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 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 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 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 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 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 @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 -J1no

(-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 _ 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




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)

= 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) 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 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 = ~ 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 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

lo

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

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)
,):':)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

= 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 _ 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] cA = ~ ( d e t A) Ci # + E m/(g)~/

,

where ~l(g ) =

~ j,k


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-

Bibliography

L. Auslander, J. Brezin, and R. Sacksteder [I]

A method in metric diophantine approximation, J. D i f f e r e n t i a l Geometry 6(1972), 479-496.

V . l . Arnol'd [~

C r i t i c a l points of smooth functions and t h e i r normal forms, Uspekhi Math. Nauk 30(1975), 3-66.

F.A. Berezin [I]

Some remarks on the associative envelopes of Lie algebras, Funkt. Analiz i ego P r i l . 1(1967), 1-14.

L. Bers, F. John, and M. Schechter [1]

Partial d i f f e r e n t i a l equations, in Lectures in Applied Mathematics Vol. I I I ,

Interscience, New York, 1964.

G. Birkhoff [~

Representability of Lie algebras and Lie groups by matrices, Annals of Math. 38(1937), 526-532.

N. Bourbaki [~

El~ments de math~matique XXXIV, Groupes et alg~bres de Lie, Hermann, Paris, 1968.

T. Bloom and I. Graham

[i]

A geometric characterisation of points of type m on real submanifolds of ~n , J. D i f f e r e n t i a l Geometry (to appear).

P. Cartier F1]

Quantum mechanical commutation relations and theta functions, in Algebraic Groups and discontinuous subgroups, Amer. Math. Soc., 1966, 361-383.

[2]

Vecteurs d i f f ~ r e n t i a b l e s dans les repr6sentations unitaires des groupes de Lie, S~minaire Bourbaki 27(1974/75), NQ 454.

203

S.S. Chern and J. Moser [~

Real hypersurfaces in complex manifolds,Acta Math. 133(1974), 219-272.

A.B. Coble [~

Cremona transformations and applications to algebra, geometry, and modular functions, Bull. Amer. Math. Soc, 28(1922), 329-364.

R. Coiffman and G. Weiss [i]

Analyse Harmonique Non-Corm~utative sur Certains Espaces Homogenes, Lecture Notes in Mathematics 242, Springer-Verlag, 1971.

N. Conze [1]

Alg~bres d'op~rateurs differentials et quotients des alg@bres enveloppantes, Bull. Soc. Math. France 102(1974), 379-415.

M. Cotlar and C. Sadosky [i]

On Quasi-homogeneous Bessel Potential Operators, in Singular Integral s , Amer. Math. Soc., 1966, 275-287.

M. Demazure

[1]

Sous-groupes alg~briques de rang maximum du group de Cremona, Ann. Scient. Ec. Norm. Sup.(4), t.3(1970), 507-588. (cf. Sem. Bourbaki, 1971/72, Expos~ 413)

J. Dixmier [I]

Cohomologie des alg@bres de Lie nilpotentes, Acta Scientiarum Mathematicarum 16(1955), 246-250.

[2]

Alg~bres enveloppantes, Paris, Gauthier-Villars, 1974 (Cahiers scientifiques 37).

J. Dyer

F1]

A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76(1970), 52-56.

G. Fano

[1]

Kontinuierliche Geometrische Gruppen, Encyclopaedie der Math. Wiss. 1111, Geometrie, 4b, § 21-22.

204 G. Favre [I]

Syst#mes de poids sur une alg@bre nilpotente, Manuscripta Math. 9(1973), 53-90.

G.B. Folland [17

[2]

A fundamental solution for a s u b e l l i p t i c operator, Bull. Amer. Math. Soc. 79(1973), 373-376. Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv f~r matematik 13(1975), 161-208.

G.B. Folland and J.J. Kohn [I]

The Neumann Problem for the Cauchy_-Riemann complex, Ann. of Math. Studies ~ 7 5 , Princeton Univ. Press, Princeton, N.J., 1972.

G.B. Folland and E.M. Stein El]

Estimates for the ~b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math, 27(1974), 429-522.

B. Gaveau [I]

Solutions fondamentales, representations, et estim~es sous-elliptiques pour les groupes nilpotent d'ordre 2, C.R. Acad. Sc. Paris 282(1976), A 563-566.

S.G. Gindikin [~

Analysis in homogeneous domains, Russian Math. Surveys 19(1964), 1-90.

L. Godeaux

[1]

Les transformations birationnelles du planMem, des Sc. Math. ××II, GauthierV i l l a r s , Paris, 1927.

[2]

Sur les transformations b i r a t i o n e l l e s de Jonqui@res de l'~space, Acad. roy. Belgique, 1922.

Goodman, R. [~

One-parameter groups generated by operators in an enveloping algebra, J. Functional Analysis 6(1970), 218-236.

[2]

Complex Fourier analysis on nilpotent Lie groups, Trans. Amer. Math. Soc. 160(1971), 373-391.

205 [3]

Differential

operators of i n f i n i t e

order on a Lie group I I ,

Indiana Univ. Math. J. 21(1971), 383-409. [4]

Some regularity theorems for operators in an enveloping algebra, J. D i f f e r e n t i a l Equations 10(1971), 448-470.

[5~

On the boundedness and unboundedness of certain convolution operators on nilpotent Lie groups, Proc. Amer. Math. Soc. 39(1973), 409-413.

F6]

F i l t r a t i o n s and asymptotic automorphisms on nilpotent Lie groups, J. D i f f e r e n t i a l Geometry (to appear).

[7]

L i f t i n g vector f i e l d s to nilpotent Lie groups, (to appear).

V.V. Gru~in [1]

E2]

On a class of hypoelliptic operators, Mat. Sbornik 83 (125) (1970), 456-473 (=Math. USSR Sbornik 12(1970), 458-476). Hypoelliptic d i f f e r e n t i a l

equations and pseudodifferential

operators

with operator-valued symbols, Mat. Sbornik 88 (130) (1972), 504-521 (=Math. USSR Sbornik 17(1972), 497-514). Y. Guivarc'h [0

Croissance polynomiale et p~riodes des fonctions harmoniques, Bull. Soc. Math. France 101(1973), 333-379.

S. Helgason

[~

Differential

Geometry and Symmetric Spaces, Academic Press, New York, 1962.

[2]

A duality for symmetric spaces with applications to group representations, Advances in Math. 5(1970), 1-154.

G. Hochschild [0

The Structure of Lie Groups, Holden-Day, San Francisco, 1965.

W.V.D. Hodge and D. Pedoe

[1]

Methods of Algebraic Geometry, I, Cambridge University Press, 1968.

L. H~rmander [1]

Hypoelliptic second-order d i f f e r e n t i a l 147-171.

equations, Acta Math. 119(1968),

206 A. Hulanicki [i]

Commutative subalgebra of LI(G) associated with a subelliptic operator on a Lie group, Bull. Amer. Math. Soc. 81(1975), 121-124.

N. Jacobson [~

Lie Al~ebras, Interscience, New York, 1962.

J. Jenkins [~

Dilations, gauges, and growth in nilpotent Lie groups, preprint, June, 1975.

K. Johnson and N. Wallach [~

Composition series and intertwining operators for the spherical principal series, Bull. Amer. Math. Soc. 78(1972), 1053-1059.

R.W. Johnson [~

Homogeneous Lie algebras and expanding automorphisms, preprint, 1974.

P. J~rgensen [~

Representations of d i f f e r e n t i a l operators on a Lie group, J. Functional Analysis 20(1975), 105-135.

A.W. Knapp [~

A Szeg5 kernel for discrete series, Proc. Inter. Cong. Math. Vancouver (1974), VoI. 2, 99-104.

A.W. Knapp and E.M. Stein ~i]

Intertwining operators for semisimple groups, Annals of Math. (2) 93 (1971), 489-578.

[2]

Singular integrals and the principal series I I I , Proc. Nat. Acad. Sc. USA 71(1974), 4622-4624

[3]

Singular integrals and the principal series IV, USA 72(1975), 2459-2461.

Proc. Nat. Acad. Sci.

J.J. Kohn [~

Pseudo-differential operators and h y p o e l l i p t i c i t y , Equations, Amer. Math. Soc. (1973), 61-70.

in Partial Differential

207

[2]

Integration of complex vector f i e l d s , Bull. Amer. Math. Soc. 78(1972), I-Iio Boundary behaviour of ~ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6(1972), 523-542.

A. Kor~nyi [I]

The Poisson integral for generalized half-planes and bounded symmetric domains, Annals of Math (2) 82(1965), 332-350.

A. Kor~nyi and S. V~gi [1]

Singular integrals in homogeneous spaces and some problems in classical analysis, Ann. Scuola Norm. Sup. Pisa, Classe di Scienze 25(1971), 575-648.

A. Kor~nyi, S. V~gi, and G. Welland [1]

Remarks on the Cauchy integral and the conjugate function in generalized half-planes, J. Math. and Mech, 19(1970), 1069-1081.

R. Kunze and E.M. Stein [~

Uniformly bounded representations I I I , Amer. J. of Math. 89(1967), 385-442.

H. Mohrmann

Eli

Ober die automorphe Kollineationsgruppe des rationalen Normalkegels n-ter Ordnung, Rendic. del Circolo Mat. di Palermo 31(1911), 170-200.

C.C. Moore [~

Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds, in Harmonic Analysis on Homogeneous Spaces, Amer. Math. Soc. (1972), 3-44.

P. MUller-R~mer [1]

Kontrahierende Erweiterungen und kontrahierbare Gruppen, J.f.d. Reine und Angewandte Mathematik 283/284 1976) 238-264.

P.J. Myrber [~

Ober die automorphen Funktionen bei einer Klasse Jonqui~resscher Gruppen zweier Ver~nderlichen, Math, Z e i t s c h r i f t 21(1924), 224-253.

208 A. Nijenhuis and R.W. Richardson, Jr. [i]

Deformations of algebraic structures, Bull. Amer. Math. Soc. 70(1964), 406-411.

R.D. Ogden and S. V~gi

[1]

Harmonic analysis and H2

functions on Siegel domains of type I I ,

Proc. Nat. Acad. Sci. USA 69(1972), 11-14. N.S. Poulsen

[i]

C~ vectors and intertwining b i l i n e a r forms for representations of Lie groups, J. Functional Analysis 9(1972), 87-120.

G. Rauch [I]

Variation d'alg~bres de Lie r~solubles, C.R. Acad. Sci. Paris 269(1969), A 685-687.

[2]

Variations d'alg#bres de Lie, Pub. Math. Univ. Paris XI (Orsay), N2 32, 1973.

B. Reed

Representations of solvable Lie algebras, Michigan Math. J. 16(1969), 227-233. C. Rockland [i]

H y p o e l l i p t i c i t y on the Heisenberg group-representation theoretic c r i t e r i a (preprint).

L.P. Rothschild and E.M. Stein [I]

Hypoelliptic d i f f e r e n t i a l operators and nilpotent groups, Acta Math. (to appear).

G. Schiffmann ~]

Int~grales d'entrelacement et fonctions de Whittaker, Bull. Soc. Math. France 99(1971), 3-72.

E.M. Stein [ i ] "Some problems in harmonic analysis suggested by symmetric spaces and semi-simple Lie groups," Actes du Congr~s International des Math~maticiens, Nice, 1970, Tom. i , pp. 173-189.

209 [2]

Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes, Princeton Univ. Press, 1972.

R.S. Strichartz [~

Singular integrals on nilpotent Lie groups, Proc. Amer. Math. Soc. 53(1975), 367-374.

N. Tanaka

[1]

On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. 10(1970), 1-82.

[2]

A differential-geometric study on strongly pseudo-convex manifolds, Lectures in Math. ~ 9 , Dept. of Math., Kyoto Univ., 1975.

F. Tr6ves [i]

Topological Vector Spaces, Distributions, and Kernels, New York, Academic Press, 1967.

M. Vergne [~

Cohomologie des alg~bres de Lie nilpotentes; application 6 l'~tude de la vari~t~ des alg~bres de Lie nilpotentes, Bull. Soc. Math. France 98(1970), 81-116.

E.B. Vinberg [~

Algebraic groups of transformations of maximal rank, Math. Sbornik 88 (130) (1972), 493-503. (=Math. USSRSbornik 17(1972), 487-496).

N.R. Wallach [1]

Harmonic Analysis on Homogen~egus Spaces, Marcel Dekker, New York, 1973.

G. Warner ~

Harmonic Analysis of Semi-Simple Lie Groups, Springer-Verlag, Berlin, 1972.

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