CR structures Someestimatesin ∂ Neumannboundaryvalueproblemforstronglypseudo-convex ¯ M K

Astérisque

M ASATAKE K URANISHI

Some estimates in ∂ ¯ _b Neumann boundary value problem for strongly pseudo-convex CR structures

Astérisque, tome 131 (1985), p. 217-235

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Article numérisé dans le cadre du programme

SOME ESTIMATES IN 9.NEUMANN BOUNDARY VALUE PROBLEM b

FOR STRONGLY PSEUDO-CONVEX CR STRUCTURES by MASATAKE KURANISHI

Introduction. We consider a system of partial differential equations of the first order

C D X f = 0 for all X € E ,

w h e r e f is an u n k n o w n complex valued function and E is a subbundle of the bundle C T M of the complex tangent vectors to a compact manifold

M , possibly w i t h b o u n d a r y . We denote by C°°(M, E) the vector space of smooth sections of E. We assume t h a t , for any X and Y G C°°(M, E) , their bracket [X,Y] is also a section of E. We also assume that E OE = { 0 } .

Let M-> Cn be a smooth embedding. Denote by E the set of all X € C T M such t h a t , w h e n considered as a complex tangent vector to Cn via the e m b e d d i n g , it is of type ( 0 , 1 ) . Then E satisfies the above c o n d i t i o n s , provided E is a subbundle. This always happens when the codimension of M i s 1. We say that E is embeddable w h e n it is locally obtained by embedding in C n .

The nature of the equation (1) depends very m u c h on its Levi-form.

N a m e l y , w e consider

(2)

C ( M , E) x C°°(M, E) 3 (X,Y) _l

i [ X,Y] mod E + E € C°° (M,C T M / (E + E)) .

We see easily that the map does not involve differentiation and actual­

ly comes from hermitian quadratic forms on the fibers of E valued in C T M / (E + E ) . H e r e , by hermitian w e m e a n w i t h respect to the bar ope-

ration induced on C T M / (E + E) b y that of C T M ,

W h e n the complex fiber d i m e n s i o n of C T M / ( E + E ) is 1 and the Levi-form is n o n - d e g e n e r a t e , w e have another interesting e x a m p l e . N a ­ m e l y , w e now consider the p r i n c i p a l b u n d l e on w h i c h its normal C a r t a n c o n n e c t i o n is d e f i n e d . Then the v e r t i c a l complex tangent v e c t o r s over E of the c o n n e c t i o n form a subbundle w h i c h satisfies our c o n d i t i o n s .

The e q u a t i o n (1) is c l o s e l y related to the complex it i n d u c e s . N a m e l y , denote b y A^(E) the b u n d l e of skew-symmetric m u l t i - l i n e a r m a p s E X . . . X E (q f a c t o r s ) •> C. Then w e have the exterior d e r i v a t i v e

(3) D : C ( W , A q ( E ) ) D : C (W,Aq(E)

just as in the case of de Rham c o m p l e x , i.e.

D u ( X 0 , . . . , X )

d x

^q ₁₌₀ D : C (W,Aq(E) D : C (W,Aq(E)D : C ( :4)

+

x

i < b

r -i^a+b

(-1) u D : C (W,Aq(E)xD : C

\>•• • >\>• • • >V •

Introduce h e r m i t i a n m e t r i c s on the fibers of E and a v o l u m e element of M . T h e n they induce a p r e - H i l b e r t structure on C°°(M, A^(E) ) . We w i s h to exhibit two formulas related to a semi-norm

(5) I I X D u l l 2 + Il X D * u | | 2

w h e r e X is an a r b i t r a r y real v a l u e d smooth function compactly suppor­

ted in the interior of M . W h e n the complex fiber d i m e n s i o n of C T M / (E + E) is 1, dim M = 2n - 1 w i t h q(n - 2 - q) > 0, and the b o u n d a r y of M satisfies rather strict c o n d i t i o n s (cf. (31)) these formulas can be combined to find an estimate of ( 5 ) . W h e n w e let X converge to a f u n c t i o n y w h i c h m a y not b e zero on the b o u n d a r y , in the limit esti­

mate w e find terms w h i c h involve integrals on the b o u n d a r y of M . Our m a i n c o n c e r n is to find an estimate of (5) such t h a t , u n d e r D-Neumann

ESTIMATES IN bh NEUMANN BOUNDARY VALUE PROBLEM

b o u n d a r y value c o n d i t i o n (cf. (3) and 3) ( 4 8 ) ) , these b o u n d a r y inte- grals are n o n - n e g a t i v e . The formulas are improved v e r s i o n s of those in (I) [ 4 ] , w h e r e one finds the details which are omitted h e r e . T h e - se formulas are not strong enough to solve the D-Neumann b o u n d a r y v a - lue problem. In the last section w e derive estimates from the a b o v e . Oui

2 2 2 hope is to find out eventually if the norm || D u | | + ||D * u | | L + C||u|| L

is compact w i t h respect to L^-norm, provided u satisfies conditions (cf. (48)) including the D-Neumann boundary value c o n d i t i o n . However it seems that our estimates are not yet strong enough to show the c o m p a c t n e s s .

Preliminary. We first fix a complex vector subbundle F of C T M (with F = F ) supplementary to E + E . Write for X,Y€C°°(M,E)

(6)

[X,Y] = i C F (X,Y) + [X,Y]F + [X,Y]__ s ß E

w h e r e

C (X,Y) € C ~ ( M , F ) , [X,Y]E €C°°(M,E) , and

D : C ( E

ec°°(M,E) .

We define E-hessian of a function f b y

(7) Hf (X,Y) = X Y f - [ X,Y' E

f

for any X, Y € C°°f M, E) . We find easily that it does not involve d i f f e - r e n t i a t i o n . When f is real v a l u e d

(8) H£(X,Y) = H £ ( Y , X ) + i C F ( X , Y ) f .

The exterior product induces an algebra structure on A ( E ) = y A ^ ( E ) , a n d

D ( u A v ) = ( D u ) v + ("1)PuA D v , uAv = M ) P q v A u

for u € C°°(M,AP(E)) and v € C°°( M, Aq (E)) . In terms of the m e t r i c w e in-

troduce the interior p r o d u c t L v : A P ( E ) + Ap"1 (E) by

(9) < u L v , w > = < u , v ^ w > .

If a € A '(E) and u € Ap (E)

(10) (u.v) L a = (u L a ) ^ v * ( - 1 ) p u A ( v L c O .

T h i s formula p l a y s a crucial role in the proof of (23) . We assume that the support of X is so small that w e can pick an o r t h o n o r m a l b a s e e 1 ' " * * , e m °^ A ^ ( E ) defined on a n e i g h b o r h o o d U of the support of X.

Let g : A \ E ) -*A^(E) be a h o m o m o r p h i s m of v e c t o r b u n d l e s over the iden­

tity m a p of M. T h e n w e let g also operate on A ^ ( E ) b y

(11) g u =

I

_k ( g e k ) A ( u L e k ) .

Thus g f = 0 for a scalar v a l u e d f u n c t i o n f. We see easily that the above g c o i n c i d e w i t h the given g w h e n q = 1. Since the r i g h t - h a n d si­

de of the above is clearly independent of the choice of o r t h o n o r m a l e^,...,e , (11) is defined g l o b a l l y . We also see easily that the ad-

* 1 1 joint of the above g is equal to the m a p induced b y g : A (E) -> A ( E ) , where * is in terms of the m e t r i c . M o r e o v e r

(12) g ( u . v ) = (g u ) A v + u A g v

(13) g(u L v ) = ( g u ) L v - u L g v .

We d e n o t e b y Y 1 , . . . , Y „ a b a s e of E dual to

e 1 ' * • * 'enT We set (14) [Y. , Y k ] =

I _f

rjk£ Y £

and define r (K) : A1 (E) -* A1 (E) b y

(15)

r ( k ) e £ " I J r k j * 6j * For K = ( k r . . . , k ) and u eC°°(U,Aq(E)) , set

ESTIMATES IN &h NEUMANN BOUNDARY VALUE PROBLEM

(16) U K = U (Yk ,..-,Yk ) .

If G is a v e c t o r field on U, w e let G operate on C (U,Aq(E)) by

;i7) ( G u ) K = G u K .

By d e f i n i t i o n w e see that the o p e r a t i o n of G c o m m u t e s w i t h the exte- rior and the interior p r o d u c t b y e^. We define Yk : c"3 (U, Aq (E) ) -»•

C ^ U . A ^ E ) ) by

(18)

0,

Y k u = Y-,Aq(E)

)

) r o o u -

It then follows (cf. §.1 I [4])

(19) D u =

1

d

^{ek Y k u .}

Hence by d u a l i t y

(20) D ' u =

d o

o

^{Cu L ek)}

.

Finally, set

(21) [Yi;Y*] =

i C p ^{D : C E)}

xw Y* + Y* + Ï

c Y* Y* + Y* +

(cf. ( 6 ) ) . We d e f i n e

q(k) : A 1(E) •* A 1(E) b y

(22) : A 1(E)

Ï

^{: A 1(E)}

A priori e s t i m a t e . Let X b e as indicated in ( 5 ) . Pick a section u of Aq(E) w h i c h is smooth on a n e i g h b o r h o o d of the support of X. To find a formula for the semi-norm ( 5 ) , it is enough to consider

* 2 ? * 2 D X D + D X D . For simplicity w e consider X instead of X for a w h i l e .

(23) PROPOSITION 1.

(D* X D + DX D*) u = A u + B u + C u , wheve A = A^ +Ag with

A1u -

I

_k

^f

^* _k ^{X Yku}

A2u

= hj3'

(Ak)+

1_

2

: A 1(E)

X V M L V

+ ek Y *

«7 X(u L (q + 1_

2

: A 1(E) A 1(E)

r*(k) ejk(u lr*(j) ek} *

B u - *-

3 J k

j ^

(XiCF(Y.,Yk) + HX (Y^Yk) ) (u L 3 C u = D (u L D X) + D X

x

^{D u .}

O u t l i n e of the p r o o f . First w o r k out the commutator r e l a t i o n b e t w e e n Y, and the exterior p r o d u c t by e- (cf. ( 1 2 ) , ( 1 3 ) ) , Similarly for the

K J interior p r o d u c t b y e-. W r i t e down ( D * X D + D X D * ) u using (19) and

( 2 0 ) . A p p l y (10) and r e w r i t e it as the sum of Ek Y*k X YK u -and terms c o n t a i n i n g u L e ^ . . T h e n our formula follows b y (21) and ( 7 ) .

We note that the above formula is a p r e c i s e v e r s i o n of the one given by J.J. Kohn in [ 2 ] .

Note that < u , u > ^ 0. In view of A ^ , w e do not have to w o r r y too m u c h about the term < A^ u , u > . W h e n w e let X converge to a func­

tion w h i c h may not be zero on the b o u n d a r y of M, D X w i l l b l o w up on the b o u n d a r y . Note that D X appears in C u . The N e u m a n n b o u n d a r y con­

d i t i o n w e consider later is exactly the one w h i c h m a k e s < C u , u > go 2 to zero. W h e n w e w a n t to obtain an a p r i o r i estimate for || D u || +

| | D * u | | 2 + C ||U||2, w e see then that the m a i n d i f f i c u l t y comes through

< B u , u > . We try to eliminate this term by taking a d v a n t a g e of the term Z j C Y j ) * XY- in A .

ESTIMATES IN bh NEUMANN BOUNDARY VALUE PROBLEM

_ -o %

We assume that M is in a m a n i f o l d M and E extends to M. Let t be a real valued function on M . For later a p p l i c a t i o n w e consider the case w h e n the b o u n d a r y of M is defined by t = 0 . H o w e v e r , there is no need to do so now. Set

[24) Y.t = a.

=

a). = a . /b

J J

=

b = (> a. a-) 1/2

§

We also define (where b * 0)

(25)

Y =Ik"kYk >

1(E)

W u j Y ~h% Yk _§

Qkj =ôkj ~wk uj d wj =^k Qkj Yk

so that

(26) W, t = 0 ,

I _

^%

w • W . = 0

3 J

d

Then we see that

(27)

£kYkXYk =

^{( Y}° ) * X Y ° +

: A 1(E)

We set

(28) W

1 ^{d K k Y k •}

When X is a vector field and f is a f u n c t i o n , w e often w r i t e [ X, f ] instead of X f , i.e. w e regard a function as a m u l t i p l i c a t i o n opera- tor .

(29) PROPOSITION 2. Assume that the support of X is contained in M' = {p € M ; b(p) * 0]. Then lw*XW. = Ar + Br + Cr, where A ' - A ' + A ' + G

J J 1 2

with

4 ' =

e (w rj

* X

(w rj

+ •lj(X B~1HT

(W .,w.

3 3

)Y° + (Y°)*

xb-1 Bt(wj3w.)),

A ' - -

(w rj (xaYv (w

+ ^K QLK QKLJ)WJ

c (w rj (w rjv xcv

QKL QKLJ> ³

G=-X ^3,K (II, t

(w rj

c (w rj

_Q3KL

xcvb cvbg

+ Q3K QJK} ³

B' =

d ixcF (W W J

,7 ,7 +x

x xw c(w rj

C •=-

c

2[Y

xc

^V(w

rj c (w rj c [vx] ^c

²

ⁱ

^{a 3}

(w rj v(w rj (w rj w (w rj

rj+

(w rjv(w rj

Outline of the p r o o f . Since the m a t r i x (Qj^) defines a p r o j e c t i o n operator

c vc

^*

cx (w rj

_vYk X Q j k Y j

(w rj (w rj

X Wj -

£j,k :x Qjk

rj (w rj

+ [ Y k > X Q j k 3YJ

- k [YJ> x Q j k

(w rj

w h e r e

R = -

(w rj

[Yk> [ Y r X Q j k v x v v ] ]

= - 2 ^k [Yk>

[w k,xU + ^i,k

( I Y k , [ Y . , X ] Q j k l " t Y k ^ C Y j

,QikU)

= - 2

c

[Yk,[Wk,X]] + (a p u r e l y imaginary n u m b e r )

(w rj

[Yk>[Yi • X ] ] Q j k

(w rj

X t Y k , [Yj)Qjk]] .

ESTIMATES IN 6B NEUMANN BOUNDARY VALUE PROBLEM

The last term of the above goes into G, and the second from the last (modulo i t ) is lj,k V[Y,[Y,,X]]Qkj w h i c h goes into B'.

The case of c o d i m e n s i o n 1 w i t h definite Levi-form. We fix a generator S eC°°(M , T M ) of F. Write

(30) C F ( X , Y ) = C S ( X , Y ) S .

To get rid of the B term in Proposition 1, we put a condition on the b o u n d a r y of M. Pick a real valued function t on M w i t h o u t any critical point on the b o u n d a r y of M and such that M is defined by tN<0 and the b o u n d a r y of M is defined b y t = 0.

(31) DEFINITION 1. Assume that M is of dimension 2n - 1 with n > 3. We say that the boundary of M is admissible when

1) There is a smooth function y on M such that

Ht = ycs

at each point on the boundary of M, provided n >, 4. If n=33 we assume further that alt the first order partial derivatives of C^ - Y also vanish at each boundary point of M.

2) At each boundary point p G M such that b(p) -0 (cf. (24))

Y(p) * 0 ,

3) For any X, Y £ C°° (M3 E) and for any p as above

X Y t(p) = 0 .

We see easily that the above d e f i n i t i o n is independent of the choice of t as w e l l as of the choice of a supplementary real vector field S.

To give an exemple of M w i t h admissible b o u n d a r y , consider a

°° n n real h y p e r s u r f a c e M 9 C of c o d i m e n s i o n 1. We w r i t e (z,w) E C , w h e r e

n-1 % ^

z € C and w E C . We assume that the origin is in M and M is given b y an e q u a t i o n :

(32) y = h ( z , z , x )

w h e r e x = Rw and y = 3 m w . We assume further that

(33) hfz.z . xl = u 1 -k

. -, h . 7 Z J z J ,k jk

m o d ( z , z , x ) 3

w h e r e (hj^) 1S a p o s i t i v e definite h e r m i t i a n m a t r i x . As is shown b y Chern and koser in [1] w e can always find a h o l o m o r p h i c chart (z,w) so that the above is valid l o c a l l y , provided M is strongly p s e u d o c o n v e x .

(34) PROPOSITION 3. For a sufficiently small r > 0 set

— 2 t=h(z3z3x) + E u -v .

Then the equation t ^0 defines a submanifold M with admissible boun­

dary .

In the following w e always assume that the b o u n d a r y of M is ad­

m i s s i b l e . We pick a smooth real valued function cp(t) on R supported in {t € R ; x < 0} such that <p(t) = 1 for t < - c^ for a p o s i t i v e number c-j . We also pick ii€C°°(M,R) w i t h compact s u p p o r t . We assume that its sup­

port is small enough so that we can find an orthonormal b a s e Y-|,... ,Yn_^

of E on a n e i g h b o r h o o d of its s u p p o r t . We set

(35) X = y cp(t) .

In the following w e outline how to get rid of the B term in ( 2 3 ) , w h i c h is the m a i n o b s t r u c t i o n to obtain an a p r i o r i e s t i m a t e .

Note that for functions f and g

ESTIMATES IN ôh NEUMANN BOUNDARY VALUE PROBLEM

(36) H f g ( X , Y ) = f H g ( X , Y ) + g H £ ( X , Y ) + (X£)(Yg) + (Yf ) ( Xg) .

Also w e see that

(37)

l(wrj* =cp

(t) H t ( X , Y ) + <p"(t) ( X t ) ( Y t ) .

Write

(38) HV(Yj,Yk) = Y C S (Y Y k ) + rjk

l(w

= 0 on bd M .

We pick our h e r m i t i a n metric to be the one defined b y Cg (which w e can always assume to be positive definite by replacing S by - S if n e c e s s a r y ) . Then w e see that

< Bu,u > = < B1 u,u > + < B~u,u >

d

^{w h e r e}

< B, u, u > = -EK

EK

< (i X S + y Y tp') (u L e k ) , u L ek >

< B^u,u > = -

^i,k

< y <p! (t)r -k (u L ek) , u L e . >

(39)

- 2 K< cpf u L Dt,u L Dy > - < cp" u L Dt ,u L Dt >

l(w

<cpHy ( Y j Y k ) (u L ek) , u L e . > .

Note that

(40)

li HZ (Wj,Wj) = Y(n-2)(i + r o ; , rQ = 0 on b d M ,

Rj l(wrj*Wj)

= (n - 2) .

and

(41) q < v,w > =

d

< v L ek ,w L ek >

w h e r e q is the degree of v and w . In view of (27) w e can always take

d

(q/(n-2))W* X W , u out of the A term, and

(1/(n-2))

Ek

< B ' ( u L ek) ,u L ev > = < BJu,u > + < B ^ U y U > , w h e r e

(42)

< B'u,u > =

Ek

< (i X S + y Y cpf ) (u L ek) ,u L ek >

< B'u,u > = < U CD ' r o

u L ek,u L ek >

+ (1/(n-2))

'h

<pHy(W, w ) (u L eR) ,u L eR > . (cf. ( 2 6 ) ) . Hence

< Bu,u > +

lk

( V ( n - 2 ) ) < Bf (u L e R ) , u L ek >

= < B ^ U j U > + < B^u,u >

Note that the right-hand side of the above does not cause any trouble at the b o u n d a r y under the D-Neumann b o u n d a r y c o n d i t i o n . By (23) and

(29) w e find then that for X as in (35)

< (D* X D + D X D *)u,u > = < Y * X Y u , u >

o o '

[43)

+

d

a-2 (2 Kl< X Y o

d d

d

b " V (W. ,W. )u > + < Gu,u > )

+

l(wr

n-2

x x

: W * j

a.

X W.

J

u ,u > + ^q n-2

E

j

< ( W . ~ ) * X W . A u , u >

+ < B2u,u > + < B^u,u > + < (A2 + C + ;q/(n-2)) ;A^ + c' ) u , u > .

Now w h e n w e calculate G m o r e e x p l i c i t l y , w e find that

G = X b 2

I L

Ht ( W W ) | ^{2 * s}

y

^{w i t h}

(44) - 4

G1 = - X b ^

l(wrj* x l(wrj(ZjHt

(Wj,Wj)

Ll ,i,s Q . - q . - a ) 11 j i s s^

+ X b 1 R

w h e r e R is b o u n d e d . T h e r e f o r e w e o b t a i n :

ESTIMATES IN bu NEUMANN BOUNDARY VALUE PROBLEM

llx 2 D u | | 2

+

II x

_1

2 D*u|| Z

= H

X 1

2 (Y +•

v o q

n-2 b"1

Ej

H* {WjJWj))u|| 2

+ (n-2-q)q (n-2)Z

II X 1

2 b " 1 I . H 1 (W W,)u|| 2

(45) ⁺ n- 2-q

n- Z

Ej ||

^X

1

2 °° ,,2 Wj u|| + q

Q-2

L l l x

1 2 W.

J

~u||2

+ < G 1 u , u > + < B 2 u , u > + < B^u,u >

+ < (A2 + C + (q/ (n-2))(A^ + C')u,u > .

Note by ( 4 0 ) , ( 4 4 ) , and 3) (31) that < b 2 G 1 u , u > v a n i s h e s w h e r e b = 0 , provided the support of X is contained in a sufficiently small neigh­

borhood of a b o u n d a r y p o i n t .

D-Neumann b o u n d a r y value p r o b l e m . The classical method of Kohn and Nirenberg (cf. [ 3 ] ) to solve the problem is to find a norm ||u|| ' on C 2 ( M , A q ( E ) ) such that

D

II

^{u|| '} is compact w i t h respect to L2~norm || u|| , 2) Il D u Ü 2 + || D*u

11

2 + C ||u|| 2>c (|| u||')2

for all u satisfying the b o u n d a r y condition : u L D t = 0 .

We apply the same method in our c a s e . H o w e v e r , the nature of the formulas in PROPOSITION 1 and 2 forces us to m o d i f y it. Firstly, since b ^ comes in our picture w h i c h is not smooth, it is more natural to enlarge the space C 2 ( M , A q ( E ) ) . S e c o n d l y , since w e localize and use d i f f e r e n t m e t h o d s to prove our estimate depending w h e r e w e a r e , w e re­

place a single norm || u|| ' b y a p r e - f r e c h e t space s t r u c t u r e .

We first study n e i g h b o r h o o d s of b o u n d a r y p o i n t s p w i t h b ( p ) = 0 . They are the characteristic b o u n d a r y p o i n t s . By the n o n - d e g e n e r a c y of

C„ and 1 ) , 2) in (31) w e see easily that :

(46) PROPOSITION 4. Let p be a char act eristic boundary point. Then, on a sufficiently small neighborhood of p, (t,o) is a chart.

(47) COROLLARY. The set' of characteristic boundary points is isolated.

(48) DEFINITION. We denote by C ' (M, ï\q (E) ) the vector space of sec- tions u of Aq(E) on M satisfying the following conditions :

1) u is C in the interior of M.

* — 1

2) D u and D u are in L ^, b u is in L^ on a neighborhood of each characteristic boundary point, and W .u is in L^ in a neighborhood of each boundary point.

3) For each C°° function f on M whose support is compact and dis- joint from the set of characteristic boundary points, f t ^ u L D t is in Lg .

We prove a priori estimate on C1 (M,Aq(E)) . The above condition 3) is the D-Neumann b o u n d a r y c o n d i t i o n . We w o r k separately on n e i g h - b o r h o o d s of interior points of M, of characteristic b o u n d a r y p o i n t s , and of non characteristic b o u n d a r y p o i n t s .

(49) PROPOSITION 5. Let X be a C°° function with compact support on M which is zero on the boundary of M. Then there are constants C > 0 and c>0 depending on X such that for any u €C 1 (M,l\q(E))

\\D u\\2 + || D*u\\2 + C \\u\\2>c(Ej \\Y.Xu\ 2

l(wx

Y* Xu\\ + \ < S Xu , X u > \ ) .

This follows by (23) b e c a u s e w e can get rid of B term (without introducing b ) by the w e l l - k n o w n method of Kohn. Note in the above

ESTIMATES IN bb NEUMANN BOUNDARY VALUE PROBLEM

that the term J\ ||Y^ Xu||2 is independent of a choice of a local or­

thonormal base Y1,...,Yn 1 and hence has a global m e a n i n g . Similarly L IIY- Xu|| 2 has a global m e a n i n g m o d u l o a term « C U u|| 2.

We next consider a small n e i g h b o r h o o d U of a b o u n d a r y point p Q . In (35) w e take y e C ° ° ( H , R ) w i t h support in U D { b * 0 } . We also replace cp(t) by cp£ = cp(t/e) and let £ -> 0. In (45) the term w h i c h contains the d e r i v a t i v e s of <p(t/e) in t and the derivatives of y is

E = < B2u,u > + < B^u ,u > + < (C+ (q/(n-2)) :f)u,u >.

Because r.k = rQ = 0 on the b o u n d a r y of M we see by 3) (48) that E con­

v e r g e s to

E1 y

= 2 & < D*u,u L Dy > - (2q/(n-2))

l(wrj[Yk

[Wk,y]]u,u >

+Ej

< [Wj,y]u , WjU >

= l(wrjxjYj,Yk]

(u L ek) , u L e . > )

+ (q/(n-2))

wrj* x

(W.,W.)u,u >

3 3 b e c a u s e (cf. (26))

y[W, ,cpf] = 0 [Y- ,tp ] [W y] = 0 . 3 £ J

Assume now that pQ is a characteristic b o u n d a r y p o i n t . We assume that U is sufficiently small so that (t,a) is a chart on U (cf. ( 4 6 ) ) . We pick y^EC°°(M,R) w i t h support in U and set

y = y -j <P ( ¹ b )

and let e-*0. Because b u is in L2 (cf. 2) ( 4 8 ) ) , w e find that E ^ converges to E' . In view of (40) and 3) (31) we then find by (45)

y 1 the following :

(50) PROPOSITION 6. Let pQ be a characteric boundary point of M. As­

sume that q(n-2-q) >0. Then there is a neighborhood U of po such that3

CO

for any Y € C (M,R_) with support in U3 there are constants C3c>0 such that

\\Du\\2 + || D*u\\ 2+C\\u\\2 ^ M

irl

II 2

Q J J b Y U I I

for any u £ C ' (M3 R_) .

We next consider a n o n - c h a r a c t e r i s t i c b o u n d a r y point pQ and pick a s u f f i c i e n t l y small U w h i c h d o e s not c o n t a i n any c h a r a c t e r i s t i c b o u n ­ dary p o i n t . Let y EC°°(M, R) w i t h support in U . T h e n the above argument proves that for any u £ C 1 (M , R)

(51) ||Du||2 + ,|D*u||2 • C | | u | |

l(wrEk

_C || Y k y u || 2 + l| W k y u | | 2 ) .

Looking at terms B and B' in (23) and (29) w e also find that

(52)

l | D u | i 2

+ ||D*u||2 + C||u||2 >=c\ < i S y u , y u > - < Y p u , y u > y |

w h e r e < u , u > ^ denotes the square of the L2~norm of the r e s t r i c t i o n of u to the b o u n d a r y of M. W i t h cp as in ( 3 5 ) , [YQ,cp] = tp1 (t)b . Hence w e see easily that

- < Y y u , y u > b d = - < b Y Y ° y u , y u > + < y u , ( Y ° f u Y b u > .

T h e r e f o r e b y (51) and (52)

(53) ||Du||2 + ||D*u||2 + C | | u||2 » c | < i b~ 1 X o

y u,y u > |

w h e r e

(54) X

o

= i b S + Y Y ° - Y Y . o

Note that X Q , Wj, Wj are tangential to the b o u n d a r y of M . We denote b y (bd)'M the set of n o n - c h a r a c t e r i s t i c b o u n d a r y p o i n t .

ESTIMATES IN bb NEUMANN BOUNDARY VALUE PROBLEM

(55) PROPOSITION 7. (bd) 1M has a foliation of codimension 1 suoh that X W . 3 W . generate the complex tangent vector space of each

o «7 «7 leaf.

Outline of the p r o o f . It is enough to show that the equat ion XQ =W_J = Wj

= 0 , w h e n restricted to (bd) 1 M , is completely i n t e g r a b l e . This fol- lows by the same calculation made in §.2 II [ 4 ] . As long as w e do not d i f f e r e n t i a t e H t ( Y . , Y , ) , the c a l c u l a t i o n there is still valid for our

J K

more general t. In view of 1) ( 3 1 ) , no m o d i f i c a t i o n is needed w h e n n = 5. For n > 4 we have to take a little more care for terms contai- ning [ Y r Y ] . Set P j = [ Y j ) Y ] - ^ i r j ¥ o k - Y r j and H1 (Yj , Y k ) = Y 6 . R + a. R w i t h a_-k = 0 on the b o u n d a r y . Then instead of the formula P.. ^ ^ £ = P j ^ ^ j £

(cf. the middle of the proof of (2.23) II [ 4 ] ) , w e have

Pi ôk£+ C V a k £ ] l(wrj* x l(wrj*jl]

Apply £j Qji ££>k Qk£ • We then find Ij Qj i Pj = 0 on ( b d ) ' M , provided n ^ 4 . This is w h a t we need. The term containing [Y^,Y] can be also handled similarly.

In view of (51) and (5 3) w e find by the above the following :

(56) COROLLARY. Let V be any complex tangent vector field on U which is tangential at the boundary to the leaves of the foliation in (55).

Then

\\Du\\2 + \\D*u\\2 + C\\u\\2 >,c\ < V y u3 y u >

I .

Note that Y ° - Y ° is tangential to the b o u n d a r y and its restric- tion together w i t h the restrictions of XQ, W j , Wj generate C T (bd M ) .

(57) PROPOSITION 8. The flow generated by i b 1 (Y° - Y°) preserves the foliation of (55).

Outline of the p r o o f . By the same c a l c u l a t i o n as in §.2 II [4] w e find that [b~ 1(Y° - Y ° ) ,W^] s 0 mod W ^ W ^ Hence [b~ 1 (Y° - Y ° ) , ] = 0 mod W^. > W ^ . Since W. , W - w i t h b r a c k e t generate XQ, our c o n t e n t i o n fol­

lows.

We are now going to prove the following :

(58) PROPOSITION 9. There is a neighborhood U of pQ in M with a chart (x3y13y')3 y 1 - (y 2> * - - >y 2n_^ > centered at p q satisfying the following : 1) U = U Ç\M is given by x4 03 2) the equations x = 0 and y1 - constants define the local fibering of (52), 3) Y° - ^ b(d/dx+

i 3/3 z/ ) + B with B - 0 at each boundary point in U3 and 4) for any u G C ' ( M 3 A q ( E ) ) with q(n-2-q) > 0

\\Du\\ \\D*u\\ 2+C\\u\\ 2ic(\\vu\\ \/2)2

where ( || ||1/2)2 denotes the integral in (x3y^) of the square of the Sobolev norm with respect to the variable y'.

PROOF. Pick a chart y! centered at p of the local fiber F^ of the

J r o o f o l i a t i o n in ( 5 5 ) . Consider the flow generated b y i ( Y ° - Y ° ) / b . Let

y = ( y p - - - > y ' ) be the p o i n t on the b o u n d a r y w i t h the p a r a m e t e r y^

o r i g i n a t i n g from yf in P . This gives a chart of bd M . We now use the flow generated by ( Y ° + Y ° ) / b to define a chart ( x , y ) . By the c o n s t r u c t i o n

Y ° - y0 = i b a / a y - + 2 B Y ° + Y ° = b a/ax

w i t h B = 0 at each b o u n d a r y p o i n t . Note also that (Y° + Y ° ) / b = 2 3/3t m o d u l o a v e c t o r field tangential to the b o u n d a r y . Hence the inequa­

lity x ^ O d e f i n e s U. Now our c o n t e n t i o n follows b y (56) and ( 5 7 ) , q. e . d .

ESTIMATES IN ôh NEUMANN BOUNDARY VALUE PROBLEM

R E F E R E N C E S

[1] C h e r n , S.S. and Moser, J.K., Real hyper surfaces in complex mani folds. A c t a Math. 113 ( 1 9 7 4 ) , 2 1 9 - 2 7 1 .

[2] K o h n , J.J., Harmonic integrals on strongly pseudo-convex mani­

folds I. A n n . of M a t h . 78 ( 1 9 6 3 ) , 112-148.

[3] K o h n , J.J. and N i r e n b e r g , L., Non-coercive boundary value pro­

blems. Comm. Pure A p p l . M a t h . 18 ( 1 9 6 5 ) , 4 4 3 - 4 9 2 .

[4] K u r a n i s h i , M . , Strongly pseudo-convex CR structures over small balls I, II, and III. A n n . of M a t h . 115 ( 1 9 8 2 ) , 4 5 1 - 5 0 0 ,

116 ( 1 9 8 2 ) , 1-64.

MaSatake KURANISHI

Columbia University in the City of New York

Department of Mathematics NEW YORK, N.Y.10027 U.S.A.