b e r t space, and operator-valued kernels. This generalization w i l l be used in Chapter IV, § I , without f u r t h e r mention (cf. Knapp-Stein [ 1 ] ) .

Chapter IV

Applications

In t h i s chapter we apply the results of the previous chapters to three areas of analysis. The f i r s t

is the study of i r r e d u c i b i l i t y

and equivalences among

p r i n c i p a l series representations f o r real-rank one semi-simple Lie groups. In the so-called "non-compact p i c t u r e " , these representations act on

L2(V ) ,

V a nil-

potent group. The " i n t e r t w i n i n g i n t e g r a l s " are s i n g u l a r i n t e g r a l operators on

V

of the type studied in Chapter I I I . The second a p p l i c a t i o n is the use of non-commutative harmonic analysis on the Heisenberg group to study the Hardy space The orthogonal projection onto the space of

H2 L2

on a Siegel domain of type I I . boundary values of

H2

functions

is a singular integral operator, and we calculate i t s operator-valued Fourier transform. (In t h i s case the boundedness of t h i s operator on using the Plancherel theorem f o r the Heisenberg group.)

L2

can be proved

The Szeg~ kernel, which

reproduces a holomorphic function from i t s boundary values, is calculated using the Fourier inversion formula on the Heisenberggroup. The goal of the t h i r d section is to establish precise r e g u l a r i t y properties f o r certain h y p o e l l i p t i c d i f f e r e n t i a l operators associated with t r a n s i t i v e Lie algebras of vector f i e l d s . This involves using the f u l l machinery of Chapte~ I I and I I I .

The basic idea, however, is quite simple. Using the l i f t i n g

theorem, one

reduces the problem to the consideration of "approximately i n v a r i a n t " operators on a graded n i l p o t e n t group. The corresponding "exactly i n v a r i a n t " operators, which are required to be s u i t a b l y homogeneous under d i l a t i o n s , have homogeneous fundamental s o l u t i o n s . Approximate fundamental solutions f o r the o r i g i n a l operators are then constructed using the group germ generated by the vector f i e l d s and the homogeneous fundamental s o l u t i o n s . The r e s u l t i n g i n t e g r a l operators are of the type studied in Chapter I I I .

The boundedness of these operators on various function

spaces y i e l d s the desired r e g u l a r i t y properties of the o r i g i n a l d i f f e r e n t i a l operators.

118

§ 1.

Intertwinin9 operators

Let

G

be a semi-simple

c e n t e r . The lwasawa decomposition o f

G

is

1.1 Bruhat decomposition and i n t e g r a l L i e group w i t h f i n i t e

formulas

G = KAN ,

where

N

is n i l p o t e n t ,

A ~IR Z

a maximal compact subgroup o f shall

restrict

i s a v e c t o r group n o r m a l i z i n g G .

The i n t e g e r

our a t t e n t i o n to the case

Z

I = I .

e i t h e r commutative o r t w o - s t e p n i l p o t e n t ,

i s the r e a l In t h i s

and the a c t i o n o f

N ,

and

rank o f

on

We

G .

case the group Ad(A)

is

K

N

N

is

will

f u r n i s h a group o f d i l a t i o n s .

Let

M and

r e s p e c t i v e l y . Then

M'

denote the c e n t r a l i z e r

M normalizes

N ,

and the n o r m a l i z e r of

A in K ,

and

B = MAN

i s a closed subgroup o f

G .

Assuming t h a t r e a l - r a n k

(G) = 1 ,

one knows t h a t

(M'/M) = 2 .

Pick

w e M'

with

w ~ M .

waw

Then -1

=a

w2 e M , -I

,

and

aeA

wMw-I=M

We d e f i n e

V = w N w- I

.

Thus

V

is a n i l p o t e n t group isomorphic to

N ,

and

w B w- I = MAV .

The map from the product m a n i f o l d diffeomorphism onto an open subset o f The Bruhat decomposition a s s e r t s

that

u

to

G given by

( b , v ) ~-~ bv

is a

G whose complement has Haar measure zero.

cosets:

G= ( B w B )

B × V

B

G

i s the d i s j o i n t

union o f

B

double

119

Multiplying

on t h e r i g h t

by

G :

Thus i f

g e G

and

n e N ,

v e V

such t h a t

w ,

we can w r i t e

(BV) ~ (Bw)

g ~ Bw ,

this

decomposition

as

.

then there exist

meM,

unique elements

aeA,

g = man v

We s h a l l g~-+ a(g)

write ,

m = m(g) g~-+ v(g)

,

a = a(g)

,

v = v(g)

.

a r e smooth f r o m t h e open s e t

respectively.

Example

Let

G = SL(2~R).

We may t a k e

K:

: (~ elR L-sin e cos

A=

Then

M = {±I}

,

a"

:a>o

and we can t a k e

We have

B =

Given

a-

: a elR ~ { o }

Then t h e maps BV

onto

g ~ - + m(g)

M , A , V

,

120

we have

g e BV <--> d # o ,

and in t h i s case

a(g) = I dl-1

m(g) = sgn(d) I ,

o

°1]

v(g) =

Id

If01

Note t h a t Bw =

-I

:

a,b elR ,

a # o

In terms of the Bruhat decomposition, we have the f o l l o w i n g i n t e g r a l formulas:

Lemma

Let

dm, da, dn, dv

denote Haar measures on

M, A, N, V r e s p e c t i -

vely ( a l l these groups are unimodular). Then (i)

left

~ f l f(man) MAN

Haar i n t e g r a l on (ii)

where

d/b

dm da dn

is a

B = MAN; fB

denotes l e f t

f ( b man) dlb = ~(a) IB f ( b ) d/b , Haar measure on

B ,

and

~(a) = Bet (Ad(a)l~)

(iii)

fB iV

Haar i n t e g r a l on Proof p r o p e r t i e s of

(i)

isa

f ( b v ) dzb dv

G . f o l l o w s immediately from the n o r m a l i z a t i o n and commutation

M, A, N .

To prove

Lebesgue measure on the Lie algebra det (Ad ml~ ) = det (Ad n i l ) = 1 ,

(ii) n

,

recall

t h a t via the exponential map,

serves as Haar measure f o r

we obtain

(ii)

from

(i)

N .

Since

and the change of

Lebesgue measure under l i n e a r transformations. The proof of

(iii)

requires a reversal of p o i n t of view. We s t a r t w i t h a

121

Haar measure direct

dg

on

G ,

p r o d u c t group

and we use

B × V .

l(f)

(f

dg

on the

Namely, we c o n s i d e r t h e i n t e g r a l

: IG f ( b ( g ) ,

continuous w i t h compact s u p p o r t on

g e BV ,

t o d e f i n e a Haar i n t e g r a l

v(g))

dg .

B x V.)

Here

g = b(g) v(g)

for

and we note t h a t

b I gv I = b(blg ) v(gv 1) •

Hence I ( f ) by

is i n v a r i a n t under l e f t t r a n s l a t i o n s by

V on

dzb dv ,

B × V .

Since

V

G ~ BV

: I I f(b,v) BV

B × V

is

dzb dv .

i s o f Haar measure z e r o , t h i s proves

1.2 P r i n c ! p a l

irreducible

I.

(iii)

We c o n t i n u e to assume t h a t

series.

L i e group o f r e a l - r a n k unitary

is unimodular, the l e f t Haar measure on

so by uniqueness o f Haar measure, we must have l(f)

Since

B and r i g h t t r a n s l a t i o n s

Let

B = MAN

representations

.

G

is a semi-simple

as in

§ 1,1.

The f i n i t e - d i m e n s i o n a l

of

are a l l

o f t h e form

B

x(man) = ~(a) o(m) , where

~

is a u n i t a r y c h a r a c t e r o f

sentation of vectors for

M . y

is non-trivial

representation

denote by

H(~)

is an i r r e d u c i b l e

and

unitary repre-

(This f o l l o w s from E n g e l ' s theorem: the space o f

r e p r e s e n t a t i o n space.) cible

A

and i n v a r i a n t

under

C o n v e r s e l y , any such p a i r y

of

the H i l b e r t

B

"f

(~,~)

hence i s the whole d e t e r m i n e s an i r r e d u -

by t h i s f o r m u l a . We w r i t e

space on which

Consider now the u n i t a r y

B ,

representation

= Ind (~) . B+G

~ ,

N-fixed

and hence

y = (~,~), y ,

acts.

and

122 By d e f i n i t i o n ,

~

a c t s on t h e H i l b e r t

T

f such t h a t

where

for all

man e B

and

x e G ,

(ii)

IV Iif(y)II 2 dy ~ IlfIl 2 <

i s Haar measure on

translations

To v e r i f y

on

V

and

~ ( a ) = Det ( A d ( a ) l ~ ) ,

.

The a c t i o n

= f(xg)

e v e r y w h e r e on

V ,

g e G ,

Example

,

If

ay+c x = ~ . .

We n o t e t h a t

One c a l c u l a t e s

,

y ~-~ v ( y g )

y ~-+ v ( y g - I )

dy = fV f ( v ( y g ) )

G = SL(2,R)

g :

y = -d/b

t h e map

has i n v e r s e

fV f ( y )

,

then

b = o ,

~(a) = 2

a(yg) =

,

.

,

result

which i s d e f i n e d a l m o s t

Also

u(a(yg))

dy .

and

v(yg) =

The t r a n s f o r m a t i o n (If

G

.

~ i s a u n i t a r y r e p r e s e n t a t i o n , we need t h e f o l l o w i n g T the decomposition x = m(x) a ( x ) n ( x ) v ( x ) ) : For any

of

H :

that

Lemma

except

Borel f u n c t i o n s

: G ÷ H(o)

~y(g) f(x)

where

of all

f(man x) = ~ ( a ) 1/2 ~(a) o(m) f ( x )

i s by r i g h t

y =

H Y

(i)

dy

(Recall

space

i.e. if

,

y ~-~ x if

is defined for all

g e MAV ,

it

y eiR

is defined for all

y.)

a =

IbT,-1 el IbY +d

which agrees w i t h t h e f o r m u l a i n t h e lemma.

Hence

u(a(yg))

= (by+d) -2 = ~d

fay+c~ k~j,

t23 Proof of lemma.

By d e f i n i t i o n ,

the element

v(x)

is characterised by the

property x e B v(x)

Hence i f u = v(yg) , gives v(ug - I ) = y .

then

y e Bug -I

,

so

ug -1 e By .

To prove the i n t e g r a l formula, we i n f l a t e performing an i n t e g r a t i o n over support in

BV ,

B .

Thus i f

Since

y e V ,

the i n t e g r a l from

f

V to

this

G ,

by

is continuous with compact

then IB IV f ( b v ( y g ) ) u(a(yg)) dzb dy = IB IV f ( b n -1 a(yg) - I m- I yg) ~(a(yg)) dlb dy ,

where

m,n

are the components of

yg

in

M, N .

By part

(ii)

of lemma 1.1,

t h i s i n t e g r a l equals IB IV f(byg) dzb dy . By part

(iii)

of the same lemma, the r i g h t t r a n s l a t i o n by

integral invariant. desired formula. Corollary

S p e c i a l i z i n g to the case

If

f e Hy ,

g e G ,

f(bv) = fl(b)

then

g

leaves the

f2(v)

II~ ( g ) f l l = IIfll .

,

we obtain the

Hence

~(g)~

is

unitary. Proof property

(i)

For of

y e V , f

write

yg = ma(yg) n v(yg) .

and the u n i t a r i t y

of

~, { ,

Then by transformation

we have

llf(yg)Ii 2 : u ( a ( y g ) ) I I f ( v ( y g ) ) I I 2 . I n t e g r a t i n g over The f a m i l y

V and using the lemma, we see t h a t {~x,~ : ~ e A, ~ ~ M}

~ (g)

is u n i t a r y ,

of u n i t a r y representations of

c a l l e d the u n i t a r y p r i n c i p a l s.eries of representations.

Q.E.D.

G is

124 1.3

Intertwining operators

we have d e f i n e d a mapping representations of

G ,

¥ = (~,~) ~-~ ~ Y

where

unitary representations of covering map" from

By means o f the p r i n c i p a l s e r i e s ,

A

Mx A

A

and

and

M .

M'/M ,

A x M to the set o f u n i t a r y

M denote the spaces o f i r r e d u c i b l e This map can be viewed as a " r a m i f i e d

onto a subset o f

covering i s the Weyl greup

from

G .

The "monodromy group" o f t h i s

and the a c t i o n o f the Weyl group is given

by c e r t a i n s i n g u l a r i n t e g r a l o p e r a t o r s on the n i l p o t e n t group In more d e t a i l , f o r the n o n - t r i v i a l on

M× A

we note f i r s t element o f

that if

M'/M ,

w e M'

V .

is a fixed representative

then we can d e f i n e an a c t i o n o f

M'/M

by s e t t i n g ( w - x ) ( a ) = X(w-law) w.~(m)

If

¥ = (~,o)

,

write

o(w'Imw) .

w-¥ = (w-y, w-o) .

Then by general r e s u l t s o f Bruhat,

one knows t h a t

(i)

~

=

~

Y

(ii)

If

Combining

(i)

~-~ ~y

carries

, <---> T'

w.y # y , and

(ii)

Mx A

,

or

¥' = w.y

then

~ T

is i r r e d u c i b l e .

we see t h a t on the s e t

into

c a t i o n p o i n t s " are the p o i n t s follows that

Y

=

Y

x = 1

if

G

Since

( w . ~ ) ( a ) = ~(a -1)

is a r a m i f i c a t i o n p o i n t . We s h a l l

theory o f s i n g u l a r i n t e g r a l o p e r a t o r s developed in Chapter I I I (a) ~y

to

c o n s t r u c t the u n i t a r y o p e r a t o r g i v i n g equivalence between and

(b) y=W.T

~w.y

when

x # w-y

determine the r e d u c i b i l i t y

the map

and i s a two-sheeted c o v e r i n g . The " r a m i f i -

{~ : w y = y} .

y = (~,a)

{¥ : w y # ¥}

;

or irreducibility

of

T

when

,

it

use the

125 From an i n t u i t i v e is the f o l l o w i n g Assume

p o i n t o f view, perhaps the most n a t u r a l s t a r t i n g

formal c h a n g e - o f - v a r i a b l e s argument:

f e H , Y

and d e f i n e

A(y) f ( x )

If

point

b = man e B ,

then f o r

= fV f(Yw-Zx) dy .

y e V we have

f(yw-lbx)

= f(ym w a w nw w - l x ) = ~(aW)I/2 ~(a w) a(mw) f(yZ nw w - l x ) ,

where we w r i t e

xg = g - l n g

,

and

Hence assuming t h a t the i n t e g r a l A(y) f ( b x )

z = mw aw .

But

aw = a -1

and

nw e V .

converges, we have = ~(a)-l/2(w.y)(b) = ~(a)i/2(w.y)(b)

Here we have used the f a c t t h a t f o r

IV f(yZ n w w - i x ) dy a(y) f ( x )

.

z = mw aw ,

IV f ( y Z ) dy = IV f ( y a-z) dy

= u(a) IV f ( y ) I t f o l l o w s from the above c a l c u l a t i o n

dy .

t h a t in a formal sense,

A(y) : Hy ÷ Hw. Y Obviously

A(y)

commutes w i t h r i g h t

To i n v e s t i g a t e to r e w r i t e

A(y)f

translations

by elements o f

the convergence o f the i n t e g r a l

as a c o n v o l u t i o n i n t e g r a l yw - I = m(yw - I )

on

V .

a(yw -1) n v(yw - I )

A(y)f

,

G .

we use lemma 1.2

Namely, we w r i t e .

126 Then f o r

f e H , Y

we have f ( y w - l x ) = ~(yw-1) 1/2 ¥(yw - I ) f(v(yw-Z)x)

where we have w r i t t e n

~(yw -1) = u ( a ( y w - l ) )

Introduce the notation

y--g= v(yg) .

y : (~)

g-1

,

¥(yw - I ) = x(a(yw-1)) o(m(yw-1)) .

Then by lemma 1.2,

(y e v)

and IV m(y-g-) dy : IV m(y) ~ ( ( y g - l ) g ) - I

f o r any integrable function in the i n t e g r a l defining

m on

A(¥) ,

V and

g e G .

dy ,

Using t h i s change of variables

we have the formal convolution i n t e g r a l

A(¥) f ( x ) = IV ~ ( ~ w - Z i I/2 y ( ~ w -1) f ( y x ) dy .

We can s i m p l i f y t h i s l a s t expression by noting that i f a ( ~ w-1 )

=

m ( ~ w-1)

= m(yw) -1

a(yw -1 )

= a(yw)

a(yw) -1

These i d e n t i t i e s f o l l o w from the inclusions y ~ w -1 e m(yw) - I a(yw) -1 N y and yw Thus i f

K

-1

is defined on

e yw M V by

K (y) = u(yw) I / 2 ¥(yw) - I

then the operator

A(~)

is f o r m a l l y expressible as

A(~) f ( x ) = IV K¥(y) f ( y x ) dy

y e V ,

then

127 Lemma

The f u n c t i o n

K T

is

Ca

on

V ~ {e}

Furthermore,

Ky(y a) = ~(a) - I ~(a) 2 K (y)

I

Ky(y - I )

(Here

ya = a - l y a Proof

If

decomposition, C~

away from

aw : wa

,

and

y e V

y = (~,o).) and

y w e BV . e .

To e s t a b l i s h -1

,

for

: K ¥ ( y ) m ~(w2) *

y # e ,

then

Thus the maps

y w ¢ Bw . y~-~ m(yw)

This g i v e s the smoothness o f the t r a n s f o r m a t i o n a e A .

Hence by the Bruhat and

y ~-~ a(yw)

are

K T

properties

of

K , T

we r e c a l l

that

Hence

y aw = a-lyw

a-I e

m(yw)a -2 a(yw) NV

so t h a t

a a-2 a ( y w) = a(yw) This g i v e s the t r a n s ~ r m a t i o n

law under

Y ~.~ya

y w e m(yw) a(yw) NV

.

Since

,

we have y -1 w e w VN m(yw) -1 a( y w ) - I w But

w V = N w ,

so t h a t y

-I

w e [w m(yw) -1 w~ a(yw) NV

This shows t h a t (y 1 w)

=

a(yw)

(y-i

=

w m(yw) - I w

Ii

w)

which g i v e s the t r a n s f o r m a t i o n since

y

is unitary.)

law under

y ~-~ y

-1

(Note t h a t

T(y) -1 = ¥(y)m

128 Example

When G = SL(2,1~ ,

a

aand

M = {l,e}

,

Y = ~

~'

where

the characters of

,

~ ( - I ) = -1 .

w : I

I'

i

o,]

a(yw) =

If

y+ = ( 4 , 1 ) ,

y_ = (~,~)

,

If

then

so t h a t

;, e i I R

A are given by,

IY

Yw = I~

,

1~ 1 y~ '

m(yw) = sgn(y) I

are the corresponding representations of

B ,

then

t h i s c a l c u l a t i o n shows that K

(y)

=

¥+

lyl -l+x

Ky_(y) = s g n ( y ) l y l -I+~

Note that u(yw) -1/2 : ]y[ .

1.4 Boundedness of i n t e r t w i n i n g operators results of Chapter I I I

to the kernels

We shall now apply the

K . I d e n t i f y V with i t s Lie algebra Y by the exponential map. Since real-rank (G) = 1 , i t follows from the properties of root systems that V = V1 0 V2 A = {a(t)

,

: t e IR+}

I29 such t h a t

ya(t)

We define d i l a t i o n s

{itY 2y

=

{6 t , t > o}

, ,

on

y e V1 y e V2

V

by

6t Y = y a ( t )

The homogeneous dimension

Q of

V

is then

Q = dim (VI) + 2 dim (V2) Let

~(a) = Det (Ad(a)In)

,

and set

IYl = ~(YW) -1/(2Q)

Lemma lyml : lyl

Proof

•

[y[ I~I

is a smooth, symmetric = ly1-1

for

y e v ,

6t-homogeneous norm on

V ,

and

me M .

As in the proof of Lemma lu3, we c a l c u l a t e t h a t when t > o

,

a((6tY)W) = a ( t ) 2 a(yw) Hence

16tYl = u ( a ( t ) ) - I / Q [Yi

But

N = Ad(w) V ,

so f o r

a e A

one has

Det (Ad(a)IN) = D e t Ad(a)Iv] - 1

Since

Det A d ( a ( t ) ) I V = t Q , 16ty I = t IYl ,

The f u n c t i o n JyJ

y ~-+ a(yw)

The M-invariance o f the transformation

t > o

is smooth on

is a smooth, symmetric norm on IY!

y~-~y~ ,

t h i s shows t h a t

V ~ {e} ,

and

a(yw) = a ( y - l w )

Thus

V .

f o l l o w s from the i d e n t i t y we have

a(ymw) = a(yw). For

13o (~)w and hence

e

a ( ~ ) w = a(yw) - I

Suppose

y(ma(t))

.

M a(yw) -1NV , Thus

= t ~/2 q(m) ,

then by lemma 1.3 one sees t h a t to the d i l a t i o n s

operator

A(y)

!~I

~t '

a

= lyl -I where

,

Q.E.D.

~ e i R .

If

~y(y) = y(wy) -1 ,

is homogeneous of degree

T

Furthermore, the kernel

K

X with respect

f o r the i n t e r t w i n i n g

is given by Ky(y) = ~ ( y ) l y l

and i s homogeneous o f degree

-Q

-Q+~ .

Recall t h a t the c a l c u l a t i o n s i n v o l v i n g

A(y)

in

vergence problems. I t is e v i d e n t from t h i s formula f o r

§ 1.3 KY

ignoved any con-

that

A(y)

is

f o r m a l l y a s i n g u l a r i n t e g r a l o p e r a t o r o f the type t r e a t e d in Chapter I I I .

The

homogeneity and smoothness c o n d i t i o n s are s a t i s f i e d

The

by

K T

f o r any

~ .

mean-value c o n d i t i o n , however, is not always s a t i s f i e d . Theorem (a) the i n t e g r a l o f

Suppose t h a t KY

over

wy # ~ .

{A < lyl < AR}

Then t h e r e e x i s t s

R > I

such t h a t

is zero f o r a l l

A > 0 .

The

operator A(y) f ( x )

= P.V. fV Ky(y) f ( y x ) dy

i s a non-zero bounded o p e r a t o r from tations

and

~

(b)

~y

to

wy = y .

A(~)

is a unitary operator.

Then the r e p r e s e n t a t i o n

the mean-value o f the f u n c t i o n

splits

Hwy which i n t e r t w i n e s the represen-

Some s c a l a r m u l t i p l e o f

Suppose t h a t

and only i f case

~w~

H

y~-~ t r

~ Y (~(w)~yw))

is r e d u c i b l e i f i s zero. In t h i s

as the d i r e c t sum o f two i n e q u i v a l e n t i r r e d u c i b l e

representations.

The p r o j e c t i o n o p e r a t o r g i v i n g the decomposition is a l i n e a r combination o f and the o p e r a t o r Remarks 1.

I

~(w) A(y) . In p a r t

we may d e f i n e an o p e r a t o r

(b) , o(w)

we are using the f a c t t h a t when which extends the r e p r e s e n t a t i o n

wy = ~ , ~

from

then

131

M to

M'

Indeed, by assumption t h e r e e x i s t s

a unitary

operator

T O on

H(~)

such t h a t o(w-lmw) = T o l ~(w) TO Since

w2 e M ,

is a scalar.

one f i n d s t h a t

T 2 o(w2) -1 o

Thus we can choose T=

e ie T

e elR

commutes w i t h

o(m)

and hence

such t h a t

o

satisfies T 2 = ~(w 2)

We set

~(w) : T . 2.

When wy -- ~ ,

a u n i t a r y map from

H Y

to

P r o o f o f theorem the kernel

~

then

~ = o ,

Hw. ~

and the o p e r a t o r sending

which i n t e r t w i n e s

~

~"

and

Y ~-~ ym ,

m e M .

is

W'T

We begin by d e t e r m i n i n g the t r a n s f o r m a t i o n

under the automorphisms

f ÷ o(w)-lf

properties

of

Note t h a t

y m w = m-1 yw mw Hence the M-component o f

ym w

is

m-1 mI mw ,

where

mI

i s the M-component o f

yw. We a l r e a d y c a l c u l a t e d in Lemma 1.4 t h a t a ( y m w) = a(yw)

Hence we o b t a i n the f o r m u l a ~

Since

IYl = lyml

a shell

Det ( A d ( m ) I v ) = i

{a ~ IYl < b}

T

~Y (y) o(m)

,

we may i n t e g r a t e

and o b t a i n the r e l a t i o n

T

(¢=) where

and

(ym) : ( w . o ) ( m - l )

= (w-~)(m -1) T

= mean-value of the f u n c t i o n

~(m)

,

lyl -x a.((y) .

this

formula over

132 Suppose now t h a t there is an

R > I

w¥ # y .

If

~ # o

L2(V) . w-~

If

and

K and

~ = o ,

o ,

by

also a p p l i e s , and

K~ .

then

(~) , A(y)

Hence

A(~)

w~ # o .

of

e x i s t s as a bounded o p e r a t o r on

Since the mean-value

T intertwines T in t h i s case. Thus Lemma I l l . 3 . 1

we must have

T = o T e x i s t s as a bounded o p e r a t o r on

The r e p r e s e n t a t i o n space o f L2(V) ~ H(~) ,

integral

e i IR,then by Lemma I I I . 3 . 1

K over {A ~ lyl ~ AR} is zero T f o r any A > o . By Lemma 1.3, the a d j o i n t kernel K (y-1)m also s a t i s f i e s t h i s T c o n d i t i o n , f o r the same value of R . The smoothness and homogeneity c o n d i t i o n s

are s a t i s f i e d by

such t h a t the

and

via the map f ~

~

and

¥

flY "

~ W'y

can be i d e n t i f i e d with

(This is the s o - c a l l e d "non-compact

p i c t u r e " f o r the r e p r e s e n t a t i o n . ) In t h i s r e a l i s a t i o n , r i g h t t r a n s l a t i o n s , and the subgroup

L2(V) ~ H(a) .

the subgroup

V

acts by

MA acts by

(ma) f ( y ) = v(a) 1/2 y(ma) f(yma) T The element

w

acts by

¥ Since

(w) f(y) = u(yw) I/2 ¥(yw) f ( ~ )

G = (MAV)LW (MAV w V) , I t is obvious t h a t

A(y)

these formulas determine

T

commutes w i t h r i g h t t r a n s l a t i o n s by

V .

By

Lemma 1.3 and the c a l c u l a t i o n above we f i n d t h a t (~) Suppose

Ky (y) ~(ma) = ~(a)(w-y)(ma) Ky (yma) f e

C ~c

(V) @ H(o)

Then

A(y)f

is given by the a b s o l u t e l y conver-

gent i n t e g r a l A(¥) f ( x ) : fV K since

Ky

has

integral

(y) [ f ( y x )

- f(x)]

zero over the f a m i l y of s h e l l s

Using equation (mm) and the i n t e g r a t i o n formula fV f(yma) v(a) dy = fV f(Y) dy

,

dy

,

{Rn < _ IYl < _ Rn+l} •

133 we v e r i f y e a s i l y t h a t i f

g e MAV, then

A(y) Rx(g) f ( x ) The p r o o f t h a t ceptual v e r i f i c a t i o n g i v e n , but take

A(y)

: ~W.y (g) A(T) f ( x )

intertwines

R (w) is more d e l i c a t e . The most conY seems t o be to r e t u r n to the formula f o r A(y) as o r i g i n a l l y

Re x > o .

converges a b s o l u t e l y f o r

Then one proves t h a t the i n t e g r a l d e f i n i n g

f

in

H" , Y

A(y)

now

where

H= : { f e C ' ( G , H ( ~ ) ) ; f(man g) = ~(a) I / 2 y ( m a ) f ( g ) } Y

.

The same change o f v a r i a b l e argument shows t h a t A(y)

and

A(y)

A(y)

: H~ + H~ ~/ w-y

,

commutes w i t h r i g h t t r a n s l a t i o n s by

G .

One proves t h a t as

Re X ~ o ,

converges to the s i n g u l a r i n t e g r a l o p e r a t o r c o n s t r u c t e d above. For d e t a i l s

we r e f e r to the l i t e r a t u r e To f i n i s h cited earlier, A ( y ) * A(y) utes w i t h

c i t e d a t the end o f the c h a p t e r .

the p r o o f o f p a r t ( a ) , we r e c a l l t h a t by the r e s u l t s o f Bruhat

~ is i r r e d u c i b l e i f w.y # T • Hence T must be a non-zero m u l t i p l e o f the i d e n t i t y o p e r a t o r , since i t comm-

~

the r e p r e s e n t a t i o n

Similary,

A(y) A(y) *

Thus w i t h a s u i t a b l e n o r m a l i z a t i o n , In p a r t

(b) ,

the r e p r e s e n t a t i o n

w.y = T ~

to

M'

i s a m u l t i p l e o f the i d e n t i t y

A(T)

implies that

operator.

becomes u n i t a r y . x = o

and

as noted in remark I .

w.o = o .

We extend

Then the c a l c u l a t i o n at

the beginning o f the p r o o f shows t h a t (o(w) T ) = o { m ) - l ( o ( w ) T ) o(m) where

T

i s the mean-value o f the m a t r i x f u n c t i o n

is a scalar m u l t i p l e o f T = o ¥

if

and o n l y i f

I ,

since

o

y ~-+ o(yw) .

Hence o ( w ) T

i s i r r e d u c i b l e . We conclude t h a t

the mean-value o f the f u n c t i o n

y ~-+ t r

,

(o(w) o(yw))

134

is zero Suppose t h i s mean-value i s z e r o . The argument above shows t h a t bounded o p e r a t o r from operator

~(w) A(y)

with

(cf.

~ Y

H¥

to

Hw¥ which i n t e r t w i n e s

~y

and

~wy .

A(y)

is a

The

i s then a bounded o p e r a t o r from

H to H which commutes Y Y remark 2 above). On the o t h e r hand, the r e s u l t s o f Bruhat i m p l y

t h a t the o r d e r o f the Weyl group ( t w o , in t h i s case) always m a j o r i z e s the number of irreducible on

components o f

~ Since o(w) A(~) is not the i d e n t i t y o p e r a t o r Y we conclude t h a t every i n t e r t w i n i n g o p e r a t o r is a l i n e a r

L2(V) ~ H(o) ,

combination o f

~(w) A(y)

and

I ,

and the i n t e r t w i n i n g

dimensional (and hence commutative). Thus

~ Y

splits

as

ring for

~ i s twoY ~+ @ ~- , where ~± are Y Y Y

i r r e d u c i b l e and i n e q u i v a l e n t . I t o n l y remains to v e r i f y t h a t i f the i n t e r t w i n i n g

c

Q

i s n o t z e r o , then

ring for

twining operator and away from

the mean value o f

{e}

~ is trivial. By the r e s u l t s o f Bruhat, any i n t e r Y i s e x p r e s s i b l e as l e f t c o n v o l u t i o n by a d i s t r i b u t i o n on V ,

T

this distribution

is the f u n c t i o n

is a c o n s t a n t . By the "unboundedness" Theorem

bounded o p e r a t o r unless

y ~-* c~(w)o(yw) ,

III.

c = o o This i m p l i e s t h a t

5.1 ,

T

T

where

cannot be a

is a m u l t i p l e o f

I ,

Q,E.D.

1.5 Examples Theorem 1.4. Suppose f i r s t either trivial

o r else

c o n d i t i o n is s a t i s f i e d

Thus

A = ~-IA(E)

Let us i l l u s t r a t e that

y = ~ ,

G = SL(2,R) . where

the r e d u c i b i l i t y Then

c ( ± l ) = ±I .

w-~ = ~

criterion means t h a t

The mean-value zero

o n l y in the second case, and we have in t h i s case

i s the c l a s s i c a l H i l b e r t A f(x)

= P.V. Z

transform:

f f(t) dt t-x

of y

is

135 A f t e r Fourier transformation so

A2 = - I .

A becomes m u l t i p l i c a t i o n by the function

The spectral decomposition of

where H2(IR) are the Paley-Wiener spaces of

A

i sgn(~),

is given by

L2

functions holomorphic in the upper

(lower) half plane, with sup 7 If( x ± iY)I 2 dy < y>o -~ The representation

~

in the non-compact picture is g i v e n by

E

~ (g) f ( x ) = (bx+d) - I f rax+cl ~x--x~-~J b if

g =

.

under

g) .

'

I t is evident from the above description that Theorem 1.4 asserts that the r e s t r i c t i o n

of

~

H±2

are i n v a r i a n t

to

H±2

is i r r e -

ducible. As an other example, consider the group

G~SL(3,~)

which leaves i n v a r i a n t

the Hermitian form z 2 z 2 + 2 Re(z I z~) where

(z 1, z 2, z3) e ~3.

,

(This group is conjugate to the group

leaves i n v a r i a n t the form

z~ Z l +

z:2 z 2 - z 3 z 3.)

The subgroups

t h i s case are the f o l l o w i n g ( a l l blank matrix e n t r i e s are zeros):

I i°

M :

me =

A :

ar = ~

N :

exp

L~° |

e -2ie

1

zo

ei

1

r -I]

i zt 1 0

,

e e IR

'

r>o

,

z e (~, t e IR

SU(2,1)

which

M, A, N, V

in

136

V:

ze¢,teR t

z=',

For t h e Weyl group r e p r e s e n t a t i v e ,

It

is then a straightforward

a(vw)

and

v~ ,

when

matrix,

t

,

where

z~':

(z:':z+it)

whose d i a g o n ~ e n t r i e s

z='~

that

then g i v e

( v w ) u -1 m(vw)

a(vw) = a r ,

where

r = 2 1 z m z + i t I -1

m(vw) = m0 ,

where

e = arg~(zmz+it~

-~=

v(~,~)

2iz = z,z-it

The a d j o i n t

action

of

Ad(ar) r > o ,

function

A

on

v(z,t)

,

where

,

T = -

V

i s an upper

and

a(vw).)

One

4t z~z-itl 2

i s g l v e n by

= v(rz,

r2t)

,

and t h e homogeneous d i m e n s i o n o f

~(ar) = r 4 ,

v = v(z,t)

V

is

Q = 4 .

and hence t h e homogeneous norm on

Iv I = u ( v w ) - 1 / 8 = !

Here

m(vw)

that

l

when

,

to determine the matrices

i s u n i q u e l y d e t e r m i n e d by t h e p r o p e r t y

triangular finds

calculation

v = v(z,t)

t/2

(u = v~

we t a k e

Izmz+itl I/2

V

The m o d u l a r

i s g i v e n by

137 The group

M = U (1)

in this

c a s e , and

on(me) = e - i n e

The a c t i o n o f

w

on

M

is trivial.

M consists of all

representations

n e

Let

Kn(V ) = ~(vw) 1/2 On(VW)

-1

veV

By the f o r m u l a s above we can w r i t e

%(v) Kn(V ) = c n tvt where %(v)

= (z*z+!t)

n

v = v(z,t)

,

Iz'~z+itl n

and

cn

i s a non-zero c o n s t a n t .

by the c h a r a c t e r Theorem Proof.

Denote by

~n

the r e p r e s e n t a t i o n

of

G

induced

ma --+ On(m ) . ~n

is reducible

We s h a l l

n # o .

For t h i s

dinates,

we can express

<---> n

show t h a t the mean-value o f

p u r p o s e , we w r i t e

Qn(V)

Thus the mean-value o f

i s even and n o n - z e r o .

b

~n

i e inO

z'ez+it = re i e .

n

(i eineds)-~ o

,

o

which vanishes p r e c i s e l y

for

n

even ,

n # o ,

<--> n

i s even,

Then using c y l i n d r i c a l

i s given by the i n t e g r a l

de

i s zero

Q.E.D.

coor-

138

§ 2

2.1

BpuRdary values o f

H2

functions

Harmonic analysis on the Heisenber 9 9roup

domain o f type I I " introduced in Chapter I I , acts simply t r a n s i t i v e l y F o u r i e r a n a l y s i s on Hardy class

G ,

on the boundary

parametrize

D .

G

Using the (non-commutative).

r e c a l l the basic facts concerning harmonic a n a l y s i s on G as IRn × Rn x ~ ,

X e IR ~ {o} ,

G acting on

L 2 ( £ n)

g = ((,q,¢)

as in

,

G .

We

§ 11.4.4, with m u l t i p l i c a t i o n

= (~+~',n+n',~+~'+ ½ ( ~ ' n ' - ~ " n ) )

•

there is an i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n

~

of

by 1 f ( x ) = e i~(~+q'x+ ~ q'~) f(x+~)

~(g)

Given

The Heisenberg group

we shall study the boundary values of functions in the

(~,n,¢)(~',n',~')

where

M of

D be the "Siegel

H2(D) .

We f i r s t

For every

§ 4.4.

Let

and

m e LI(G) ,

x-y = ~ xiY i

,

( x , y e IRn) .

we d e f i n e the o p e r a t o r

#(x)

on

L2(£p)

by the

operator-valued integral @(x) = ~ X(g):~ re(g) dg G Here

dg

is Haar measure on

G (=Lebesgue measure on IR2n+l

in the above

c o o r d i n a t e s ) . The Plancherel formula is J im(g)l 2 dg = I G IR where

]]TI]~S : t r (TmT)

lle(~)l]~sdu(x)

is the square o f the H i l b e r t - S c h m i d t norm, and the

Pl ancherel measure d~(~) = c n lXt n d~ ,

with

c n = (2~) -n'1

and

d~

Lebesgue measure on IR .

I f we define

L2(G)

to

t39 X ~-~ T(~) on IR

be the H i l b e r t space o f a l l measurable, o p e r a t o r - v a l u e d f u n c t i o n s such t h a t

[ IIT(x)ll~s d~(x) <

(where

T(~)

is an o p e r a t o r on

u n i t a r y map from Let

p

L2(G)

onto

L~IRn)) ,

then the map

~ ~-+ ~

extends to a

L2(G) .

be the r i g h t r e g u l a r r e p r e s e n t a t i o n o f

G on

L2(G) :

p(g) ~(X) = ~(xg) The F o u r i e r transform o f valued f u n c t i o n

~(g)

p(g)

is then l e f t m u l t i p l i c a t i o n

by the o p e r a t o r -

:

(p(g)~)~(~)

= ~(g)

~(~) .

To study the " t a n g e n t i a l Cauchy-Riemann" equations s a t i s f i e d by the boundary values o f holomorphic f u n c t i o n s , we shall need to extend presentation o f the Lie algebra o f (any Lie group) of

C=

~

of

G .

G on a H i l b e r t space

vectors f o r

~

If

x

H(~) ,

p

to a re-

is any u n i t a r y r e p r e s e n t a t i o n we define the subspace

H~(~)

by

H=(~) = {v e H(~) : g ~-+ ~(g)v This is a dense subspace o f

H(~) .

Given

is a

Ca f u n c t i o n }

v e H=(,)

and

X e ~ ,

we d e f i n e

~ (X) v = ~ t t= ° ~(exp t X) v

Then

~ (X) : H=(x) + H=(~) ,

homomorphism from

~

and the map

to operators on

X ~-~ R (X)

H=(z) .

is a Lie algebra

Hence i t extends uniquely to an

a s s o c i a t i v e algebra homomorphism from the c o m p l e x i f i e d universal enveloping algebra

U(~)

also denote by Give

i n t o the algebra o f l i n e a r transformations o f

H~(~) ,

~

H=(~)

the topology defined by the f a m i l y of semi-norms

which we

140

v ~- II~(T)vll, as

T

ranges over

U(~) .

Then

H~(~)

is a Fr#chet space. We denote by

the space o f continuous c o n j u g a t e - l i n e a r f u n c t i o n a l s on H~(~) ~ H ( ~ )

H=(~) .

H'~(~)

The i n c l u s i o n

and the canonical isomorphism between a H i l b e r t space and i t s

a n t i - d u a l then provide an i n c l u s i o n o f

H(~)

into

H-~(~) :

U(~)

such t h a t

H~(~) ~ H(~) ~H-=(~) . Let

T ~

T

be the canonical i n v o l u t i o n on

By taking a d j o i n t s , we o b t a i n a r e p r e s e n t a t i o n The u n i t a r i t y of

~

implies that

~ (X)

X ~-+ ~_~(X)

X = -X of

g

for

x e g.

H-~(~)

on

.

is skew-symmetric, and hence

(~®(T)u,v) = (U, ~_~(T*)v) if

u e H~(~),

v e H-=(~) ,

In the case Ha(p)

x = p

consists o f a l l

left-invariant

is the r i g h t r e g u l a r r e p r e s e n t a t i o n o f functions

m on

G such t h a t

differential

operators

T

G .

G of the form

differential

C~

on

~ Tif i ,

operators on

o f the Heisenberg group, decreasing

T e U(g) .

C~

a l l d i s t r i b u t i o n s on left-invariant

and

H-~(~ ~)

where

G (finite

H~(~ ~) = S ( ~ n ) ,

f u n c t i o n s , and

The space fie

G ,

the space

T m e L2(G) H-=(p) L2(G)

and

for all

consists o f Ti

are

sum). For the representations

the Schwartz space o f r a p i d l y -

is the space o f tempered d i s t r i b u t i o n s

on ~n .

Let

m e H~(p) .

(*) (a.e.

Then the F o u r i e r transform

~(~) : H(~ ~] ÷ H~(~ ~) [d~).

Furthermore, f o r every

is the o p e r a t o r - v a l u e d f u n c t i o n

T e U(g) ,

~ ~-+ ~ ( T )

formula,

(**)

@ is a smoothing o p e r a t o r :

f II~(T) ~(x)ll~s

tR

d~(x) <

@(~) ,

the F o u r i e r transform of and by the Plancherel

p (T)~o

141 Conversely, any measurable operator f i e l d for all

T e U(g)

@ which s a t i s f i e s

is the F o u r i e r transform of a f u n c t i o n

To o b t a i n the SzegU kernel f o r F o u r i e r i n v e r s i o n formula f o r

G .

H2(D) ,

If

A

(*)

and

(~*)

m e H~(p) .

we shall need the (non-commutative)

is a bounded operator on a H i l b e r t

space, denote i t s absolute value by IAI = (A~A) 1/2

IAI

Then

is a non-negative s e l f - a d j o i n t

is H i l b e r t - S c h m i d t ,

{e n}

A

is nuclear i f

IAI 1/2

i.e.. z ( i A I e n, e n)

if

operator, and

< ~

,

is an orthonormal basis. In t h i s case t r ( A ) = z(Ae n, en)

is defined independently of the choice of basis. Define ~-~ A(~)

LI(G)

from

to be the space of a l l measurable, operator-valued f u n c t i o n s

IR to

L2(IR n) IIAIIz

= S

such t h a t tr(IA(x)l)

dp(~)

<

R

This is a Banach space in the norm Given

A e LI(G) ,

function

m on

(~)

(*m*)

n u l l f u n c t i o n s , as always). A

to be the

G given by

This defines a map from

and

(identifying

we define the inverse F o u r i e r transform of

~(g) = S t r ( ~ ( g ) IR

@(~) = A(~)

~AI1

a.e..

LI(G)

A(~)

into

d~(~)

C(G) .

I f i t happens t h a t

For example, i f we s t a r t w i t h

holds w i t h

m e H~(p) ,

q~ e

then

L2(G) ,

r# e El(G) ,

A(X) = @(X) . (For proofs of the assertions of t h i s

s e c t i o n , see the notes and references at the end o f the Chapter.)

then

142 2.2

Tangential

Chauchy-Riemann e q u a t i o n s

D = {(z,w): in

(~n+l ,

Im w >

§ 11.4.4.

The complex v e c t o r f i e l d s

are t a n g e n t t o Thus, i f

H f

!

1 < k < n

~w

~z k

~n+1

and span the a n t i - h o l o m o r p h i c is a function

t a n g e n t space a t each p o i n t o f

which i s h o l o m o r p h i c on a neighborhood o f

M

in

then

(,)

L-k f = o

These are the " t a n g e n t i a l Recall the l i f t i n g

on

M

where

x

map

W carrying

p

of

We know t h a t

functions

g .

If

(Xk = Re(Lk)

'

{ P k ' Qk : 1 < k ~ n}

Yk = Im(Lk)

as i n

P(Ak) Wf = o ,

where Ak = Pk " i Qk

on

M

to

functions

on

G :

,

W intertwines

X(Pk ) = Xk "

(tt)

1 <_ k < n

i s the L i e a l g e b r a homomorphism from

§ 11.4.4 .

tation

,

Cauchy-Riemann e q u a t i o n s . "

Wf(u) = f ( e ~ ( U ) . o )

in

w e ¢}

: Im(w) = IIzll 2} ,

Lk =--~-~~ i z k

M .

z e cn ,

,

w i t h boundary M = {(z,w)

as in

llzll2

Consider the domain

g k

to vector fields and the r i g h t

on

M

defined

regular represen-

are the elements o f

g

such t h a t

X(Qk ) : Yk

§ 11.4.4)

,

then e q u a t i o n s

1 < k £ n

,

(t)

become

148 Suppose now we s t a r t w i t h a f u n c t i o n (~)

~(Ak) m = o

Here

p = ~_~ ;

i.e.,

,

m e L2(G) , i <_ k < n

we consider these as equations in the space

introduced in the previous paragraph. By d e f i n i t i o n , e H~(~) ,

(~(A~) ~,m) = o

(The usual t e s t - f u n c t i o n as saying t h a t

Theorem

(~)

space

t h i s means t h a t f o r a l l

Cc(G )

Suppose

m e L2(G) .

a.e.

for

(ii)

Range @(~) ~ V ~

function Proof

Then

satisfies

~(~) = o

V~

i _< k < n

is dense in

#

(i)

where

,

H~(~

holds in the usual d i s t r i b u t i o n

the Fourier transform

~(Ak)~ ~

H-~(p)

one has

(~)'

<~

and consider the eguations

so t h i s is the same

sense.)

m satisfies

the equations

(~)

the conditions ~ < o ;

a.e.

for

~ > o

is the one-dimensional subspace of exp [- ~ llxll2]

,

,

L2(IRn)

spanned by the

(the "vacuum s t a t e " )

By the r e s u l t s of the previous s e c t i o n , the F o u r i e r transform of

is

~1(Ak)~(x)

.

Hence

(~(A k) ~, ~) : I t r ( ~ ( A ~ ) IR We claim f i r s t

t h a t equations

(~)

Range

(~)

; (A) ; (>,):':)d~(>.)

are e q u i v a l e n t to

^

for

i < k < n =

and

m(~) i a.e.

Range

~ .

I t is immediate from the Plancherel formula t h a t converse f o l l o w s from the existence of " s u f f i c i e n t l y ~(~) .

Specifically,

if

f e L2(IR n)

and

(~)

implies

(~) .

The

many" operators of the form

h e Cc(IR ) ,

then the operator-

144 valued f u n c t i o n ~-+ h(x)

g~f

is the F o u r i e r transform of some rank one on sation of

L2(IR n) H~(p)

a c t i n g by

(Here

v ~-~ ( v , f ) g . )

g ~ f

is the operator of

This f o l l o w s from the c h a r a c t e r i -

given in the previous paragraph, since the elements of

act in the representations coefficients.

e H~(p)

xx

as d i f f e r e n t i a l

With t h i s choice of

~ ,

operators w i t h polynomial

we have

~ ( A ~ ) ; (X) ; (X)* = h ( X ) ( ~ ( A ~ ) g ) £

Hence by the Plancherel formula,

(~)

U(£)

(;(X)f)

.

implies t h a t

/ h(~)(,X(A~)g,~ ; (X)f) dp(X) = o R

f o r a l l such of

h ,

f , g, h

(Recall t h a t

i ~ k < n

~ (A k) •

~(X)f) = o

and a l l

To i n v e s t i g a t e

f e L2(IRn),

(~),

By the a r b i t r a r i n e s s

on

a.e. (X)

g e S (IRn),

we need the e x p l i c i t

Going back to the formulas f o r x~(Pk) = ~

acting

.

t h i s is e q u i v a l e n t to the conditions (~(A~)g,

for

tr(g ~ f ) = (g,f))

S (IRn),

Ak = -Pk-iQk ,

where

'

~®(Qk) = i x x k

{x k}

Ak) = - aXk + Xxk Ak = Pk - iQk '

~(Ak)

=

~

+ Xxk

,

we have

(~).

form of the operators we c a l c u l a t e t h a t

,

are the coordinate

t h i s gives the formula

For the operator

~

which is p r e c i s e l y

functions

on

IRn .

Since

145 Case 1:

If

~ < o ,

l a t e d by

~(A~)

.

+ ~x k, - ~x3k + ~x~ = 2% I

contains the f u n c t i o n s

~k

p(x)

Range

---> ~(~) = o If

when

~ > o ,

is a n n i h i l a t e d by

p

L2(IR n) ,

the f u n c t i o n

~1(Ak) .

Ciioirollary

Hence i t

(i)

v~ and

(ii)

L2(G) .

space commutes w i t h l e f t

translation.

P~

=

v~

Hence

On the o t h e r hand, p(x) v ~ ( x ) ,

(~

is

of the Theorem, and completes the proof.

m e L2(G)

which s a t i s f y

I < _ k < n

,

If

~ e L2(G) ,

P onto t h i s sub-

then the F o u r i e r transform

where

I'0 ~

,

if

~ < o

is the f u n c t i o n

expF-

,

~-ilxll]2

and

W~ Z is one-dimensional

The orthogonal p r o j e c t i o n

c(~) v~ ~ v~ Here

S (IRn),

W~ contains the functions

p(o) = o .

The space o f f u n c t i o n s

Px ~(~) ,

W~

in

in t h i s case. This shows t h a t

is a closed subspace o f

is

~llxll 2) is

is orthogonal to

P(Ak) m = o ,

P ~

is an a r b i t r a r y polynomial.

as is well-known, so t h a t

vx(x) = exp{-

is any polynomial such t h a t

e q u i v a l e n t to c o n d i t i o n s

p

x < o .

and is spanned by the f u n c t i o n

of

and is a n n i h i -

,

where

the same argument as in Case 1 shows t h a t where

S (IR~ ,

~ (Ak)

exp(~llxl~),

This set of f u n c t i o n s is dense in

Case 2:

is in

i n d u c t i v e l y t h a t the subspace

W~ :

(~)

expI~llxll2)

Using the commutation r e l a t i o n s

~k

one v e r i f i e s

the f u n c t i o n

)~ >

,

0

and

c(x)=

EIvxl[-2 = (~/~)n/2 .

146 Proof:

This follows immediately from the Plancherel theorem and the theorem

j u s t proved. Definition functions

Let

m e L2(G)

Remark

Define

H~(G) ~

be the closed subspace of

which s a t i s f y L2(H)

on M via the map u ~

L2(G)

consisting of a l l

(~).

by transporting the Haar measure on

e~(u) • 0 .

If

H~(H)

G to a measure

is the subspace of functions in

L2(M) which s a t i s f y the tangential Cauchy-Riemann equations

(~)

(in the d i s t r i -

bution sense), then

where

W is the l i f t i n g

§ 11.4.4.

map from functions on H to functions on

(In this notation,

2.3

as in

b = boundary.

Projection onto

P : L2(G) + HE(G )-

G ,

H~G_(_~_ #S a slngular integral operator

be the projection operator in Corollary 2.2.

we want to use the Fourier inversion formula on

G (§2.1)

Let

In this section

to show that

P is a

l i n e a r combination of the i d e n t i t y operator and a singular integral operator of the type studied in Chapter I I I . To i l l u s t r a t e the method, we f i r s t classical Paley-Wiener space Fourier transform

f(~)

H~(IR) ,

vanishes f o r

orthogonal projection, then for

consider the analogous problem f o r the consisting of functions x ~ o .

If

P+ : L2(IR) ~ H~(IR)

eo

d~ o

£->0 £>0

!i

e -~(~+ix) d~t

~(x) dx

whose

is the

m e C#(~R) we can w r i t e , by the classical

Fourier inversion formula, 1

f e L2(IR)

147

=

,,II dx +ix

lira

~-~o c>o To w r i t e t h i s l i m i t

as a s i n g u l a r i n t e g r a l , we observe t h a t f o r any f i x e d

1 R dx i ~ l i m T ~ £ ~+ix = ~

dx 1 x-~+l : ~ -

R > o ,

'

~>0

independent of

R .

change of v a r i a b l e

(Write the i n t e g r a l x ÷ cx.)

Since

1 P+~(o) = l i m Tl~ f

as an i n t e g r a l over

[o,R]

and make the

m has compact support, we thus can w r i t e

c÷ix

-

dx

+ "2"1

~(0)

c>O

: ~_T~II 9(x)-m(O)x dx + ½ 9(0)

But the f u n c t i o n

x

-1

has mean-value zero, so t h i s l a s t equation can also be

w r i t t e n as P+~(o) = ~ I

lim ~0

Finally,

~

using the t r a n s l a t i o n - i n v a r i a n c e

P+ = ½ [I-iA] where

A

.9~.(.x.l. dx + ~1 9(0) • x

I X~> E

is the c l a s s i c a l

P+ ,

we conclude t h a t

,

H i l b e r t transform ( c f . § 1.5).

Coming back to the Heisenberg group we f i r s t

of

G and the p r o j e c t i o n

P onto

H~(G)

observe t h a t by the (non-commutative) Fourier i n v e r s i o n formula and

C o r o l l a r y 2.2, co

P~(e) = I t r ( P x ~(X)) d~(X)

,

0

if and

~ e C~c (G) o Here

d~(x) = c n Ix I n dx

is the Plancherel measure f o r

G ,

148 PL ~(~) =c(~)(vk @ v~) ~(X) ^

=c(x) v~ ~ ~(x)* v x v~(x) = exp[- Tx

where

ilxli2]

and

,

c(x) = (X/~) n/2

Hence

t r (Px #(~)) = c(~)(~(X) Vx , v~)

= C(X) ~ m(X)(~(g-l)

VX' v~) dg .

G Next, from the e x p l i c i t

form of the representations

T:x

in

§ 2.1,

routine c a l c u l a t i o n shows that (~(g)

v~, v~) = c(~) -1 exp[ix~ - ¼ (II~l~ + llnII2)~ ,

where the coordinates of

g

P~ as the i t e r a t e d i n t e g r a l

are

(¢,n,¢)

,

as in

§ 2.1.

Thus we can w r i t e

(non-absolutely convergent)

co

(*)

P~(e) = ~ (f e -xT(g) e(g) dg] dp(~) o G

,

where

x(g) if

= ~- (ll~II 2 + llnll 2) - ic

g : (C,n,c) To interchange the order of i n t e g r a t i o n in

factor

exp(-E~) ,

E > o ,

(m) ,

we introduce a convergence

so t h a t

Pm(e) = lim f (7 e-~(c+T(g)) d~(~)] dg . c÷O G 0 E>O

Using the formula f o r the Plancherel measure to evaluate the inner i n t e g r a l , we obtain the formula (me)

P~(e) : lim c f ~(g) dg ~+o G (~+x(g)) n+l E>O

'

149 c = n I ( 2 ~ ) -n-1

where

I t remains to rewrite

and

(~)

considered i n Chapter I I i o

~ e C~c (G)

as a p r i n c i p a l - v a l u e

For d i l a t i o n s

on

i n t e g r a l o f the type

G we take the one-parameter

group o f automorphisms whose a c t i o n in canonical c o o r d i n a t e s i s

~t (~'~'~) : ( t ~ ' t n ' t 2 ~ ) when

t > o ,

The function

'

T introduced above is then homogeneous of degree 2 :

~(~tg ) = t2~(g) . We d e f i n e a homogeneous norm on

G by s e t t i n g

Igl = I T ( g ) l 1/2

= [2 + 4-2(1i~112 + ii~i12)211/4 -1

This norm i s smooth and symmetric, (Recall t h a t

g

The homogeneous dimension o f

and the f u n c t i o n

G

is

Q = 2n+2 ,

has c o o r d i n a t e s

(-~,-n,-~),)

K(g) = T(g) - n - I

i s homogeneous o f degree -Q .

Obviously

K(g-Z) ~ = K(g)

so

K

is

C co

on

G ~ {e}

°

Also

,

i s Hermitian symmetric.

Lemma. Proof

K has mean-value z e r o . Introduce " c y l i n d r i c a l "

p = ( I / 4 ) ( I I ~ I I 2 + llnl[2) .

c o o r d i n a t e s on

,

dm i s the measure on the u n i t sphere in

constant.

G by s e t t i n g

The change o f measure is then

d~ dn d~ = c p n - i d~ do d~ where

K

In these c o o r d i n a t e s , we have

R2n ,

and

c

is a p o s i t i v e

150 K(g) dg : c f f ( p - i ~ ) -n-1 pn-1 do d~ E

a~Ig ~b where

E clR 2

is the h a l f annulus a

4

<~

2

+

p2

<_

b4

,

p>O

Changing to p o l a r coordinates in the

b 2c f a

Writing

,

p%

plane, we get the i n t e g r a l

)n-i

dr de} T "

{ i e i ( n + l ) e (cos e o

cos e = ~1 (e I'e + e - i e )

and using the binomial expansion, we f i n d t h a t

the integrand is a sum of even harmonics e x p ( 2 i k o ) , 1 < k ~ n, i n t e g r a t e s to zero over

[o, 3

,

and hence i t

Q.E.D.

We can now s t a t e the main r e s u l t of t h i s s e c t i o n . Let us denote by

K the

s i n g u l a r convolution o p e r a t o r K m(x) = P.V. } m(xy) K(y) dy . G By the lemma j u s t proved and the r e s u l t s o f Chapter I I I , to a bounded o p e r a t o r on Theorem

we know t h a t

K extends

L2(G) .

The p r o j e c t i o n

P onto

H~(G)

is a l i n e a r combination

of

K and

the i d e n t i t y o p e r a t o r . Proof express

Since

P ~(e)

P

and

in terms of

the same c y l i n d r i c a l lim

K commute with l e f t K m(e)

and

translations, it

~(e) ,

for

is enough to

~ e C~ (G) .

By using

coordinates as in the lemma, we f i n d t h a t

! ,,,,dg

~+o Ig1_R ~+=~g~jn+1

a

~>0

e x i s t s and is independent of o~

e Cc (G) ,

R .

we thus o b t a i n from

Since

(~-m(e))K

(~'=:'=) the equation

is an i n t e g r a b l e f u n c t i o n i f

151

Pm(eJ = c f (m(g) - m(e)) K(g) dg + a m(e) . G But is

K

has mean-value z e r o , so the f i r s t

cKm(e) .

,

For the t h r e e - d i m e n s i o n a l

i s the i n t e r t w i n i n g

tation

equation

the p r o o f .

Remark K

side of this

This shows t h a t P = cK + a l

finishing

term on the r i g h t

~2

of

{(z,w)

=

(n=1)

,

the o p e r a t o r

a s s o c i a t e d w i t h the r e p r e s e n -

.

Szeg~ kernel f o r

D

§ 1.5

o p e r a t o r s t u d i e d in

SU(2,1)

2.4

Heisenberg group

H2(D)

: z e ~n

be the Siegel domain o f t y p e I I

Let

,

,

w e ~

,

i n t r o d u c e d in

Im(w) > Izll 2}

§ II.

4.4 .

(z,w) e D ,

If

we

can w r i t e ( z , w ) = ( z , wo + i t ) where

Im w o = llzll 2

f

D ,

on

and

The Hardy space

(:'~)

Thus the p o i n t

we d e f i n e the f u n c t i o n ft(Z,Wo)

functions

t > o .

f

H2(D)

on

D

ft

= f(z,w ° + it)

on

Given a f u n c t i o n

(Z,Wo) e M .

M by

,

t > o

i s then d e f i n e d to be the space o f a l l

holomorphic

such t h a t

sup I I f t ( m ) l 2 dm t>o M

Recall t h a t the measure group ( E q u i v a l e n t l y ,

,

dm

<

i s the image o f Haar measure on the

we can p a r a m e t r i z e

M by

IRn x]R n × R

Heisenberg

v i a the

152

map

(z,w) ~-+ (Re z, Im z, Re w) In t h i s section we w i l l

to show t h a t :

(i)

and use Lebesgue measure in the parameters.)

use the Fourier a n a l y s i s on the Heisenberg group

the boundary values of f u n c t i o n s in

space

H~(M)

already studied;

H2(D)

is a H i l b e r t space, and the mapping from

is an isomorphism onto

(ii)

H~(M) ;

To i l l u s t r a t e plane

function

f

in

¢ .

side of

(*)

,

to i t s boundary f u n c t i o n f

can be recovered from

formula.

Starting with

Im z > o

f

the f u n c t i o n

the method, we f i r s t

{Im z > o } c

comprise the

w i t h norm given by the l e f t

(iii)

i t s boundary values by an i n t e g r a l

H2(D)

consider the c l a s s i c a l m e H~(]R) ,

case o f the h a l f -

we obtain a holomorphic

by the Fourier i n v e r s i o n formula:

,, 1 f ei~Z 2~£~j f ( z ) =~-~z d~

,

Im z > o

0

(Recall t h a t

~(~) = o

for

~ ~ o .)

I f we define

is the inverse F o u r i e r transform of the f u n c t i o n

ft(x)

= f(x+it)

e x p ( - t ~ ) #(~) ,

,

then

ft

so t h a t by

the Plancherel theorem,

t>oSUp llftllL2(~R)= IImlIL2(IR). In p a r t i c u l a r , ft ÷ m in

L2

the set of f u n c t i o n s as

t + o .

{ft}t>o

is bounded in

L2(IR ) ,

Conversely, given a holomorphic f u n c t i o n

upper h a l f - p l a n e w i t h the property t h a t the set

{ft}t>o

f t ÷ m as

t + o .

Finally,

to represent

f

is bounded in

we can use a weak-compactness argument to obtain a boundary f u n c t i o n such t h a t

and

f

in terms of

in the L2(IR),

2 m e H+(IR) , ~

instead

^

of

m ,

we i n v e r t the order of i n t e g r a t i o n

now, since

Im z > o) f(z)

=

(no convergence f a c t o r is needed

to get the formula ~1

~

f ( f e i ~ ( z - x ) d~} m(x) dx

--~0

=TTTI

dx

153 The f u n c t i o n

(2~i) -1 (x-z) -1

kernel expressing

f(z)

is the Szeg~ kernel in t h i s case (the reproducing

in terms o f the boundary values o f

We now return to the space

H2(D) ,

r e p l a c i n g F o u r i e r analysis on IR Heisenberg group

G .

g

and carry out a s i m i l a r a n a l y s i s ,

Recall t h a t from § I I . 4.4 the map from

G t6

H

is given

+ ttntt2)

1 z j ( g ) : g (~j - i n j )

when

.

by (non-commutative) F o u r i e r analysis on the

in coordinates by i w(g) : c + ; (tl~ll 2

(**)

f)

has canonical coordinates

,

(~,n,~) elR n x IR x IR .

We shall w r i t e

z(g) = ( z l ( g ) . . . . . Zn(g)) e cn , and i f

z , z ' e cn , z.z'

we set =

Note t h a t as a real

n ~

j=l

C=

z.z[

JJ

manifold,

g , t ~ - ~ (z(g) where

g e G and The f i r s t

,

D is isomorphic to

w(g) + i t )

G x IR+ via the map

,

t > o

step in the a n a l y s i s is to show t h a t by a n a l y t i c c o n t i n u a t i o n o f

the F o u r i e r i n v e r s i o n formula f o r

G ,

we can synthesize f u n c t i o n s in

s t a r t i n g from the F o u r i e r transforms o f f u n c t i o n s in

H~(M) .

H2(D) ,

For t h i s , we

need the f o l l o w i n g consequence o f Theorem 2.2: Lemma

The F o u r i e r transform of the space

valued f u n c t i o n

~-+

v~. R wx ,

where

v~, w e L2(IRn)

(i)

v~(x) : exp [- T llx

(ii)

w~ = o

(iii)

the f u n c t i o n

and

if

2 G) Hb(

consists o f a l l o p e r a t o r satisfy

x < o ~, x ~ - + w x ( x )

is measurable on IR+ × IRn

154

I I lw~(x)l 2 o IRn

(Here

~n/2 dx d~

v ~ w denotes the operator on If

m e H~(G)

and

#(x) = v~ @ wx ,

H i l b e r t space isomorphism from dv = c ~n/2 dX dx

L2(~Rn)

H~(G)

<

given by

then the map

onto

(v ~ w)(u) = (u,w) v m ~-+wx

L2(IR+× IRn ,'

(dX = Lebesgue measure on

~) ,

.)

defines a where

IR, dx = Lebesgue measure on

£n

c = constant). Proof of Lemma

Conditions

(i)

Theorem 2.2. To v e r i f y c o n d i t i o n norm of the operator

vx @ wx

and

(iii) is

,

(ii)

are d i r e c t consequence of

we observe t h a t the H i l b e r t - S c h m i d t

[Ivx[ I llw~II .

Hence by the Plancherel

formul a,

G

Im(g)12 dg = c n 7 llv~[]2 o

= c [ I

llwxll2 n dx

[W~(X)12 n/2

dx d~ ,

o IRn since

Nvxil2 = (~/~)n/2

.

(Here

c = n/2

Cn ,

where

c n = (2~) -n-1 .)

Together

w i t h the Plancherel theorem, t h i s proves the Lemma. The main r e s u l t o f t h i s section is the f o l l o w i n g Plancherel measure f o r

Theorem

Suppose

(d~ : Cnl>,Ind~,

is the

G) :

m e H~(G) .

Then f o r every

t > o ,

the operator-valued

function Ft(~ ) : e-~t ~(~) is in

(~f) then

LI(G).

I f the f u n c t i o n f(z(g),

f

w(g) + i t )

is defined on the Siegel domain = [ tr(~(g) o

f e H2(D)

and

Ft(~))

d~(~) ,

D by

155

sup IIftI[L2(H) : II~ IIL2(G) t>o Furthermore,

Wft ~ ~

in

L2(G )

as t ~ o

. ^

Proof

By the Lemma j u s t proved, we can w r i t e tr(~(g)

( i n n e r product in

L2(IRn))

#(~)) = ( ~ ( g ) .

m(~) = v~ ~ w~ ,

and hence

v~, w )

Using the e x p l i c i t

form of the r e p r e s e n t a t i o n

we c a l c u l a t e t h a t (~(g)

V~, W~) : e i~w-~z'z I

e

T

w~-~-Tdx

,

[Rn

where

w = w(g)

and

z = z(g)

are defined by

the integrand on the r i g h t side o f The trace norm o f the o p e r a t o r

IIFtll I

(~)

This makes i t evident t h a t

is a holomorphic f u n c t i o n on

Ft(x )

D .

is given by

= e -Xt IIv>ll l~v~II = e ->,t II#(~)i12 2

I t f o l l o w s by the Plancherel theorem t h a t formly in

(~m) .

g e G and u n i f o r m l y f o r

t

(~)

is a b s o l u t e l y convergent, u n i -

in compact subsets o f

(o,~) .

Hence

f

is the l i m i t of holomorphic f u n c t i o n s , u n i f o r m l y on compact subsets of

f

is holomorphic. C l e a r l y

form is of

e -~t ~(~) .

L2(M ) ,

Remark in

H2(D) ,

H2(D)

and

Wft

is the

L2

Hence by the Plancherel theorem

Wft -~ m in

L2(G )

as

t~o

ft

{ft }

is a bounded subset

Q.E.D.

H~(G)

to functions

via the F o u r i e r i n v e r s i o n formula. To see t h a t we o b t a i n a l l o f If

f e H2(D) ,

then f o r each

s a t i s f i e s the t a n g e n t i a l Cauchy-Riemann equations (since

holomorphic). Hence closed subspace o f such t h a t

,

so

G whose F o u r i e r t r a n s -

By the Theorem, we can pass from f u n c t i o n s in

in t h i s way, we can argue as f o l l o w s :

t > o ,

f u n c t i o n on

D ,

{Wf t}

is a bounded subset of

L2(G ) ,

there is an element

H~(G) .

Since

f

H~(G)

is is a

~ e H~(G) and a subsequence

tk

156 Wftk ÷ m

On the other hand, since

f

weakly

is holomorphic, the Cauchy-Riemann equations give

the r e l a t i o n ~ Wf t = i ~~ Wft Taking Fourier transforms, we conclude that ^

~

(Wft)

^

= -X (Wft)

and hence (Wft)^(~)

= e - x t ~(x)

The proof of the Theorem then shows that We conclude our study of the space terms of the boundary values of e H~(G) ,

we w r i t e

f ,

f

is obtained from

H2(D)

by r e w r i t i n g

eliminating

~(~) = v~ Q w~ , w~ = c(~) ~^ ( ~ ) *

@ by formula

(~). (~)

the Fourier transform.

in Given

by the Lemma. Then

v~ ,

where

c~ = llv~l1-2

Hence we can w r i t e tr(~(g)

#(~)) = c ( ~ ) ( ~ ( g ) v ~ ,

#(~)mv~)

= c(~)(#(~)~(g)v~,

,

vx)

= C(~) I ( ~ ( # - I g ) v ~ ' V~) ~(y) d# . G (The last step is

a-priori

true i f

m also is in

have the formula c(~)(~(g)v~, Hence formula

(~)

v~) = e i~w(g)

of the Theorem can be w r i t t e n as

LI(G). )

But from

§ 2.3 we

157

f(z(g),

w ( g ) + i t ) = c n !~ expE-~t+i~w(y-ig)]

= n!c

(This l a s t i n t e g r a l and

m e L2(G) ,

earlier

m(7)~n d¥ dx

I [ t - i w ( y - l g ) ] -n-1 m(~) d~ . n G

is e a s i l y seen to be a b s o l u t e l y convergent, f o r any

so by Fubini and dominated convergence t h i s j u s t i f i e s

t > o the

steps. )

To w r i t e t h i s formula i n terms o f the holomorphic coordinates on the ambient space

$n+l ,

we c a l c u l a t e t h a t w(-f-lg) = w(g) - w - ~ -

2i z(g) • z - ~

Hence i f we denote the boundary values of

f

on

,

M by

f

a l s o , then we have

the i n t e g r a l formula f ( z , w ) = dn f [ w - w ' - 2 i z . T ~ - n ' l M where

(z ' ,w')

are the coordinates on

Thus the Szeg~ kernel f o r the space S ( p , p ' ) = dn ~ - w ' - 2 i where

p = (z,w) e D and

M ,

H2(D)

e M

dm

,

dn = (-1) n + l n !

and

is the f u n c t i o n

z.z-'] -n-1

p' : ( z ' , w ' )

f(z',w')

,

(2~i) -n-1

158

§ 3

3,1

D

operator (with

C~ c o e f f i c i e n t s )

is said to be h T p o e l l i p t i c

equation

operaitoirs

Fundamental s o l u t i o n s f o r homogeneous h y p q e l l i p t i c

be a d i f f e r e n t i a l that

Hypoelliptic differential

Df = g

on a

C~

i f every d i s t r i b u t i o n

operators

manifold

solution

Let

H f

Recall

to the

satisfies Sing Supp(f) = Sing Supp(g) .

Here

Sing Supp(f)

denotes the siingular support o f

the open set on which

f

is a

f

,

i.e.

the complement o f

C~ f u n c t i o n .

From the p o i n t o f view o f a n a l y s i s on n i l p o t e n t groups, one o f the most. i n t e r e s t i n g examples i s an o p e r a t o r o f the form D=X where

Xo, X I , . . . X n

theorem o f

o

+

n z j=l

X~ , J

are real v e c t o r f i e l d s

on a m a n i f o l d

M .

L. H~rmander t h a t such an o p e r a t o r is h y p o e l l i p t i c

algebra generated by

Xo, X1 . . . . ,X n

This is p r e c i s e l y the i n f i n i t e s i m a l

I t i s a fundamental p r o v i d e d the Lie

spans the tangent space a t each p o i n t o f transitivity

H .

hypothesis t h a t was the s t a r t i n g

p o i n t o f our c o n s t r u c t i o n s in Chapter I I . The f i r s t transitive

step i n our a n a l y s i s o f h y p o e l l i p t i c

L i e algebras o f v e c t o r f i e . l d s w i l l

operators associated with

be to study the corresponding

o p e r a t o r s on a graded n i l p o t e n t group. In f a c t ,

it

s t r u c t u r e and the hypothesis o f h y p o e l l i p t i c i t y

t h a t we need at f i r s t .

Assume § 1.1.1 .

V

is o n l y the graded v e c t o r space

is a r e a l , graded v e c t o r space w i t h d i l a t i o n s

An o p e r a t o r

D on C~(V)

will

D(~o~t) = t~(D~) o B t

{a t } ,

be c a l l e d homoqeneous o f degreie

as in ~

if

159

for all

~ ~ C~(V) .

Theorem

Suppose

homogeneous o f degree Assume t h a t

D

DK = 6 .

~ ,

and i t s

unique d i s t r i b u t i o n

is a

with

K

on

V

d e f i n e d by a f u n c t i o n

supported at

k(x)

{o}

There e x i s t s

are both h y p o e l l i p t i c .

and the c o n d i t i o n which i s

k

DKo = 6

C~

is locally

On t h e subspace

away from

distribution

analysis,

define distributions

~ e Cc(U)

.

If

K

is

and homogeneous o f

> - Q .)

of

D

and i t s

transpose,

imply that

< a}

o f z e r o , which s a t i s f i e s

C~ f u n c t i o n

U) ,

U Ht

Ko

and t o c o n s t r u c t

Of c o u r s e ,

is invariant ,

0 < t 2 1

K

the topology coincide.

Ko ,

i s homogeneous o f degree

on

< Ht , for

~ - Q and s a t i s f i e s

Ko , d e f i n e d on some

t o p o l o g y and t h e

K = Ko - Ho

Since t h e s e t

a

and t h a t no d i s t r i b u t i o n

N = {~ ~ C~(U) : D~ = O} ,

homogeneity p r o p e r t i e s ,

h o(X ) dx

V .

Then t h e r e e x i s t s

imply that

0

integrable,

The i d e a o f t h e p r o o f i s to t a k e the d i s t r i b u t i o n

distribution

which i s

U ;

distribution

particular

~ > o

The h y p o e l l i p t i c i t y

functional

U = {Ixl

on

V

O) .

a distribution

neighborhood

2)

at

(Sketch)

t o g e t h e r w i t h some g e n e r a l

i)

Dt

can be homogeneous o f degree

P r o o f o f Theorem

o p e r a t o r on

which i s homogeneous o f degree

,

(Note t h a t

differential

0 < ~ < Q = homogeneous dimension o f

Hypoellipticity

~ - Q .

C~

transpose

(6 = d e l t a f u n c t i o n

Remark

degree

D

which a p r i o r i

ho ~ N

such t h a t t h e

a - Q (H o

also satisfies

under d i l a t i o n s

{6 t

has no

being t h e DK = 6 .

: 0 < t < I}

,

we can

by

~ > = < KO, ~ > - t - a < Ko, ~ ° ~ 1 / t > , were homogeneous o f degree

~ - Q ,

then

Ht

would be

160 zero.

In any e v e n t , the assumption t h a t

the f a c t t h a t the d e l t a f u n c t i o n

D

i s homogeneous o f degree

i s homogeneous o f degree

-Q ,

~ ,

and

imply that

DHt = a - 6 = 0 . Hence by h y p o e l l i p t i c i t y

of

D ,

Ht

i s o f the form

h t ( x ) dx ,

We want to show t h a t ko(X ) dx ,

l i m h t e x i s t s . Away from O, Ko t+o k o e C~(U~{o}) . The f o r m u l a above f o r Ht

where

with

h t e C~(U)

i s given as can be w r i t t e n

in

p o i n t w i s e terms as

h t ( x ) = ko(X ) - tQ-~ k o ( 6 t x ) , for

x e U ~ {o}

and

o < t < I .

Hence

hs(X ) - hr(X ) = r Q-~ ko(~rX ) - sQ-~ ko(6sX )

= r Q-~ h s / r

if

o < s ~ r < 1 .

Taking

(Since

h t e C~(U) ,

s = r 2 , we o b t a i n the r e c u r s i v e

(6rX)

this

this

relation,

(*)

where

2n-i s k=o

t ~ - ~ Ht

and hence by p r o p e r t y (relative

derivatives

•

r k(Q-~) h r ( a k x) , r

2n

The mapping

C~(U)

x = o.)

we f i n d t h a t

hr ( x ) = n

rn = r

f o r m u l a a l s o holds a t

relation

hr2(X) = r Q-~ h r ( a r X ) + h r ( X )

Iterating

,

2)

is evidently

above the

t~-*

ht

(0,1)

C~(U) .

Hence i f

In p a r t i c u l a r , c < a ,

then

into

D'(U

i s c o n t i n u o u s from ( 0 , I )

t o the t o p o l o g y o f u n i f o r m convergence o f f u n c t i o n s

on compact subsets o f U) .

compact subset o f

map

c o n t i n u o u s from

and t h e i r

{h t : # ~ t ~ ½}

is a

, into

181

sup

Ixl~ 1/4~t~1/2

lht(x) l = C

2n But i f

o < s < 1/4 ,

1/4 } r < 1 / 2 .

By

we can w r i t e (~)

this

sup l h s ( X ) l ]xI~s

Since

~ < Q ,

the serles

sup

for

some i n t e g e r

n

and

g i v e s t h e bound

< C -

z rk(Q-~) k~o

on t h e r i g h t

Iht(x)l

c o n v e r g e s , and we o b t a i n

a u n i f o r m bound

< c

ixi~ E o
Using t h i s

s = r

-

u n i f o r m bound, we f i n d

from the equation for

hs - hr

that

the

by d e f i n i t i o n

of

limit

ho(X ) :

exists,

uniformly

lim hr(X ) r÷o

on compact s u b s e t s o f

U .

Hence t h e l i m i t

Ho = l i m H r r+o

exists

as a d i s t r i b u t i o n .

If m0 ,

we d e f i n e

K = Ko - Ho ,

then

DK = 6 .

Furthermore,

we have

< K, ~ >

f r o m which i t whole p o i n t

is clear

that

=

l i m r -~ ro.~

K

o f t h e argument j u s t

is

<

K mo61 o' /r

>

homogeneous o f d e g r e e

g i v e n was t o p r o v e t h a t

the equation

< K, ~ > = t m < K, ~o~ t > ,

~ - Q this

on

limit

U .

(The

exists.)

Using

162

we e x t e n d

K

u n i q u e l y to a d i s t r i b u t i o n

The uniqueness o f

K

C~ f u n c t i o n s

t h e r e are no

Examples 1.

is clear

on

V

on

V ,

homogeneous o f degree

from t h e h y p o e l l i p t i c i t y

of

which a r e homogeneous o f degree

The most c l a s s i c a l

D ,

m - Q .

since

~ - Q < 0 .

case o f t h e theorem i s t h e L a p l a c e

operator

n =

)2

z

(~/~x i

i=1 on

V = IRn ,

The f u n c t i o n

n > 2 , where the d i l a t i o n s k(x)

in t h i s

llxll

c2 log

[Ixll ,

< Q

,

i s t h e E u c l i d e a n norm.

(When

n = 2

the fundamental

solution

is

which is n o t homogeneous. This shows the n e c e s s i t y o f t h e c o n d i t i o n

in t h e t h e o r e m . )

2.

Another classical

hypoelliptic

A - ~/~x o where

scalar multiplication.

case i s

o n IIxl[ 2-n where

are o r d i n a r y

A

so t h a t

IRn .

Here t h e d i l a t i o n s

(Xo,X 1 . . . . . Xn) = ( t 2 x o, t x I . . . . . t Xn)

Q = n + 2 .

k(x)

In t h i s

i

3. Laplacian

a r e g i v e n by

,

case

c n x o n / 2 exp [-llx[l / 4 x o] ÷2 0

where we have w r i t t e n

IRn+l

,

i s t h e L a p l a c e o p e r a t o r on

6t

o p e r a t o r i s t h e h e a t o p e r a t o r on

12112 = x~

+...+

,

xo > o

~

XO < 0

:,

X2n .

On the H e i s e n b e r g group o f dimension

2n + I

,

t a k e the " s u b e l l i p t i c "

163

A

where

IX i ,

n~> X~ i=l ]

Y~ = 6 i j Z

Q = 2n + 2 , norm

=

+

Y~ I

,

as usual. The fundamental s o l u t i o n f o r

is

Cn n i x i 2 -x

j u s t as in the Euclidean case, f o r a s u i t a b l e choice of homogeneous

Ixl

The corresponding "heat o p e r a t o r " - a/sx o + A

by H~rmander's theorem, as is i t s transpose

4.

A

a/~x

is also h y p o e l l i p t i c ,

+ A .

o

Example 3 can be generalized to any graded n i l p o t e n t group with the

property t h a t the elements of degree one generate

the Lie algebra. Take f o r

A A act

the sum of the squares o f the elements of degree one in some basis, and l e t via the r e g u l a r r e p r e s e n t a t i o n .

3.2

P r i n c i p a l parts of d i f f e r e n t i a l

operators

Let

V

be a graded

n i l p o t e n t Lie algebra. Assume t h a t x

: v

+

L(M)

is a p a r t i a l homomorphism from manifold

H .

Let the spaces

V

i n t o the Lie algebra o f

D0(X)m

of d i f f e r e n t i a l

be defined as in D e f i n i t i o n 3, § 111.5.3 . possibility

operators of

T(V6)

be the tensor algebra over

VC

V extend to automorphisms of T(V6) =

where

T(m)(v~)

(T(m)(v~)

and

m .

V)

The d i l a t i o n s

T(V@) and define a grading

are the elements o f homogeneous degree

ni ,

x-degree

s @ T(~)(V~) m>o

is spanned over

is o f degree

A of

(the f r e e a s s o c i a t i v e non-

commutative algebra generated by the c o m p l e x i f i c a t i o n o f on

x-degree ~ m

In t h i s section we want to explore the

o f assigning a " p r i n c i p a l p a r t " to an operator

Let

{6t }

C~ vector f i e l d s on a

¢

by the formal monomials

n l + . - - + n k = m .)

algebra, the l i n e a r map x

m ,

r e l a t i v e to

VlQ...~v k ,

where

{~t } . vi e V

By the universal property of the tensor

extends uniquely to an algebra homomorphism

164 X : T(V¢) where

÷

O(M)

,

D(M) denotes the algebra of d i f f e r e n t i a l operators with

on H .

Evidently

C~ c o e f f i c i e n t s

x(T(m)(v~)) c DO(X)m .

Definition:

If

A has p r i n c i p a l part

A e DO(X)m , and

~(P)

P e T(m)(v¢) ,

then we shall say that

if

X(P) - A e DO(X)m_l

Remarks

This d e f i n i t i o n is somewhat ad hoc ,

vious short comings. The f i r s t

and suffers from two ob-

is that the p r i n c i p a l part, i f i t e x i s t s , is not

uniquely determined. One could t r y to remedy this by defining the p r i n c i p a l part as an element of the quotient

DO(X)m / DO(X)m_l ,

but this seems quite u n t r a c t i b l e .

The second, more serious, l i m i t a t i o n is that the p r i n c i p a l part a homogeneous polynomial in the vector f i e l d s

X(v)

with constant d o e f f i c i e n t s .

More generally, one would l i k e to allow c o e f f i c i e n t s from with the classical theory of e l l i p t i c

is required to be

C~(M) ,

in analogy

operators. To prove r e g u l a r i t y theorems f o r

such operators, one would need to strengthen the results of

§ 3.1

to obtain

fundamental solutions depending smoothly on parameters. The class of operators in Chapter I I I ,

§ 5.3

would have to be s i m i l a r l y generalized. This can be done in

some cases, but f o r the present exposition we shall ignore such generalizations. Suppose now that operator

x(P)

A e DO(X)m has a p r i n c i p a l part

X(P) .

To the

there corresponds a l e f t - i n v a r i a n t d i f f e r e n t i a l operator on

homogeneous of degree

m.

Namely, the operator

dR(P) ,

P under the extension of the r i g h t regular representation o f the tensor algebra. Of course,

dR(P)

which is the image of dR to a representation

is uniquely determined by the image of

P in the universal enveloping algebra of the Lie algebra

V .

(Recall that the

universal enveloping algebra is canonically isomorphic to the quotient of modulo the ideal generated by the elements

V ,

uQv - v@u - [u,v]

,

with

T(V)

u,v e V .)

165 Let local

p

be a measure on

coordinate

system.

If

M

w i t h smooth, nowhere v a n i s h i n g

~, ~ e CT(H ) ,

< ~,~ > : f m(x) { ( x ) Define the operator

At

m, ~ e CT(H ) .

such t h a t

v t = -v

Lemma part

X(P t )

for

If

d#(x)

by t h e i d e n t i t y

Let

p ÷ pt

A e D0(X)m

a v e C~(H) .

part

Xi

r-step

+ av

this

n ~ i=l

of

T(V{)

x(P)

,

then

At

has p r i n c i p a l

nilpotent x

,

p r o v e s t h e Lemma.

n x i=l

o f t h e form

on

M

and

ai,

b e C~(M) .

L i e a l g e b r a on g e n e r a t o r s by

x(wi)

and the p r i n c i p a l

P =

one sees t h a t

2 Xi + a i Xi + b

are v e c t o r f i e l d s

homomorphism

A e D0(X)2 ,

Iterating

by p a r t s ,

C o n s i d e r an o p e r a t o r

A =

partial

has p r i n c i p a l

Using an i n t e g r a t i o n

Example

free,

be t h e unique a n t i - a u t o m o r p h i s m

v e V .

x(v) t : -x(v)

where the

,

.

Proof

where

i n any

set

< A~,~ > : < ~, At ~ >

for all

density

= Xi

part of

wi @ wi

(cf. A

w I . . . . . wn ,

Chapter II, is

X(P)

Take f o r

,

§ 1.2, where

V

the

and d e f i n e t h e Example). Then

166 In case

r = 1 ,

For any

r ,

If the then

Xi

the o p e r a t o r

the o p e r a t o r and t h e i r

~(P)

dR(P)

is a l i n e a r

isomorphism from

same n o t a t i o n s as i n

Suppose

(Here

.

V

§§ 111.5.3

A e DO(~)m

span the t a n g e n t spaces on

V, M, ~

Let

M ,

be as in

§ 3.2 .

We

~(v) x

onto the t a n g e n t space a t without further

is a d i f f e n t i a l

i)

O<m
2)

the o p e r a t o r s

dR(P)

Q = homogeneous dimension o f

R2

by H~rmander's theorem.

the map

which is homogeneous o f degree

RI ,

r

V ,

x .

We adopt here the

comment.

o p e r a t o r on

Jq which has p r i n c i p a l

We make the f o l l o w i n g assumption on the c o r r e s p o n d i n g o p e r a t o r

Theorem and

~+

on

f o r the same reason.

x e M ,

V

dR(P) ,

is hypoelliptic

C o n s t r u c t i o n o f a par a m e t r i x

assume f u r t h e r m o r e t h a t f o r

I(P)

is the o r d i n a r y Laplace o p e r a t o r on IRn .

commutators up t o l e n g t h

is also hypoelliptic,

3.3

part

dR(P)

Given

of type 1

m on

and

V :

dR(P t )

are h y p o e l l i p t i c

on

V .

V .)

a e C#(X) ,

there exist operators

A 1, A2

of type

m

such t h a t

A1 = a I + R1

A 2 A = a I + R2

Proof by Theorem 3.1 .

Let

kI

be the fundamental s o l u t i o n

Pick a f u n c t i o n

b e C~(X)

so t h a t

and s e t Kl(X,y) : a(x) k I (e(y,x))

b(y)

f o r the o p e r a t o r b = 1

dR(P)

given

on the s u p p o r t o f

a ,

167

Then

K1

i s a kernel

of type

By the r e s u l t s o f follows:

If

~

m ,

§ 3.2 ,

and we l e t

Ty

i s a f u n c t i o n on

f = kI ,

is a differential

X

o f t h e form

o p e r a t o r on

we see t h a t the f i r s t

a

and

~(P)x K I ( x ' Y )

"

on

AI

o f Theorem 2 o f

operator

At

equation,

~(x) = f(o(y,x))

+ (Tyf)(e(y,x))

of order

~ m - 1

,

as

then

, at

v = 0 .

Taking

formula gives a delta function, Hultiplying

kI

and

by the t r u n c a t i o n

§ 111.5.3 .

This proves the f i r s t

the h y p o t h e s e s , by Lemma 3.2 .

R

e q u a t i o n . The

This g i v e s t h e second

Q.E.D.

Suppose the L i e a l g e b r a

degree one. Assume t h a t

f e LP(x)

g e S~(X) .

Proof

Then

Given

is g e n e r a t e d by i t s

elements o f

,

a f e S~+m(X)

a ,

V

satisfies

Af=g

I ,

§ 111.5.3

The same is t r u e f o r the a c t i o n o f the l o w e r o r d e r term

also satisfies

Corollary

where

.

of

b o n l y c o n t r i b u t e s a n o t h e r k e r n e l o f t y p e I i n the c a l c u l a t i o n

of

by v i r t u e

~

term o f t h i s

the second term gives a k e r n e l o f type i functions

be the c o r r e s p o n d i n g o p e r a t o r .

we can e x t e n d f o r m u l a ( I I I )

(X(P)m)(x) = ( d R ( P ) f ) ( e ( y , x ) ) where

A1

for every function

we c o n s t r u c t o p e r a t o r s

A

a e C#(X) .

of type

m and

T

of type

such t h a t

AA=aI+T

Then by C o r o l l a r y 111.5.4 , Iterating

t h i s argument

§ 3.4

we have

T f e Sp .

Hence

a f e S~.

k + m times proves the C o r o l l a r y .

Local r e g u l a r i t y

now we o n l y assume t h a t

P m and A g e Sk+

We c o n t i n u e i n the same c o n t e x t as

§ 3.3 ,

but

168 : V

is a transitive

partial

point of

Let the spaces

M .)

be d e f i n e d as i n

S~,loc (H'l)m

L(M)

homomorphism ( s u r j e c t i v e onto the tangent space a t each DO(X)m o f d i f f e r e n t i a l

§ 111.5.3 ,

Definition

÷

If

Definition

1 < p < ~

consists of all

and

operators of

x-degree 2 m

3 .

m is a non-negative i n t e g e r , then

distributions

f

on

M such t h a t

DO(l) m f CL~oc(M)

(i.e.

if

vi e V

are o f degree

ni

,

and

n1+..-+n k 2 m ,

then the d i s t r i b u t i o n

derivative

l(v I) ...

Remark

l ( V k ) f e L~oc(M ) . )

I t i s c l e a r from the formula f o r the transpose of a vecto~ f i e l d

that this definition

is independent o f the choice o f measure

In o r d e r to apply the r e s u l t s o f to know how the spaces

Sp m,loc

§ 3.3

~

on

M .

to the present c o n t e x t , we need

behave under l i f t i n g s

of transitive

partial

homomorphisms. Fix a point

xo e M ,

w : c~(M)~ where M ,

~

is a sufficiently

and r e c a l l the l i f t i n g

c~(~)

map

,

small open set around

0

in

V .

For any f u n c t i o n on

let Wf(v) : f(eX(V)xo )

Let

A : V ÷ L(~)

f e L#oc(M) ,

be a l i f t i n g

define

Wf

of

~

as c o n s t r u c t e d in Chapter I I

to be the d i s t r i b u t i o n

Wf(v) dv

on

~ .

.

For

169

Lemma I .

There is an open set

P

:

C~(~) + C~(Mo)

y

:

v÷c=(a)

Ho

around

xo ,

a s u r j e c t i v e l i n e a r map

and a l i n e a r map ,

such t h a t (1)

< Wf, ~ > = < f , Pm >

(2)

~(v)

P~

=

Here

f e L#oc(H), ~ e C#(~)

over

~

and

H ,

Remark

P(A(v)

m + y(V)m)

and the p a i r i n g in

(i)

is given by i n t e g r a t i o n

respectively.

Recall t h a t in the case t h a t

W is the map l i f t i n g

~

is an exact homomorphism, then

functions from a homogeneous space to the group, The o p e r a t o r

P is " i n t e g r a t i o n over the cosets" in t h i s case, and is a standard tool in harmonic a n a l y s i s .

Proof of Lemma 1. coordinates given in is a f u n c t i o n o f and view

x' If

We use the d e s c r i p t i o n of the range of

§ 11.2.4 :

There are coordinates

x' = (x I . . . . . Xm)

only

as g i v i n g coordinates on f e L#oc(H )

and

(m = dim H) .

{xj} Set

on

W in local ~

so t h a t

x" = (Xm+I . . . . . Xn) ,

g o.

m e C#(~) ,

then w i t h t h i s i d e n t i f i c a t i o n

we can

write < Wf' ~ > = H

Here J>O

dx'

and

dx"

f(x')

~(x~x") J(x~x") dx' dx"

denote Lebesgue measure in the respective coordinates, and

is a s u i t a b l e change of measure f a c t o r . Thus i f we define P~(x') = f ~(x',x") J(x',x") dx"

,

Wf

170 then c o n d i t i o n (1) i s o b v i o u s l y s a t i s f i e d . t a k i n g f u n c t i o n s o f the form To v e r i f y

The s u r j e c t i v i t y

of

P

i s e v i d e n t on

m l ( X ' ) ~2(x") .

( 2 ) , we express the v e c t o r f i e l d s

z(v)

and

A(v)

in these

coordinates in the form

~(v) : a ( x ' )

~/~x'

A(v) : a ( x ' )

~/~x' + b ( x ' , x " )

(such an expression is e q u i v a l e n t to the r e l a t i o n by p a r t s in the i n t e g r a l

for

P(A(V)~)

,

~l~x"

A(v) W = W ~ ( v ) )

we o b t a i n r e l a t i o n

.

Integrating

(2), finishing

the

proof. If on

f

is a distribution

Q by formula ( I ) .

Then

on

on

d e n s i t y f a c t o r s in the measures on and

A(V)

Q , H

since and

Q ,

on

is surjective.

the i n t e r t w i n i n g

A~o i n j e c t i v e l y Because o f the r e l a t i o n between

a v e C=(~o )

Let

f

be a d i s t r i b u t i o n

f e

Proof f

g a L~oc(Mo)

on

and

M° .

Then

f

The Lemma i s c l e a r l y true when ~(v) f

are both l o c a l l y

in

m= o , Lp .

~ ) ~ ~ Cc(

°

~(v) P~ > : < g, P~ > Using r e l a t i o n

(3)

by d e f i n i t i o n

of

Wf .

Then t h e r e is a f u n c t i o n

such t h a t < f,

for all

P

as a d i s t r i b u t i o n

W z(v) = (A(v) + av)

Lemma 2

Suppose

Wf

now becomes

(3) where

we now d e f i n e

W maps the space o f d i s t r i b u t i o n s

i n t o the space o f d i s t r i b u t i o n s

~(v)

H O

we f i n d t h a t

171

< W f , A(V)~ > : < W g - a v, ~ > which shows t h a t implication

~

A(v)W f e L~oc(~ ) .

t h e r e is a f u n c t i o n

f e k~oc(Mo)

G e L~oc(Q )

< f, ~ e C#(~) .

taking

m o f the form < f, g e L~oc(Ho) .

A(v)W f e L~oc(~ ) ,

m l ( X ' ) m2(x") ,

P

~(v)~ > = < g,~ > ,

if

homomorphism from

§ 1.2).

V

"principal

part"

A

~(P) ,

where

g e S~,loc(H,~ ) ~

Proof point

xo

Then i f ,

it

P

A

follows that

The r e s u l t i s l o c a l ,

satisfies

on

of

~

~ .

Theorem" o f

which is also a

This allows us to t r a n s f e r

o p e r a t o r on

operators

satisfies

H ,

.

V):

such t h a t

dR(P)

the equation

f e S~+m,loc(M,~ ) ~

A

has a

m ,

0 < m< Q .

and

dR(P t )

Af=g , (Here

are

where i < p < ~ .)

so we may work in a neighborhood o f a f i x e d

Theorem can be a p p l i e d . Since

t e n t Lie a l g e b r a , there e x i s t s a p a r t i a l Thus

A

V

g e n e r a t o r s , w i t h i t s standard

is homogeneous o f degree

differential

f e L~oc(H )

on which the L i f t i n g

n

(Q=homogeneous dimension o f

be a d i f f e r e n t i a l

Assume t h a t the l e f t - i n v a r i a n t hypoelliptic.

and

,

In t h i s case the " L i f t i n g

to v e c t o r f i e l d s

the r e s u l t s o f § 3.3, as f o l l o w s

Let

~ e C~(Ho)

to the main r e s u l t o f t h i s s e c t i o n . Assume now t h a t

Chapter I I ,

Theorem

(x',x")

By i n d u c t i o n t h i s f i n i s h e s the p r o o f .

Chapter I I asserts t h a t there e x i s t s a l i f t i n g partial

in l o c a l c o o r d i n a t e s

we f i n d by H~Ider's i n e q u a l i t y t h a t

r - s t e p n i l p o t e n t Lie algebra on

(cf.

then (3) shows t h a t

such t h a t

Using the formula f o r

We turn f i n a l l y

gradation

and

~(v) P~ > = < G,~ >

for all

is the f r e e ,

t h i s argument gives the

.

Conversely, i f

where

Iterating

,

V

is a free nilpo-

homomorphism A which is a l i f t i n g

the hypotheses o f § 3.3 .

of

~.

172

By r e l a t i o n : Wf

satisfies

(3)

o f t h i s s e c t i o n , we f i n d t h a t the d i s t r i b u t i o n

an equation o f the form Df = g

where

g = Wg ,

and

D

,

is an o p e r a t o r w i t h p r i n c i p a l

part

A(P) .

By Lemma 2

and the C o r o l l a r y to the Theorem o f § 3.3, we conclude t h a t

e S~+m,loc(~,A ) •

Hence

f e S~+m,loc(H,x ) ,

ExamI ] ~

Let

A

Q.E.D.

be the o p e r a t o r considered in the example at the end

o f § 3.2, and assume t h a t the v e c t o r f i e l d s commutators up to l e n g t h Assume also t h a t e i t h e r n > 2 ,

A .

n

dim H > 2 ,

generators s a t i s f i e s

For example, i f

and t h e i r

s u f f i c e to span the tangent space a t each p o i n t o f o r else

and the homogeneous dimension

algebra on to

r

{X i : 1 ~ i < n}

Q of the f r e e ,

Q> 2 ,

f e L~oc(M )

dim H = 2

and

r > 1 .

Then

r-step nilpotent Lie

so the theorem j u s t proved a p p l i e s

i s such t h a t

Af e L~oc(H ) ,

then we can

conclude t h a t

Xif

also, for nilpotent

i < i, j ~ n .

,

X i X j f e L~oc(H)

Note t h a t t h i s r e s u l t i n v o l v e s no e x p l i c i t

groups o r s i n g u l a r i n t e g r a l o p e r a t o r s .

mention o f

M.

173 Comments and references for Chapter IV

§ 1

We have largely followed Knapp-Stein [1], to which we refer the reader

for more examples and references. The l i t e r a t u r e on intertwining operators for semi-simple groups is extensive. For the basic analytic properties of the " i n t e r twining i n t e g r a l " , cf. Kunze-Stein LFI] and Schiffmann [1]. For recent developments, we c i t e Johnson-Wallach [1], Helgason [2], Knapp-Stein ~ ] ,

[3]. See also Stain's

survey t a l k LF1], Wallach [1], and Warner [ I ] .

§ 2.1

The use of non-commutative Fourier analysis on the Heisenberg group

to study the space

H2(D)

was f i r s t

done by Ogden and V~gi [1], who consider the

general "Siegel domain of type I I " . For a survey of the unitary representation theory of nilpotent groups, cf. Moore [1]. The Ca regularity theory is treated; e.g. in Goodman [1], [2],

§ 2.2 of

[4], Poulsen [1], and Cartier [1], ~ ] .

The Theorem is due to Ogden-V~gi [1]. Our proof, using the behaviour

Ca vectors under direct integral decomposition, is s l i g h t l y d i f f e r e n t .

"tangential Cauchy-Riemann equations" can be written more i n t r i n s i c a l l y of the operator

The

in terms

~b ; cf. Folland-Kohn [1].

§ 2.3

The Theorem is due to Kor~nyi-V~gi LFI]; cf. Kor~nyi-V~gi-Welland [ i ] .

§ 2.4

For the general theory of the Szeg~ kernel of a domain, cf.

Gindikin [I] and Stein [2]. For the Szeg~J kernel in the case of Siegel domains, cf. Kor~nyi [ i ] . The construction of the Szeg~J kernel given here, using the Fourier analysis on the Heisenberg group, avoids the problem of proving a priori that functions in

H2(D)

have a boundary integral representation. For connections

between Szeg~J kernels and representations of semi-simple groups, cf. Knapp L1].

§ 3.1

For the h y p o e l l i p t i c i t y

of second-order operators, cf. H~rmander ~ ]

174 and Kohn [ ~ ,

[2]. The Theorem and i t s proof are taken from Folland [2]. The

results from functional analysis cited in the proof can be found in TrOves [1], § 52. The heat equation, r e l a t i v e to the s u b e l l i p t i c Laplacian in examples 3 and 4, has been studied by Hulanicki [ i ] , Folland [2], and J6rgensen [1]. The fundamental solution for the s u b e l l i p t i c Laplacian on the Heisenberg group was calculated by Folland [ i ] .

Gaveau [ ~

has used stochastic integrals to calculate fundamental

solutions on two-step nilpotent groups. Rockland [I] has shown that on the Heisenberg group, a homogeneous, l e f t - i n v a r i a n t d i f f e r e n t i a l operator is h y p o e l l i p t i c , provided i t s image in every n o n - t r i v i a l irreducible unitary representation has a bounded inverse. See Gru~in [1], ~ ] for examples of h y p o e l l i p t i c operators with polynomial c o e f f i c i e n t s .

§ 3.2

For the relations between tensor algebras, universal enveloping

algebras, and l e f t - i n v a r i a n t d i f f e r e n t i a l operators on a Lie group, cf. Helgason [ i ] . The study of operators whose principal part is a polynomial in the vector fields

~(v)

with variable coefficients has been treated, in special cases, by

Rothschild-Stein [1].

§§ 3.3-3.4

These results are due to Rothschild-Stein [ ~ ,

extending s i m i l a r

results of Folland-Stein ~ ] . The proofs given here are somewhat d i f f e r e n t , since we have again t r i e d to emphasize the s i m i l a r i t y with the case of functions on a homogeneous space, as in Chapter I I . For the classical e l l i p t i c

r e g u l a r i t y theory we refer to Bers-John-Schechter

LI]. For applications of the r e g u l a r i t y results here to the study of the

~b

operator, we refer to Folland-Stein ~ ] and Rothschild-Stein [1]. For e l l i p t i c r e g u l a r i t y in the context of unitary representation theory, cf. Goodman [4].

Appendix Generalized Jonqui#res Groups

In Chapter I we r e s t r i c t e d our a t t e n t i o n to the group of automorphisms of the ring

P of polynomial functions, and we embedded every simply-connected n i l p o t e n t

Lie group as a subgroup of such a group. For geometric reasons i t is desirable to consider a larger group, namely the group of automorphisms of the f i e l d of r a t i o n a l functions (the Cremona group). In this appendix we want to construct certain ( f i n i t e dimensional) Lie subgroups of the Cremona group, extending the constructions of

§ 1.1.

In f a c t , the construction works over any f i e l d of

c h a r a c t e r i s t i c zero. We w i l l assume in these notes that the c o e f f i c i e n t f i e l d is the complex numbers, however, since we have systematically avoided any mention of algebraic groups up to t h i s point. The f i r s t

step in t h i s analysis is to replace the one-parameter d i l a t i o n

group on the vector space

V by an

n-parameter d i l a t i o n group

The generators of this group span a commutative subalgebra a d j o i n t action of In

§ A.1

h on

Der(P)

(n = dim V).

h c Der(P) ,

and the

is diagonalizable.

we study the vector f i e l d s with polynomial c o e f f i c i e n t s which are

eigenvectors f o r

ad(h) .

dimensional subalgebras of t h e i r structure in

§ A.3 .

In

§ A.2 ,

Der(P) ,

we construct a family of maximal f i n i t e each of which contains

h ,

and determine

(The n i l p o t e n t algebras studied in Chapter I occur

as subalgebras of these maximal algebras.)

In order to achieve

maximality, we

have to include vector f i e l d s which generate b i r a t i o n a l (but not everywheredefined) transformations of

V .

In

§ A.4

we construct groups of b i r a t i o n a l

transformations corresponding to these maximal subalgebras.

176

A.1

Let

P

Root space d e c o m p o s i t i o n o f Der(P)

be the a l g e b r a o f p o l y n o m i a l f u n c t i o n s

complex v e c t o r space V~ ,

1 < i < d .

Fix a basis

{x i }

Thus the monomials

the v e c t o r f i e l d s = Der(P)

V .

{~a~ i

(~i = Dx i )

for

on a f i n i t e - d i m e n s i o n a l V

and dual basis

{ a : a e INd}

: a e ~ d , 1 < i < d}

{~i }

for

P ,

and

are a basis for

are a b a s i s f o r the L i e a l g e b r a

"

Consider the commutative s u b a l g e b r a

The a c t i o n o f

h

on

=

span { ~ i ~ i

P

is d i a g o n a l i z a b l e .

Hi ~a

=

Hence i f

we d e f i n e l i n e a r

and s e t

M = {Ua : a e ~d}

:

1 < i < d}

Indeed, i f we w r i t e

Hi = ~ i ~ i

,

then

ai ~a

functions ,

Sa '

a e INd ,

on

_h by

~a ( H i ) = ai

then

P

=

z H

(~ e M)

ff

: {f e P : H f = p(H)f

where

Thus

dim H

= i

,

and

H

has b a s i s

The a c t i o n o f

adh

on

g

(a

(H e £ ) } if

.

p = ~_

can a l s o be d i a g o n a l i z e d .

Indeed, we have the

commutation r e l a t i o n s

(I)

Bali , ~j~j]

(6ij

= Kronecker d e l t a ) .

~a,i

(Hj) = 6ij

- aj

,

_9. =

Hence i f and s e t

Z gx

=

(6ij

- aj)

~a~ i

we d e f i n e the l i n e a r L = {~a,i

functions

: 1 <_ i _< d, a e ~d}

(~, e L) ,

~a,i ,

then

on

h

by

177 where ~_~ = {X e_g

:

From the Jacobi i d e n t i t y

IX,HI

= ~(H)X}

and the formula

H(Xf) : XHf + D , ~ f

,

we o b t a i n

~

is

the r e l a t i o n s

(2)

[g-K ' - g ~ ]

g-K+~

~

~ H _~ H_x (Define

H

= 0 ,

~

= 0

I t is e v i d e n t t h a t itself,

so t h a t

h

if

~ ~ M or

h = ~o

~ ~ L.)

and t h a t the n o r m a l i z e r of

is a Cartan subalgebra o f

g .

h

in

The r o o t spaces

gx ,

~ # 0 ,

are o f two b a s i c types:

dim

Proposition

If

(I)

For a l l

i ,

g~=

d

(II)

There i s an index

and

~

~ e L ,

In case

= H_~ . i

such t h a t

i # j

.

(I)

,

there are (II),

vector fields

~aa i

Definition

(I)

the p r o p o s i t i o n .

(i)

choices o f

are a l l

We w i l l

call

linearly

write

while

dim~

that a e INd

satisfies

if

elementary

if

i s a non-

(I) or (II).

and

i

so t h a t

a,i

.

The corresponding

independent and span

~ < 0

x(Hj)

= 1 .

~ e L

t h e r e is a unique choice o f

positive integer for all We w i l l

d

~(Hi) = 1 ,

In t h i s case

I t i s c l e a r from formula

w h i l e in case

(II)

i s a n o n - p o s i t i v e i n t e g e r . In t h i s case

~(Hi)

positive integer for all Proof

then one o f the f o l l o w i n g s i t u a t i o n s occurs:

~ # 0

and

~

,

~(Hi)

x = Xa,i

'

Q.E.D.

is a non-

i . ~ e L

it

satisfies

condition

(II)

of

178

Remarks

The d i c h o t o m y d e s c r i b e d

geometric properties

1.

Suppose

Furthermore,

of the vector fields

~ = ~a,i

since

is independent of

i n the p r o p o s i t i o n

~(Hi) ~i

is elementary. = i

'

in

,

this

so t h a t

g~ ,

Then

forces

is r e f l e c t e d

in the

as f o l l o w s :

~

has b a s i s

ai = 0 .

X~ = ~a~ i

.

~a

Hence t h e f u n c t i o n

t h e f l o w g e n e r a t e d by

X~

is

I ~i ÷ ~i + tCa ~j÷~j We s h a l l

P

call

this

, j~i

an e l e m e n t a r y f l o w ,

as e l e m e n t a r y automorphisms.

and r e f e r

t o the induced automorphfsms o f

(More g e n e r a l l y ,

an automorphism o f

P

defined

by S

~i ÷ c~i + f ( ~ l . . . . . ~j ÷ ~j

where

f 2.

,

j ~ i

i s a p o l y n o m i a l and Suppose

x = 0 .

~i-1' ,

c # 0 ,

Then

~i+1 . . . . . ~d )

is called

g-o = ~ '

a Jonqui#res

transformation.)

and the v e c t o r f i e l d

z c i Hi

generates

the f l o w

Ci ÷ ~i exp(tci) 3.

Suppose

x < 0

automorphism o f

P .

and

~ 0 .

ai = -~(Hi)

and

X e ~

To see t h i s ,

,

X # 0 .

use t h e f a c t

Then f o r any

b elN d ,

Then that

X

does n o t g e n e r a t e an

X = ~aH ,

one c ~ I c u l a t e s

where easily

H e g-o that

Xn (~b) = Cn ~na + b

where

c n = B(H)(B(H)

linear

forms on

that

corresponding

-B(H) / m(H) ~ ~ .

not act locally define

h

+ m(H))-,.(B(H) to

+ (n-1)m(H)) a , b e INd .

This c h o i c e makes

nilpotently

on

an automorphism o f

P .

P ,

.

Here

m,B e M

are t h e

But we can always p i c k

cn # 0

and the f o r m a l

for

all

series

n .

Hence

z(i/N!)× n

b

so

X

does

does n o t

179

4.

When ~ < 0 ,

it

is p o s s i b l e f o r c e r t a i n vector f i e l d s in

~X

generate automorphisms o f l a r g e r algebras o f f u n c t i o n s than the algebra

to P .

The

simplest example, perhaps, is obtained by taking

X = CaH as above and r e q u i r i n g

that

The c a l c u l a t i o n above then

H be orthogonal to

~ ,

i.e.

~(H) = 0 .

shows t h a t z(Z/n!)xn~b = ~b exp [B(H)~ a] .

Hence

X generates an automorphism o f the algebra o f e n t i r e functions

series with i n f i n i t e 5.

radius of convergence) in t h i s case.

When the local f l o w defined by

X e~

is r a t i o n a l ,

a one-parameter group of automorphisms o f the f i e l d example, l e t

(= power

× = ¢i H ,

then

X

generates

R of r a t i o n a l f u n c t i o n s . For

where

H = z ciH i The local f l o w generated by

,

X

cI

= i

is obtained by s o l v i n g the system of d i f f e r e n t i a l

equations

i

x~ ( t ) = C k X l ( t ) x k ( t )

x k (o)

~k

One f i n d s t h a t -c k x k ( t ) = ~k ( l - t ~ l ) Hence i f

ck e Z

,

then the map

~k ÷ Xk

is r a t i o n a l f o r each

r a t i o n a l inverse -c k ~k = Xk ( l + t X l ) Hence

exp t X

is defined f o r a l l

t

as an automorphism o f

R .

t ,

with

180

A.2

Maximal f i n i t e - d i m e n s i o n a l

su b a l g e b r a s .

Having s t u d i e d v a r i o u s o n e - p a r a m e t e r groups o f automorphisms generated by v e c t o r fields

in the r o o t spaces

g~ ,

we t u r n t o the problem o f f i n d i n g

finite-dimen-

s i o n a l L i e groups o f automorphisms o f the p o l y n o m i a l a l g e b r a , o r o f the a l g e b r a of rational

functions.

subalgebras

m

As a f i r s t

s t e p , we s h a l l

o f the L i e a l g e b r a

~

look for finite

dimensional

which are maximal i n the f o l l o w i n g

two

senses: (i)

m ~h

(2)

m

We w i l l

,

so t h a t rank (m) = dim V .

is n o t c o n t a i n e d in any l a r g e r f i n i t e - d i m e n s i o n a l c o n s t r u c t an i n f i n i t e

Choose an element

H e ~

where the

ni

are p o s i t i v e

g

integral

(H i = ~ i ~ / ~ i

i n t e g e r s and

)

in the sense t h a t

,

nI = 1 .

By permuting the basis

Hi

we a r r a n g e t h a t 1 . . . n. .I. where

1 ~ q ~ d

nq

i s a d m i t t e d . Then

<

nq+ 1 H

_ < - - . <_ n d = r

,

g e n e r a t e s the o n e - p a r a m e t e r group o f

linear transformations ni at x i = t Let

V = VI Q---Q Vr

xi

Let

Vk = span {~i

hyperplane in taking all

~m .

,

(t # 0).

be the a s s o c i a t e d d e c o m p o s i t i o n o f

Vk = span {x i : ni = k} We s h a l l

.

f a m i l y o f such maximal a l g e b r a s as f o l l o w s :

which i s p o s i t i v e

H = ~ niH i

subalgebra o f

V ,

where

: n i = k}

be the dual space. The element construct a finite-dimensional

the r o o t spaces on o r above t h i s

H

also defines a

algebra

hyperplane, together with

dimensional r o o t spaces l y i n g one step below the h y p e r p l a n e .

~

by q

one-

,

181 In more d e t a i l ,

define

k_ = E {~_~ : < ~,H > = k}

I~ Theorem Der(P)

{~H : ~ e V}~=

m = n

+

z n_k k>o

i s a maximal f i n i t e - d i m e n s i o n a l

subalgebra of

.

Proof

Since

[_g_~, g_.~] _~ g__~+ ,

I n k, n j ] Furthermore,

H(~i) = n i

g

,

it

is clear that

nk+ j

so t h a t

n

c

n_l •

Hence ~_,nk~ To show t h a t

m

G

nk_ I

i s a L i e a l g e b r a , we must prove t h a t

In-, n ] c We note f i r s t (~jH)

with

nka k = n i for all

k

that nj

=

[h, n_~ 1

.

If

with

This vanishes unless since

Since

nk

n__

~ # 0 ,

i s the sum o f the r o o t spaces

then

is increasing

We c a l c u l a t e

~jH] = ~ i j

i = j,

g_~ has b a s i s in

k ,

this

~a~ i

,

forces

where ak = 0

the commutator to be

~aH

and in t h i s case

H ( ~ j ) = ~j , [n_,n_]

Each space

,

since

~a = ~k

f o r some

k

between

n. = 1 . J

The f a c t t h a t

This shows t h a t

n_ ,

g_~c ~

nk > ni .

[~a~ i ,

1 and q ,

~

ai = 0 .

and

n_

m ~k

1 < j <_ q ,

gives

: 0 .

is a Lie subalgebra of is certainly

Der(P).

finite-dimensional.

Indeed,

~k

for

k > o

is

182

spanned by t h e e l e m e n t a r y v e c t o r f i e l d s

s njaj For each

(k,

i)

equation.

Furthermore,

(aa i

with

ai = 0

and

number o f

a e INd

satisfying

= ni - k .

t h e r e are o n l y a f i n i t e since

ni ~ r

,

we have no s o l u t i o n s

if

this

k > r .

This

shows t h a t nk=O,

k>r, that

and completes the p r o o f

It

subalgebra

h ,

m .

Y ~ m ,

it

then

would be t h e sum o f i t s

X and Y

X = ~i H .

< ~, H > = -1 .

subalgebra of

intersections

g

~ .

which

is maximal, i t

suffices

w i t h the r o o t spaces

t o show t h a t i f

do n o t g e n e r a t e a f i n i t e - d i m e n s i o n a l

Since If

~

Y ~ m ,

X and Y

would thus have t o have in

algebra containing

Since any such s u b a l g e b r a would a l s o c o n t a i n t h e Cartan

Hence t o prove t h a t

where

is a finite-dimensional

remains t o show t h a t t h e r e i s no f i n i t e - d i m e n s i o n a l

properly contains

g~

m

we have

< ~, H >

~ - I

f o r some

n ,

but

Lie subalgebra of .

Also

generated a finite-dimensional

(adX) n (Y) = 0

Y e g~

X e ~

,

g , where

L i e a l g e b r a , we

since this

element l i e s

~+nu Since

g~+n

~m

f o r any

we need o n l y p r o v e t h e f o l l o w i n g

n ~ 1 ,

lemma

to c o m p l e t e the p r o o f o f t h e theorem:

Lemma

If

Proof.

There are two cases to c o n s i d e r .

vector field, calculate

Y e ~

then

Y ~ m

Y ~ m ,

then

means t h a t

~X,~

If

# 0 ,

Y = ~a~ i

where

X = ~i H .

i s an e l e m e n t a r y

z nka k = n i + m where

m > 0 .

We

t h e commutator [Y,X]

which i s c l e a r l y Y = ~az ,

but

~a H

: 6ii

not zero.

where

[y,~

If

Z = z CkH k

:

~a

- m ~l~a~i Y

,

i s n o t an e l e m e n t a r y v e c t o r f i e l d ,

and

(ClH _ bZ)

a # 0 .

,

In t h i s

case

then

183

where

b = z akn k .

This commutator can vanish o n l y i f

t h i s would i m p l y tha~

c I # 0 ( s i n c e b # O) ,

o n l y p o s s i b l e when t h e r e e x i s t s In t h i s

case we would have

Remark

( t # O)

on

n:

j ~ q

Y e m ,

The element

xi + t nix i

a

H

and hence

such t h a t

b = I .

aj = I

In p a r t i c u l a r , But t h i s

and

ak = 0 ,

is k # j

p r o v i n g the lemma.

generates the o n e - p a r a m e t e r group o f d i l a t i o n s

~R

z

ClH = bZ .

(= span~ { x i } )

~k

.

The s u b a l g e b r a

cm

k>l i s the c o m p l e x i f i c a t i o n was s t u d i e d in Example HI = xB x

§ 1.3 Let

and

o f the a l g e b r a a s s o c i a t e d w i t h t h i s d i l a t i o n o f Chapter I .

n = 2 ,

H2 = y~y

(X(H1), x(H2) )

and w r i t e Identify

in the p l a n e . Then {(1,-n),

(-m,1),

x = ~1 '

the r o o t s L

Y = ~2 "

X e L

consists of all

(-m,-n)

:

y

(-m,1)

:

xm~y

(-m,-n)

:

xm+lyn~ x ,

n ~ 1 ,

the a l g e b r a d e f i n e d by as f o l l o w s

(cf.

n

(l,-n)

H

h

has b a s i s point

points

: m,n e IN} . vector fields:

x

set

xmyn+l~y .

H = H1 + nH2 = x~ x + ny~y ,

as in the theorem. Then the r o o t s o f

Fig I ) : Roots

n = 1:

Then

w i t h the i n t e g r a l

The c o r r e s p o n d i n g r o o t spaces are spanned by the f o l l o w i n g

Fix an i n t e g e r

group which

Root

Vectors

I (-1,0) ,

(1,0)

xH ,

~x

(-1,1)

,

(i,-i)

X~y ,

Y~x

(0,-1) ,

(0,1)

yH ,

~y

and l e t h

on

m m

be are

.

184 n=3

n=2

n=l

" ""

x

3h - x\ h ~ -xh - -

.

o = elementary r o o t spaces

.

.

•

.

xy2h

h

•

y2h

\

xO~x

~

o,,2~ ~ J~x

• = non-elementary root spaces 0

y3h_

0

Fig 1

Root spaces f o r

Der(P)

(P = polynomial algebra in two v a r i a b l e s )

Y2~ x

"

185

(-I,0)

,

(1,0)

(-k,l)

,

o~k~n

xH

,

~x

n > I:

Note t h a t One v e r i f i e s

When

in the case

that

n = 1 ,

m = s/(3)

n > 1 ,

m

xk~y , o~k~n

the n e g a t i v e o f e v e r y r o o t

in this

case

is the semi-direct

:

(cf.

§ A.3)

is again a root.

.

sum o f

(xH) ~ h ~ (~x)

and = span { x k ~ y :

Here

~

i s t h e sum o f

is also a root.

h

and the r o o t spaces

One v e r i f i e s

the

sl(2)

is not a root. via the

The i d e a l ~

(n+l)

ad(H2) = - I

~

~

A.3

(cf.

Structure Der P

in

H

functions

on

such t h a t

-~

,

xH, ~ x '

and

h+H1,

i s the sum o f t h e r o o t spaces

m~

irreducible

that

ad(r)

of

s/(2)

representation

gl(1)

factor

such t h a t acts on ,

u

while

.

Let

m

be the maximal f i n i t e - d i m e n s i o n a l

§ A . 2 , d e f i n e d by a c h o i c e o f

,

1 = n I <_ n 2 <_ . . .

H e h ,

where

<_ n d = r .

determines a decomposition

: n i = k}) V .

~

the

of m

V = V1 (~-.-(~ V r (V k = span { x i

those

while

§ A.3)

H = ~ niH i This choice of

for

i s c o m m u t a t i v e , and one c a l c u l a t e s

- dimensional

on

subalgebra of

~ g/(1)

p a r t spanned by

spanned by H2 .

m~

that

r = sl(2)

with

o~k~n} .

,

, and d e t e r m i n e s a g r a d a t i o n

{H n}

o f the p o l y n o m i a l

is

186

We have Hn = Z HX

Since

hcm

,

(< H,X > = n)

we have a r o o t - s p a c e

m = h + ~ mX

where

mx = m v ~ _ x

h .

'

and

By t h e c o n s t r u c t i o n

,

decomposition

(X e A)

A = A(H)

is a finite

m ,

is evident

of

it

A = {X e L : < H,X > >_ O} t W { - ~ j

where

xj(Hi)

= ~ij

and

m

q = dim VI

=_g}<

We w a n t t o s e p a r a t e Lie algebra.

For this

it

m

if

.

s e t o f non - z e r o l i n e a r that

: i < j

>

< q}

,

0

a semi-direct

is natural

f or ms on

In p a r t i c u l a r ,

< H,X >

into

.

sum o f a r e d u c t i v e

and a n i l p o t e n t

to examine the set

Ar = A ~ ( - A ) .

Since

< H,X > ~ -1

,

every root

in

Ar

Consider first

the case

Lemma 1.

~o = h_+ E {-gx : X e A r

Set

where t h e i s o m o r p h i s m i s d e f i n e d decomposition

Proof = ni

< H,X > = 0

or ~ 1 .

< H,X > = 0 :

r r ~ s ~ gZ (Vk) -o k=l

z njaj

must s a t i s f y

and

< H,X > = O} .

Then

,

by t h e a c t i o n

of

g# (V k)

on

~x

has b a s i s

P

i n d u c e d by t h e

V = z Vk .

If and

< H,X > = 0 , ai = 0 .

If

X ~ 0 ,

then

-x e A

also,

and hence ~a = ~j

(n i = n j )

.

this

forces

~a

~a~ i

,

where

t o be l i n e a r ,

187

The v e c t o r f i e l d s ~o

{~iaj

: n i = n j = k}

is t h e sum o f t h e s e s u b a l g e b r a s ,

N e x t , we o b s e r v e t h a t

span t h e L i e a l g e b r a o f

gZ(Vk)

,. and

Q.E.D.

the commutative subalgebra

n__+= P l = {Dx : x e V 1} is isomorphic to

VI

as an

~

module. The c o n t r a g r e d i e n t

module i s i s o m o r p h i c t o

n_=VlH

Lemma 2

Set

r = h + z mx

(x e Ar)

Then r = n_ @ r o @ n

+

One has the commutation r e l a t i o n s

I

=

=0

In, = Proof.

It

is clear

r o o t s spaces o f t h e dient

h-module,

with

x e Ar

it

< H,X > = + i

from t h e remarks above t h a t with

must t h e r e f o r e .

< H,~ > = -1 .

n_

Since

i s t h e sum o f t h e n+

is the contragre-

be t h e sum o f the r o o t spaces o f t h e

This e x h a u s t s t h e p o s s i b i l i t i e s

for

X e Ar .

x e Ar

Hence

r = n_ @~o @~+

Using t h e d e s c r i p t i o n relations. x e V1

and

In p a r t i c u l a r , ~ e VI

,

of

-[o i n lemma 1, we e a s i l y

we n o t e t h a t

[D x , ~

we have

F-Dx , ~H] = < x , ~ > H + ~Dx

Remark

= Dx

One e a s i l y

verifies

that

e

h

(q = dim V1) :

verify if

t h e commutation

x e VI .

Hence f o r

188

L ~ sZ(q+l) and hence

~

A e gZ(V1),

is reductive. x e VI ,

Dx - ~H - z a i j

e gZ(V2)

(View

~ e V~ .

sZ(q+l)

Such a m a t r i x

~i @j - t r ( A ) H

,

To c o m p l e t e o u r a n a l y s i s

if

gZ(Vr)

~...@

(aij)

as the m a t r i c e s

(~ _ErA )

,

where

corresponds to the vector field is the matrix of

of the Lie algebra

Au = A ~ Ar

,

m ,

A

on

VI

.)

we s e t

,

and d e f i n e =

Z

~

XeA u Since

< H,k > ~ 0

for all

U

=

x e Au ,

Z -~k k>o

--

we have

'

where ~k = :C {g_~ : ,1 e Au

It

is clear

from lemma 2

and t h e d e s c r i p t i o n

of

~1 Thus i t

that n+ ,

=

Lemma 3

Define

~k = ~k

if

< H,~ > = k}

k > 2

Furthermore,

.

by

§ 1.3

we see t h a t

{X e ~1 : X

remains t o d e s c r i b e

and

vanishes at

O}

~o •

H° = 0 o

Hm ° = z HjH k

and

( j + k = m , j<m, k<m)

Then u 0 : {X e Der(P)

Proof

Denote by

w

: X HmC_ H°m f o r a l l

the set of vector

fields

X

m}

such t h a t

X Hm c H° -

for all

m .

is obviously

a Lie algebra.

Since

m

189

Hk = { f

it

is clear that

under

adh , Let

Hk

o Hk

and

,

are invariant

under

~ .

Hence

w

is invariant

and i s thus t h e sum o f r o o t spaces.

X = ~aa i

be a r o o t v e c t o r in

w .

to the gradation

{H n}

degree z e r o , r e l a t i v e

E njaj We c l a i m t h a t would have

e P : Hf = k f }

aj = 0

~a = ~j

Conversely,

,

if

then we c l a i m t h a t

Since ,

is a vector field

of

we have

= ni

for all

j

and thus

such t h a t

n i =nj

X~ i = ~j ~ H°ni

X = ~a~ i

is a vector field

njaj

,

= ni

X e w .

~a e

X

aj = 0

Indeed, it

if

,

.

Indeed i f

not,

then we

a contradiction.

such t h a t

n i = nj

is clear that

, in this

case

H° ni

so t h a t

X HmcH°

H

ni

-

m-n i

But we see from t h e d e f i n i t i o n

that

H°n Hm~H ° m+n T h i s proves t h a t

Finally, is not a root,

if

X e w

~

i s any r o o t o f

h

then by t h e p r o o f o f lemma i

E nia j T h i s proves t h a t

~

= ni

= w ,

We can now combine a l l

,

such t h a t

on

aj = 0

g~

for

has b a s i s

< H,~ > = 0 , ~a@i

,

where

n i = nj

Q.E.D.

t h e p i e c e s and o b t a i n t h e s t r u c t u r e

of

m:

and

-X

190 Theorem

Let

Ar = A n ( - A )

A

and

be the non-zero r o o t s of

Au = A ~ Ar ,

m= r Q u

( v e c t o r space d i r e c t sum) ,

~

is a r e d u c t i v e subalgebra o f

(ii)

u

is a n i l p o t e n t

ideal of

m

(iii)

~

acts l o c a l l y n i l p o t e n t l y

on

We have

u = u

a t z e r o , since

~_ui, u ~ ~ i + j

by v e c t o r f i e l d s field

r

~r °

and

with

~a

~k = ~k

when

non-linear .

k ~ 2

[ • o~ ' But

~o

vanishes i s spanned

The b r a c k e t w i t h any v e c t o r

thus vanishes a t zero. To v e r i f y t h a t

--O

+ ~I +"'+

- - O

i s a Lie a l g e b r a , we only need check t h a t

~a~ i ,

into

P .

(ii) u

A

m

a l r e a d y proved (lemma 2 and remark).

To v e r i f y t h a t

Divide

and

(i)

--

m .

(~ e Ar)

(i)

Proof.

on

and s e t

r : h + z m~

Then

h

and

~o

u

i s an i d e a l , we f i r s t

given in lemmas 1 and 3

~-ro' Uk] ~ k

By homogeneity,

Recall t h a t coefficients,

'

~___+,Uk~~-~k+l "

check c e r t a i n cases w i t h

k = 0,1,2

n + = D1 .

we see t h a t

~-+'

make i t

Since

for

~k =

uo]

k ~ 2 ,

vanishes a t

0 ,

in

u

--O

A.2

have n o n - l i n e a r

and hence

-CUl

= 0

we only need

.

Since the v e c t o r f i e l d s ~n+,Uo]

Uo~

clear that

1 <_ k < r

The c a l c u l a t i o n made i n the p r o o f o f Theorem

~ -,

observe t h a t the d e s c r i p t i o n s o f

shows t h a t

191 Furthermore,

n__ = V~ ~ ,

It follows that order at

0 .

~_, ~]

so the vector f i e l d s in vanishes at

and t h a t

This shows, by the d e s c r i p t i o n s o f

This completes the proof t h a t (iii)

0 ,

Let

u

u

--0

have q u a d r a t i c c o e f f i c i e n t s . ~_, ~ and ~ i

vanishes to second above, t h a t

is an ideal

Pm = ~ Hk (k ~ m)

the decomposition

~-

be

V = V1 @...~ Vr ,

the f i l t r a t i o n

on

P associated w i t h

We s h a l l prove t h a t

um (Pm) : 0 Consider f i r s t

the a c t i o n o f

uo .

By lemma 3

we o b t a i n

1

~o (Hm) ~_z where the sum is over a l l ~ o (H1) = 0 ,

HkI" -. Hkl+ I

k I . . . . . k/+ 1

,

such t h a t

~

< m and

E ki = m .

Since

it follows that m

% (H m) : o Next, we have ~k

with

Pm~Pm_k ,

so t h a t

kI kr ~1 " ' " ~r

PmCPm-k

k = k1+---+k r

F i n a l l y , the commutation r e l a t i o n s

~_ui , uj~ c u i +j

show t h a t i t s u f f i c e s to consider ordered products k kI kr ~o ° Ul " ' " Ur Pm "

ko + k I + - - - + k r >_ m .

with

Since on

Pr

V~P r ,

By the above

t h i s is zero.

we o b t a i n , in p a r t i c u l a r ,

by n i l p o t e n t t r a n s f o r m a t i o n s . Hence ~

f i n i s h e s the p r o o f o f

(ii)

and

(iii)

.

a faithful

r e p r e s e n t a t i o n of

is a n i l p o t e n t Lie algebra. This

192

A.4

Birational

rational

f u n c t i o n s on

We s h a l l

denote by

~(~) = ~

where

V ,

Aut(R)

i.e.

If

~1 . . . . .

~(~1 . . . . . ~n )" generators

~n

the q u o t i e n t f i e l d

and automorphism

m.)

Aut(P)

,

for

for

A transformation

Fi

functions into R

(we view

R

R = Rat(V)

V

,

and

be the f i e l d

o f the i n t e g r a l R

functions

There i s a n a t u r a l

f,g e P .

t h e n we can i d e n t i f y

R with

is determined by i t s

function

in

n

variables.

a c t i o n on the

C o n v e r s e l y , given r a t i o n a l

then these e q u a t i o n s d e f i n e a unique homomorphism ~)

This homomorphism w i l l

GI . . . . . Gn

m

Aut(R)

.

are c a l l e d b i r a t i o n a l

be the L i e a l g e b r a c o n s t r u c t e d i n

o f a p o s i t i v e i n t e g r a l element

Aut(R)

Now the group

m of R

be an ,

i.e.

find

such t h a t

Ci = G i ( m ( ~ l ) . . . . . m(~n ))

Let

inclusion

: F i ( ~ I . . . . . ~n ) ,

as an a l g e b r a o v e r

Hence the elements o f

P .

(We r e q u i r e t h a t

automorphism p r e c i s e l y when we can s o l v e these e q u a t i o n s r a t i o n a l l y rational

domain

of

and can be expressed as

is a rational F1 . . . . . Fn ,

Aut(R)

m e Aut(R)

~(~i) where

c

m e Aut(P)

is a basis

CI . . . . . ~n '

Let

the group o f automorphisms o f

f o r any s c a l a r

m ( f / g ) = mf/~g ,

Transformations

Aut(R)

H e h .

§ 2 ,

" transformations relative

to a choice

We would l i k e to " e x p o n e n t i a t e "

m

i s n o t a L i e g r o u p , in any r e a s o n a b l e sense.

in

193 For example, as we noted in

§ A.I

,

a derivation of

R ( o r P)

n e c e s s a r i l y generate a one-parameter group o f automorphisms o f i s no f u n c t o r mapping Lie subalgebras o f d e r i v a t i o n s o f Aut(R).

For the a l g e b r a

m ,

m which are e i g e n v e c t o r s f o r whose r o o t s

~

R

Thus t h e r e

however, we a l r e a d y know t h a t the v e c t o r f i e l d s ad h

generate r a t i o n a l

reasonable to expect t h a t everY element of R .

R .

to Lie subgroups o f

flows ( i n f a c t ,

are non-negative generate polynomial f l o w s ) .

o f automorphisms o f

does not

~

those is

generates a one-parameter group

We s h a l l v e r i f y t h i s by an e x p l i c i t

the i n f o r m a t i o n about the s t r u c t u r e o f

Hence i t

in

m obtained in

c o n s t r u c t i o n , using

§ A.3.

Recall t h a t by Theorem A.3, m : n- ~ ro ~ n+ ~ u If

Xeu

,

then

eX

is d e f i n e d as an automorphism o f the polynomial a l g e b r a .

We set U = {e x : X e u} By the p r o o f o f Theorem 1 . 1 . 4 , i t t h a t the

map X ÷ e X

f o l l o w s (using the l o c a l n i l p o t e n c y o f

is a bijection

from

a complex, simply-connected n i l p o t e n t Let

N+

~x ~-~ Px

N_

=

f(v+x)

,

defines a bijection

to the v e c t o r group Let

Lie group on

be the group o f t r a n s f o r m a t i o n s pxf(V)

The map

u to U ,

~ on P)

and aefines the s t r u c t u r e o f U .

{Px : x e V1} ,

where

f e R from

~+

to

N+ ,

and

N+

is isomorphic

VI .

be the group o f t r a n s f o r m a t i o n s

{~

: ~ e V1} ,

where

~

is

determined from i t s a c t i o n on the l i n e a r f u n c t i o n s by the formulas

~(~) The map

~H ~-+ ~_~

= ~(I+~) -k

,

defines a bijection

isomorphic t o the v e c t o r group

VI .

~ e V~ from

n_

onto

N_ ,

and

N_

is

(Note t h a t the f l o w generated by the v e c t o r

194 field

CH

is

°-t~

'

and

o~

is a b i r a t i o n a l

To c o n s t r u c t the subgroup first

corresponding to the subalgebra

~o '

our

impulse, based on lemma A . 3 . 1 , would be to take the d i r e c t product o f the

groups on

Ro

t r a n s f o r m a t i o n , by Remark A . 1 . 5 . )

GL(Vk) ,

V .

w i t h the a c t i o n on f u n c t i o n s being induced by the l i n e a r a c t i o n

This guess i s indeed c o r r e c t f o r

action of Let A e GL(Vk)

GL(VI) Gk ,

must be " t w i s t e d "

k ~ 2 ,

(cf.

However, f o r

For the case

on the subspaces

k = i

,

k = 1

the

remark a f t e r lemma A . 3 . 2 ) .

be the group o f automorphisms

a c t i n g as the i d e n t i t y

Gk = GL(Vk) .

k ~ 2 .

we l e t

GI

f ~ - + foA - 1 , w i t h

Vj , j ~ k .

Thus

be the group o f automorphisms

whose a c t i o n on l i n e a r f u n c t i o n s is (m)

when

~ ~-~ (det A) -k ~oA"1

~ e V~ .

the subspaces

Here Vk ,

A e GL(V1) , k > 1 .

D

i s the f i n i t e

k>2

such t h a t

view

GL(Vl) and

and

A

,

c e n t r a l subgroup

Vk ~ O} .

is extended to act as the i d e n t i t y on

Thus

G1 = GL(VI)/D where

,

(Here

{ml : ~q+l = i and k q

q = dim Vl)

D as subgroups o f

.

SL(q+I,C)

A i:01

It will

: I for all

be convenient to

via the embedding

,

A e GL(V1)

(det A)-

Now define

be the subgroup of that

( d i r e c t product as groups) ,

R° = GI x G2 x . . . x Gr Aut(R)

G is isomorphic to

generated by r/D ,

where

N F

,

GL(Vk) ,

Furthermore, the groups

k ~ 2 , N±

and

GI ,

and

N+ .

N+ .

We s h a l l show

GL(Vr) ,

(q = dim Vl) -

we a l r e a d y have an isomorphism w i t h Gk ,

t o c o n s t r u c t an isomorphism between N_ ,

and

i s the d i r e c t product

F : SL(q+I,$) x GL(V2) . . . . . For each f a c t o r

Ro ,

and l e t

k > 2

Gk .

m u t u a l l y commute. Thus we o n l y need

SL(q+I,$)/D

and the group generated by

195

We s h a l l v i e w

as t h e m a t r i c e s i n b l o c k form

?1 = S L ( q + l , $ )

det(g) = I

where

x e V1 , ~ e V~ , A e End(V1)

can be e x p r e s s e d

in t e r m s o f t h i s

,

and

,

The c o n d i t i o n

~, e ~,

det(g) = 1

d e c o m p o s i t i o n as

X d e t A - < ~, a d j ( A ) x > = 1 , where

adj A

is the " a d j u g a t e " o f

A

(transposed cofactor matrix).

= adj(a) (Recall t h a t SL(q+I,$)

AA = AA = ( d e t A) I . )

Write

.

We can imbed the v e c t o r groups

VI ,

V~

into

as t h e subgroups

V+ =

,

X e

0

Thus

V+ _

are commutative subgroups o f

Theorem 1. r e s t r i c t i o n to

,

which are n o r m a l i z e d by

There i s a unique homomorphism

from

to

The k e r n e l o f

D .

~

is

GL(Vk)

is the homomorphism o n t o

On the basis o f t h e p r e v i o u s d i s c u s s i o n , on

?

onto

GL(V1) .

G whose

V+_ is

and whose r e s t r i c t i o n

Proof

sI

define

~

FI .

Consider the subset o f

On t h i s

set we have the unique f a c t o r i s a t i o n

£1

where

Gk

d e f i n e d above.

the o n l y problem is to det A # 0

in

(mm) .

196

CA X ~\, 'k{< x] Hence i f

~

exists,

it

{A ~ L~

(9)

/ =

if

I\~A-1

i s g i v e n on t h i s

s e t by

x~ x> = ~~A-I

~(A) PA-Ix

Since the subgroups uniquely,

0\i i/A i] <0

I

V±

and

GL(VI)

such a homomorphism e x i s t s .

i s the o n l y p o i n t a t which we w i l l suffices,

f o r the e x i s t e n c e o f

phism from

I' I

to

The a d j o i n t A- I

,

for

Given

Aut(R)

action of

x e VI ,

w i t h the a d j o i n t

~ ,

GL(V1)

(If

x,~

are near

on

(?)

Px ~

Both s i d e s o f their

action

= ~

(?)

on a l i n e a r

Px ~

then

~A-1

V+ _

is

this

determines

(~)

be

$) ,

defines a local

it homomor-

x ~ - * ( d e t A)Ax ,

-1

~ ~-. ( d e t A)

shows. Since t h i s

agrees

reduces the problem t o the f o l l o w i n g :

we form the p r o d u c t

/A

A

x\

transformation

~(A)

function

sending

is invertible.).

y + y + < ~,y > x .

We must show t h a t in t h i s

case

PA-lx

are automorphisms o f C e Vk .

R , We f i n d

so i t

suffices

that

(~) : (~ + < ~,x >)(1 + < ~,x > + ~ ) - k

On the o t h e r hand, we have

(~)

is simply-connected (this

as a m a t r i x c a c u l a t i o n

i s the l i n e a r 0 ,

to show t h a t

G1 on N+ ,

x e V1, ~ e VT,

A = I + x ~ ~

rI

formula

.

~ e V~ ,

action of

Since

rI ,

need t h a t the s c a l a r f i e l d

01

where

generate

to c a l c u l a t e

197

o A_I x ( A ) p - I (~) : ~A-1 + < ~,Ax > ~A - I + < ~,Ax > A x (~A + d e t A) k

But

det(l

+ x @ ~) = 1 + < C,x > , A- I

< ~ , x > # -1 ,

we can w r i t e

= I - (1 + < ~,x >)-1 x @

= ( d e t A) A - I

= ( 1 + < C,x >) I - x @

Using these f o r m u l a s , we e a s i l y of

and i f

verify

(?)

,

and hence e s t a b l i s h

the e x i s t e n c e

~ . Obviously

normal

Ker ~ c r I ,

subgroup.

Remark

If

This

and hence

implies

that

g =~

Ker ~ c {~I

Ker{~)

e rI

= D,

: q+l

= I}

,

since

it is a

Q.E.D.

,

then the calculation made in the proof of theorem 1 shows that

~(g)¢

= CA -1 + < ¢,Ax > ~A - I

+ < ¢,Ax >

(CA + d e t A) k

if

C e Vk .

(Here

A

i s e x t e n d e d to be

This f o r m u l a remains meaningful recall

that if

det A = 0 ,

I

on

even when

then the c o n d i t i o n

Vk

A

for

k > 1 ,

is singular. det g = I

by d e f i n i t i o n . )

To see t h i s ,

forces

< ~,Ax > = -1

In p a r t i c u l a r ,

~A # 0 ,

so the d e n o m i n a t o r o f

(~)

i s never i d e n t i c a l l y

For the n u m e r a t o r , we w r i t e

~A - I + < ~,Ax > ~A - I

If

~ e V~

with

k > i

,

To examine t h e b e h a v i o r on

then VI

= ( d e t A ) - I ( ~ A + < ~,Ax > ~A)

< ¢,Ax > = 0 ,

pick a basis

and {x i}

~A -1 = ~ , for

VI

by d e f i n i t i o n . and dual b a s i s

zero.

198 {~i }

for

V1 .

Using t h e c o n d i t i o n

~i # + < ~ i '

det g = 1 ,

we c a l c u l a t e

t h a t when

~ = ~i

Ax > cA = ~ ( d e t A) Ci # + E m/(g)~/

,

where ~l(g ) =

~ j,k

<

~j,x

> det

> < ~,x k

cji

cz

j k Here

[cij ]

classical

i s the c o f a c t o r

fact

determinants"

in determinant

it

A

proof,

~/(g)

(~)

= 0 .

on

~ ,

of

u

m ,

det

(~)

But i t

is a

the "compound

by

det A .

rank A < 1 ~

i s an i r r e d u c i b l e

(For a

all

2x2

polynomial,

is everywhere regular ~

from

V_GIV +

to

£

on

rI

.

is in fact

o f a group o f automorphisms c o r r e s p o n d i n g

we o n l y need t o put t o g e t h e r G

Aut(R)

normalises

U ,

and

the groups

G ~ U = {1}

.

and we v e r i f y

easily

determined in

nilpotence

u of

To p r o v e t h a t

G

and

Hence

t o the

U , as f o l l o w s :

M = G U

is a

.

From t h e p r o o f o f Theorem A . 3 , we see t h a t

The passage from local

that

are d i v i s i b l e

det A = 0 ~

Since

basis.

.

Theorem 2

Proof

that

to this

formulas")

~l

the extension of

To c o m p l e t e o u r c o n s t r u c t i o n

subgroup o f

relative

("Jacobi's

t h e n u m e r a t o r in

continuation,

Lie algebra

A

ml ")

This shows t h a t

g i v e n by

theory

use the f a c t

vanish ~

must thus d i v i d e

By a n a l y t i c

of

a p p e a r i n g i n the f o r m u l a f o r

non-computational minors of

matrix

ClkJ

that

Lemma A . 3 . 3 to u

U on

follows

Ad(N+) we f i n d

stabilises that

directly

Ad(Ro)

ad(~±) ~ .

acts n i l p o t e n t l y

By the s t r u c t u r e

also stabilises

from t h e s e c a l c u l a t i o n s

u .

and t h e

P

G ~ U = {1}

,

we o b s e r v e f i r s t

one has G ~ U~RoN+U

.

that

since

GnUcAut(P)

,

'

199 (In formula

(~)

above, i f

~ # 0

then

Tc(g)c

is not a polynomial f u n c t i o n on

V.) Set n+ + u

T = N+U o

From lemmas 2 and 3 of the previous s e c t i o n we see t h a t

is a nilpotent

L i e algebra which acts l o c a l l y n i l p o t e n t l y

on

£ .

Hence

T = exp (n+ + u) On the o t h e r hand, we e v i d e n t l y have

n_+ + u = U_o + n _ l +n_2 + . . . +

I f we d e f i n e the spaces in

n_r

H° o f n o n - l i n e a r homogeneous polynomials o f degree m

m as

lemma 3, and s e t

Qm = H° m + Pm-I

,

m > I

then we can d e s c r i b e the Lie algebra

,

~+ + u

as

~+ + ~ = {X e Der(P) : X Pm ~Qm (cf.

lemma A.3.3

and

c h a r a c t e r i s e the group

§ 1.1.3) T

.

Ro

E x p o n e n t i a t i n g t h i s d e s c r i p t i o n , we thus can

as

T = {~ e Aut(P) Now the group

f o r a l l m} .

: ( m - l ) Pm ~ Qm f o r a l l m} .

acts l i n e a r l y

ant. The above d e s c r i p t i o n o f

T

on

V

,

l e a v i n g each subspace

Vk

invari-

makes i t e v i d e n t t h a t

Ro/n T = {1} . Finally, ~+ + ~

we have onto

Example

T .

N+h U = {1}

because the e x p o n e n t i a l map i s a b i j e c t i o n

This completes the p r o o f .

Let us r e t u r n t o the examples

and employ the same n o t a t i o n . The a l g e b r a integer (View

n . x,y

from

When n = I ,

then

(dim V = 2)

a t the end o f

§ A.2 ,

m i s determined by a choice o f p o s i t i v e

M = SL(3) ,

as inhomogeneous c o o r d i n a t e s f o r

acting projectively

~2.)

on

{x,y}

.

200

When

n > 1 ,

then

M = G U ,

where

G = SL(2) x GL(1) Here

Col

$

e G

acts by t h e b i r a t i o n a l

transformation

(b + d x ) ( a + cx) - I CoY (a + cx) -n

The group

U

consists

÷

(c i e ~) .

The group

of all

transformations

y + c I x +...+

M

cn xn

is the classical

" J o n q u i ~ r e s group o f o r d e r

n ."

Comments and references f o r Appendix

The study of f i n i t e - d i m e n s i o n a l Lie subgroups of the ( i n f i n i t e - d i m e n s i o n a l ) group of b i r a t i o n a l transformations of an a f f i n e space has a long h i s t o r y ; cf. Fano [ i ] .

The classical "Jonqui~res groups" in two variables occurred in the

c l a s s i f i c a t i o n by Enriques of a l l f i n i t e - d i m e n s i o n a l groups of b i r a t i o n a l transformations in two variables. They were studied in more d e t a i l by Mohrmann [1] and Godeaux [1], and "automorphic functions" on these groups were considered by Myrberg [1]; cf. the survey a r t i c l e by Coble [ I ] . In recent years the subject has been g r e a t l y extended by Demazure [1] and Vinberg [ i ] .

The algebras

and groups we construct here f u r n i s h a class of examples

f o r Demazure~ general theory of "Enriques systems". Several of our proofs are special cases of his general methods. Since there exists no c l a s s i f i c a t i o n of Enriques systems, as contrasted to the c l a s s i f i c a t i o n of root systems f o r semisimple algebras, i t is perhaps useful to have such examples constructed e x p l i c i t l y . The fact that the Lie algebras

m are maximal seems to be new. The subalgebra

of vector f i e l d s homogeneous of degree zero has appeared also in Arnol'd [ i ] ,

~o in

connection with the c l a s s i f i c a t i o n of normal forms f o r smooth functions at a critical Pedoe [ i ] ,

point. For "Jacobi's formulas", used in the Remark in Chap. 2, § 8 .

§ 4 ,

cf. Hodge-

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Subject Index almost orthogonal o p e r a t o r s automorphisms o f polynomials Bernoulli operator birational transformation boundary values Bruhat decomposition Campbell-Hausdorff formula canonical coordinates coboundary o p e r a t o r comultiplication Cremona group C~ v e c t o r dilations distance function

85 8 42 192 154 118 13,51 9 22 14 175 139 1 71

elementary r o o t .......... automorphism e x p o n e n t i a l map

177 178 49

faithful representation filtration: polynomials .......... Lie algebra .......... C~ f u n c t i o n s .......... d i f f . Operators free n i l p o t e n t Lie algebra fundamental s o l u t i o n

16 2 12 5 105 36 159

g r a d a t i o n : Lie algebra ......... polynomials ......... v e c t o r space Hall basis Hardy space Heisenberg algebra .......... group homogeneous: b i l i n e a r map ........... diff. operator ........... dimension ........... distribution ........... function ........... norm ........... polynomial ........... vector field hypersurface hypoelliptic diff. operator

5,13 2 I 37 151 11 63 20 158 68,76 68 68 3 1 7 53 158

infinitesimal transitivity intertwining integral

158 125

Jonqui~res group .......... transformation

200 178

kernel o f type

103

s

length o f f i l t r a t i o n l i f t i n g theorem Lipschitz condition

12 39 77

maximal subalgebras mean value measure o f homogeneous type

180 7O 74

o p e r a t o r o f type s order of vector field

104 6

parametrix p a r t i a l homomorphism Plancherel formula principal part of diff. operator principal series representation ......... irreducibility criterion 2 p r o j e c t i o n Hb

166 36 138 164 123 130

real rank r o o t spaces

118 177

Siegel domain s i n g u l a r kernel Sobolev spaces space o f homogeneous type s u b e l l i p t i c Laplacian Szeg~ kernel

146

61 78 108,168 76 162 157

t a n g e n t i a l Cauchy-Riemann equations t r a n s i t i v e p a r t i a l homomorphism transpose o f d i f f . o p e r a t o r

142 168 165

unboundedness o f s i n g u l a r i n t e g r a l s

97

vector fields: ............. .............

4 54 54

polynomial c o e f f i c i e n t s holomorphic anti-holomorphic