Calculus on Heisenberg Manifolds. (AM-119), Volume 119 9781400882397

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Annals of M athem atics Studies Number 119

Calculus on Heisenberg Manifolds by

Richard Beals and Peter Greiner

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1988

Copyright © 1988 by Princeton University Press ALL RIGHTS RESERVED

The Annals of M athem atics Studies are edited by W illiam Browder, Robert P. Langlands, John Milnor and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and D avid Vogan

Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding m aterials are chosen for strength and durability. Paperbacks, while satisfactory for personal collec­ tions, are not usually suitable for library rebinding

Printed in the U nited States of America by P rinceton University Press, 41 W illiam Street Princeton, New Jersey

L ibrary o f C o n g ress C a ta lo g in g -in -P u b lic a tio n D a ta Beals, Richard, 1938Calculus on Heisenberg m anifolds / by Richard Beals and Peter Greiner. p.

cm. - (The Annals of m athem atics studies ; 119)

Bibliography: p. Includes index. 1. H ypoelliptic operators. 2. Calculus. 3. Differentiable m anifolds.

I. Greiner, P. C. (Peter Charles), 1938-

II. T itle. III. Title: Heisenberg manifolds. IV. Series: Annals of m athem atics studies ; no. 119. Q A 329.42.B42 1988 515.7’2 4 2 -d c l9 ISBN 0-691-08500-5 (alk. paper) ISBN 0-691-08501-3 (pbk.)

88-9939 CIP

To our parents

Alice Beals Anthony and Ildiko Greiner

Contents Preface

ix

Introduction

3

Chapter 1. The M odel O perators 1. Differential Operators and Their Models 2. Estimates for the Model Operator

11 11 16

Chapter 2. Inverting the M odel O perator 3. Operators, Symbols, Composition and Invariance 4. Inverting P \ in Normal Form: Symbols 5. Inverting P \ in Normal Form: Kernels 6. Inverting P \ in Skew-Symmetric Form: Symbols 7. Inverting P \ in Skew-Symmetric Form: Kernels 8. Solution of the Model Operator in General Form

23 23 29 38 50 61 65

Chapter 3. Pseudodifferential O perators on Heisenberg Manifolds 9. Standard Pseudodifferential Operators 10. V-Pseudodifferential Operators 11. Group Structures on R d+1 12. Composition of ^/-Invariant Operators 13. The ^-Composition of Homogeneous Symbols 14. The Composition of V-Operators 15. Kernels of V-Operators 16. The Proof of the Invariance Theorem (Theorem 10.67) 17. Adjoints of V-Operators 18. Hypoellipticity and Parametrices for Second Order Differential Operators 19. V-Operators on Compact Manifolds - Hilbert Space Theory

80 80 90 100 105 116 119 128 135 138 140 146

viii

Contents

C hapter 4. A p p lic a tio n s to th e db-C o m p le x 20. 21. 22. 23.

The c^-Complex and Hypoellipticity of □&. Condition Y(q) Param etrix for C P artial Inverses and Projections in Case of a Non-degenerate Levi Form 24. on a Compact Manifold 25. The P artial Inverse and Associated Projections for [H&)?

151 151 156 161 165 168 173

B ib lio g ra p h y

181

I n d e x o f T e rm in o lo g y

189

L ist o f N o ta tio n

192

Preface We study certain hypoelliptic second order partial differential operators. The m otivating example, which we treat in detail, is the Kohn operator on a C R manifold. A full asym ptotic calculus is developed for an algebra of pseudodifferential operators which includes a param etrix for □&. Our intention has been to develop a calculus which is as explicit as pos­ sible. We calculate the leading term s of param etrices in the hypoelliptic case and the leading term s of partial param etrices and of Cauchy-Szego projections in certain nonhypoelliptic cases. C om putation of further term s involves com puting convolutions of homogeneous functions on certain twostep nilpotent groups; we develop a calculus for such convolutions. We have attem pted to make the exposition concrete, reasonably selfcontained, and accessible to readers interested either in several complex variables or in partial differential equations. It is with pleasure th a t we take this opportunity to express our appre­ ciation to A nnette Yu for the excellent job she did in typing the original m anuscript. New Haven, Conn. U.S.A. Toronto, Ont. C anada

July 1985

Calculus on Heisenberg Manifolds

Introduction T he operator □& is a second order partial differential operator associated to the N ( m ) . All the local Holder and Lp regularity results for A can be deduced from this construction. For example, if Aw belongs locally to L p,

Introduction

4

where 1 < p < oo, then every second order derivative of u belongs locally to LP. This kernel m ethod gives the kernel k as an asym ptotic sum (5)

k ~ ko

ki + &2 + • **

where ko is known and the kj are successively less singular. Here we can take kj to be the kernel of QoR?, but it is difficult to establish directly what form the kj take. In the symbol method one looks instead for the symbol: the function (or distribution q (x , •)) such th at (6 )

Qu(x) = f

J R-

e*'**«(ar,0«(f) 7 ^ ^ -

\Zir)

Thus q (x , •) is the partial Fourier transform (7)

q(x,$) =

J

e~t('z k ( x, z) dz .

Symbols are m anipulated much more readily than kernels. The operator A itself has symbol (8)

„(A ) = £

aj k ( x )€j€k + lower order.

According to the symbol calculus for classical pseudodifferential operators the symbol of the composition QfQ fl of operators having symbols q' and qn is asym ptotically (9)

gW ' ~ £ ^ ° P e V ) ( ^ ? " ) a

One seeks an expansion of the symbol q of A - 1 : (10)

~ go + 0i + 02 + • • •

where qj(x, •) is homogeneous of degree —2 —j in the £ variables. Then necessarily 00(2 ,£) = an(^ remaining term s qj are easily com puted by recursion. Pseudodifferential m ethods involve the Fourier transform , so they are intrinsically L 2 theories. Nevertheless in this classical case (10) immediately gives a kernel expansion of the type (5). Here kj is determined from qj and it can easily be shown to have the form (11)

k j ( x , z ) = k j ( x , z ) + pj(x, z )lo g |z |

Introduction

5

where Ar'-(x, •) is homogeneous of degree 2 —n + j and p j ( x , •) is a polynomial homogeneous of degree 2 — n -f j . One then recovers the Holder and Lp regularity. The symbol m ethod can be adapted to study complex powers A 5 or the heat operator ^ + A, giving enough information on the kernels of these operators to allow one to connect local and global geometric invariants.

Contents of C hapters 1-4 In C hapter 1 we introduce a class of second order partial differential op­ erators P. This class includes both Dj and the heat operator + A. Associated to such an operator P on a manifold M is a codimension 1 subbundle V of the tangent bundle T M . In tu rn we associate to V and to a point y E U , where U is a coordinate neighborhood identified with a subset of R d+1, a group structure on R d+1 having y as identity element. This group is isomorphic to one of the groups H m x R d - 2 m ; where H m is the (2m -f l)-dim ensional Heisenberg group; m — m(y) may vary with y. There is then a left invariant operator P y which is a best approxim ation to P at the point y. This “model o p e r a t o r P y plays the same role for P as the translation invariant operator A x plays for A. A basic concern is whether P is “hypoelliptic with loss of one derivative”. This means th a t if P u is locally in L 2, then each first derivative of u is locally in L2, a loss of one derivative relative to the result for A. If so, one wants to construct a parametrix Q for P (an inverse modulo C°°). We show in §2 th a t the model operator P y satisfies the basic estim ate for hypoellipticity with loss of one derivative if and only if it is invertible. C hapter 2 is devoted to calculating the inverse Qy of the model operator P y whenever it exists. Both the symbol and the kernel of Qy are found explicitly. In appropriate noninvertible cases we calculate the symbol and the kernel of a partial inverse and of the projection onto the kernel of P y. C hapter 3 begins with a sum m ary of results from the classical theory of pseudodifferential operators. We then introduce the algebra of V-operators and develop its calculus in somewhat greater detail than in our paper [1]. As an application we show th a t an operator P of the type considered in C hapter 1 is hypoelliptic with loss of one derivative if and only if each of the model operators P y is invertible. If so, the inverses Qy give the principal term of a full param etrix Q which is a V-operator. This implies in particular

Introduction

6

th a t the kernel of Q has an asym ptotic expansion in local coordinates oo

( 12)

k(x, y) ~

oo / ; ( * , -V ’*(y)) + ^ P j ( * . - V ’®(!/) log j ——d j =0

Here ipx : U —►R d+1 is a known coordinate map, f j ( x , •) is a function with parabolic homogeneity of degree j , P j ( x , •) is a polynomial with parabolic homogeneity of degree j,\\ || denotes a parabolic homogeneous norm, and the principal term /_ = - E J=1

J

“f M

^

) ' - « < » )£

k=l

is in normal f orm i f A = (ajk) 25 given by (1.26) and (1.27).

C hapter 1

16

§2 Estimates for the Model Operator T he second order operator P is said to be hypoelliptic with a loss of one derivative in U if (2.1)

u G V' ( U) ,

P u € H { oc( U ) - + u € H { + 1( U ) , s € R .

Here H*oc(U) denotes the localization of the standard Sobolev space H s . Let ||w||^ denote the norm (2.2)

N |2 + £ P

> ||2+ £

j=0

H X JX M I2,

« € C e~ ,

j,k = 1

where || . . . || denotes the L2-norm. Let 7I denotethe closure of C%° in the | | . . . ||« norm. (2.3) THEOREM : The operator P is hypoelliptic with loss of one derivative in U if and only if for each y E U the model operator P y is bijective from 7i to L 2. In various guises and in various degrees of generality, this theorem can already be found in the literature, e.g. Hormander [1]. We shall prove it later, and also show th at computing the inverse of the model operator allows us to compute the principal term of a complete asym ptotic expansion for the param etrix of P . In this section we derive the necessary and sufficient condition for the desired invertibility of P y . We set Z+ = { 0 , 1, 2, . . . }. (2.4) D e fin itio n : Given y E U , the singular set A y is defined as follows. Let (bjk) be the matrix with elements (1.11) and let ± i a i , . . . , ± i a n , aj > 0,

(2.5)

be the nonzero eigenvalues of the skew symmetric matrix with elements (bjk — bkj), repeated according to multiplicity. Then Ay = R

(2.6)

(2.7)

n = 0;

Ay = j ± J 2 ( 2aJ + ! )aJ : a G Z+ } *•

(2.8)

if

j-1

Ay = [ a € R , | A | > ^

2n = d> >

t

cij 1

if

2n < d.

j=l J

(2.9) T h e o re m : The model operator P y , given by (1.13) satisfies the apriori estimate

17

The Model O perators |M |« < C\\P*u\l

(2-10)

u E C ™,

if and only if X(y) does not belong to the singular set Ay . From now on we leave off the superscript y from X j, j = 0 , 1 , . . . , d and from P y - i.e. Xo, . . . , Xd are given by (1.10) and P by (1.13). First we note th a t the norm (2.2) is essentially independent of the choice of X \ , . . . , X d • In fact let e = (ejk) denote a nonsingular linear transform ation of H d and set (2.11)

Y0 = X 0, d

Yj = J 2 e j k X k , j =

(2.12)

k= 1

Then INI HX = IMP + E l P > l | 2 + E j— o

\\YjYku\\2

j,k = 1

(2-13) < c ( i M i 2 + E i i x jU |i2 + E '

j= 0

\\x >x *u \\2

j,k= 1

Since e is nonsingular, this shows th at \\u \ \ n =

( 2 -1 4 )

I M k ,x

.

«

An integration by parts yields ||X ju ||2 < ||u|| ||X ?u||, j = 1 , . . . , d, so d

(2.15)

||u||* « ||u|| + ||X 0«|I + J 2 WX JX ^ \ \ jyk = l

Next we show th a t (2.16)

\ \ Xj Xku f < C(\\Xfn\\2 + ||X 2u||2 + ||XoU||2),

a G C%°. We start with X j X k u — X k X j u -f" c X ou , X 0XjU — X j X qii ,

Chapter 1

18

where c denotes a generic constant. Then integrating by parts we have

( X j X ^ X j X k u ) < \(XjXku,Xifc^tOI + M X j X ^ c X o u ^ ^ \(XkX j X ku , Xj u)\ + K^-XfcU.cXoti)! ^ MX j X l ^ X j U ^ + \(cx 0x ku , x j u )\ cX qu^ ^ \\Xfu\\ 2 + \ \ X l u f + 2|(cX 0u, X kXj u) |

(2 17)

< | X ? u ||2 + ||X M |2

+ i||X j X ,U||2 + 2c||XoU||2, which proves (2.16). Now (2.15) and (2.16) show th at (2.18)

||u||* « ||u|| + ||X 0U|| + £ | | * 2«||,

« 6 Ct“

;= i

Next we consider the behavior of X j, j = 1 , . . . , d under an orthogonal change of the variables (a?i,. . . , Xd) — x ' . Thus, let R = (rt*j) G O(d) and set y0 = * 0,

(2.19) (2.20)

B = R*B' R,

y1 = Rx' . where

B = (bj k ).

Then X q = Y0 and 9 9 X j - d x j + 2 ^ bjkXkdxo

dxj + 2 E (

X ) rm j b'mnrnky : k

J J f c = lm , n = l

/

(2 .2 1 ) A

/ d

i r r'i v%; +

1=1

1 A 5 2 r5 16'm2/m%o

19

The Model O perators

Consequently the orthogonal change (2.19) of the variables x\ induces a new set of vector fields 2/1, . . . , where B is replaced by B f . (2.14) applies and (2.22)

I M k .x ~

Proof of Theorem 2.9. As noted in (1.22), by making a quadratic coordi­ nate change we may assume th at B = (bjk) is skew-symmetric. After an orthogonal change we may assume the model vector fields are in normal form, i.e. f v

(2.23)

_ d 0 ~ dx0 ’

X j — IhTj ~ 2Xn+jaj l h ^ ’

^ —3 —

X n + j = d x n+j + 2 XJ a j d x ^ ’

Xk =

,

1 —3 —

2n < k < d.

This is justified by (1.30) and (2.22). It is convenient here to make one more quadratic change of the variables of the kind (1.18) and (1.19) with #?>«+.? ~ Q n + j j ~ ~^a j >

1 —j — n5

and other coefficients zero. After this change we have

Xj =

, dx j )

0< j 1 ^ j ^ n OXn+j OX0 Now we are ready to derive the estim ate (2.10). By (2.18) it suffices to show th at (2.24)

IMI2 + ||X 0W||2 + £

||X 2U||2 < C \ \Pn \ \\

j =1

u E C%°. First we assume th a t not all a j ys vanish in (2.23). We take the Fourier transform in the variables (x0; x n+ i, . . . , X2n ]®2n+i, • • •, %d) and denote the dual variables by ( r ; £ i , . .. ,£n ; 771, . . . , r}d_ 2n) = ( t , ^ ) . By Plancherel’s theorem (2.24) is equivalent to the following family of inequal­ ities for (r, £ ,17) E R x R n x R d_2n: n

(2.25)

,

im i 2 + E d i ^ M i 2 + life + j=l ^ C\\PT,t,nv\\2,

ii2) + K

v € Cc(R n),

d-2n

\

+ £ Vi) i hi2 J=1

Chapter 1

20 where (2.26)

Dj =

M j v ( x ) = xj dj v( x) , n

(2.27)

PTt(,v = Y , { D ] + & + r M j ) 2} + A t +

\ V \2 .

j- 1

The case r — 0 can be easily handled by taking the Fourier transform in the variables x i , ... , x n . Hence we may assume r / 0. Then a translation in variables x \ ) . . . , x n will eliminate the occurrences of £ i , . .. ,£n in (2.25) and (2.27). Furtherm ore, let Vpv(x) = pn/2v(px),

(2.28)

p> 0

be unitary dilations in L2(R n ). Then (2.29)

V ~ 1Mj Vp = p ~ l Mj.

V - ' D j V p = pDjt

Taking p — | r |_1/ 2 we see th a t (2.25) (with £ = 0) holds in general if and only if it holds in the two cases: r = ±1. By these observations we have reduced (2.25) to n

/

IMI2 = £ { I P ^ I |2 + llM. H |2} + ( l + (2.30)

d—2n Y

j

toil

j=l v J=1

< C \ \ P ? v \ \ 2, v e C ? ( R n) where (2.31)

P ± = £ { L >2 + M 2} + M 2 ± A = Q + M 2 ± A. j =1

Let Qj = D j -f Mj2, a Hermite operator. Then (2.32)

Qj = (Dj - i M j Y ( D j - i Mj ) + dj > aj

so D j + M f < D f -f D j M ? D j + M f -f M j D ) M j — D j Q j D j -j- M j Q j M j

(2.33)

— Qj +

j

+ [M j, Qj]M j

= Q) + [£>,-, M j]D j + [ Mj , D]}Mj = Q 2 — 2idj Mj Dj + 2ia,j Dj Mj = Q) + 2a2 < 3Q2.

The Model O perators

21

Adding, we get n

n

(2.34)

+

« € C e° °(R "),

where the last inequality holds because of (2.32). On the other hand, the leftmost term in (2.34) dom inates the rightm ost term , so (2.30) is equivalent to

(2.35) ri e n d ~ 2 n , V €

HQHI + (i + M2)IMI < c\\{Q + M2 ± ahi, C ~ ( R n ). n

We note th at Q — J 2Q j is an n-dimensional Hermite operator. Q is j= i

separable and Qj, j = 1 , . . . , n has spectrum {(2 k + 1)aj, k = 0 , 1 , 2 , . . . } . Therefore Q has spectrum (2.36) Now A G sp(Q) if and only if there is a \ G L 2 PI C°°, \ / 0 such th at (Q — X)\ = 0. Similarly A ^ sp(Q) if and only if Q — A has an inverse which is bounded on L 2. (a) 2n = d. Then (2.35) becomes (2.37) (2.37) implies th a t ±A ^ sp(Q). On the other hand, if ±A ^ sp(Q)} then Qv = (Q ± A)v

Xv

= [ I t H Q ± \ ) ~ 1] ( Q ± * ) v , which implies (2.37), since ||i;|| < C ||Q i;||. Thus, if d — 2n, (2.35) holds if and only if ±A ^ sp(Q), i.e. A G Ay of (2.7). (b) Suppose 2n < d. We want (2.35) to hold with C independent of 77 G H d~ 2n. The positivity of Q implies th at Q 2 + r f < (Q -f ?72)2. Thus (2.35) holds with C = 8 whenever \rj\2 > m ax(2|A |,l). As for the remaining compact set of 77 we can use the argument in (a) to see th at (2.35) holds if and only if (2.38)

~ \ v \ 2 ± A ^ sp(Q),

n e R d“ 2n

which is equivalent to A ^ Ay, where Ay is given in (2.8).

Chapter 1

22

(c) Finally, if aj = 0 , j = 1 , . . . , n, then (2.24) implies the following inequald

ity

7-2 + 5 Z toil4 - c '(Ar + tol2)2> j =1 r] G R d and r G R . Such a constant C, independent of 77 and r, can be found if and only if A ^ R . Thus we have proved Theorem (2.9). We note th a t P y can be extended continuously to H. Then (2.10) is valid on all of 7i. Now (2.10) implies th a t P y is one-to-one on H with closed range in L 2. Hence we have (2 .3 9 )

C o r o l l a r y : P y has a bounded left inverse on L 2 i f and only if

A(y) i A y .

CH A PTER 2

Inverting the Model Operator §3 Operators, Symbols, Composition and Invariance To compute the inverse of a model operator, we shall first compute the symbol of the inverse, in the sense of pseudo differential operators. We begin w ith a brief introduction to the subject. Let V = {#} be a finite-dimensional real vector space with dual space V* — {£} and pairing (x,£) on V x V f . If Q : C°°(V)

C°°(V)

is a linear differential operator with sm ooth coefficients, its symbol is the function q : V x V' -► C, q(x, $) = e - ( (x)[Qe(\(x),

e( (x) =

.

Thus [