GLOBAL REGULARITY FOR d ON WEAKLY PSEUDO ... REGULARITY FOR d ON WEAKLY PSEUDO-CONVEX MANIFOLDSt1)...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 181, July 1973

GLOBAL REGULARITY FOR d ON WEAKLY

PSEUDO-CONVEX MANIFOLDSt1)

BY

J. J. KOHN

ABSTRACT. Let M' be a complex manifold and let l/cc M' be an open

pseudo-convex submanifold with a smooth boundary which can be exhausted

by strongly pseudo-convex submanifolds. The main result of this paper is the

following: Given a 3-closed (p, t7)-form a-, which is C on M and which is

cohomologous to zero on M then for every m there exists a (p, q — l)-form

Z2, . which is C on M such that ¡lu, , = a.\m) 17777

1. Introduction. In this paper we study global regularity properties of the

Cauchy-Riemann equations in serveral complex variables. This work deals with

forms on manifolds but in this introduction we will describe our results for the

special case of functions on domains in C".

Given a domain M CC C" and functions a., /= 1, ■••,«, on M we consider

the inhomogeneous Cauchy-Riemann equations:

(1.1) Z2_ = a , j = 1, . . . , n,z . '

1

where z , • • • , z are coordinate functions on C", x . = Re (z .), y . = Im (z .),1 ' ' 77 '; 11 i

u = VAduJ dx . - \j— Idu/dy ) and zz- = V2{du/dx . + \f^ldu/dy .). We are con-z j i i z j l i

cerned with the dependence of u on a; in particular, if the a . are "smooth"

can we find a "smooth" u satisfying (1.1). Since the system (1.1) is elliptic,

the interior regularity is well understood—for example if the a. £ C°°(zVl) then any

solution zz of (1.1) is in C°°{M) since u satisfies the equation Au = 4~S. a . .,z>

So that our question really is about smoothness at the boundary of Al, here we

cannot expect that every solution u is smooth, since u + h is also a solution

whenever h is holomorphic on M. We will assume that bM, the boundary of M,

is smooth; more precisely, we assume that in a neighborhood (/ of bM there

exists a real-valued function r £ C°°{U) such that r > 0 outside of Al, r< 0 in-

side Al and dr 4 0 everywhere, so that bM is given by the equation r = 0. We

further assume that Al is pseudo-convex, which means that if P £ bM and

{a1, ■ ■ ■ , a") £ C" such that 1 r {P)a> = 0 then

Received by the editors July 27, 1972.

AMS {MOS) subject classifications (1970). Primary 32C10, 35N15.(1) During the preparation of this paper the author was partially suppotted by an

NSF project at Princeton University.

Copyright © 1973, American Mathematical Society

273

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

274 J. J. KOHN

(1.2) Zz - ÎP)a'ïïk > 0.z .z

1 k

This hermitian form is called the Levi-form. Pseudo-convexity is a natural con-

dition which, among other things, implies that the system (1.1) always has a solu-

tion, provided the a. satisfy the necessary compatibility conditions:

(1-3) jzL kz.a _ = a

k s

Now we want to distinguish between two types of regularity theorems: glo-

bal and local. By global regularity we mean that, given smooth cl , on M, there

exists a smooth u on M satisfying (1.1). By local regularity in a neighborhood

ii of P £ bM, we mean that, given a ., there exists u satisfying (1.1) such that

the set of points in U at which the cl . are smooth contains the set in U at whichr i

u is smooth.

Local regularity holds whenever the Levi-form (1.2) is positive definite at

P, in fact if the Levi-form is positive definite at every point of bM then the

unique solution zz of (1.1) which is orthogonal to the holomorphic functions pos-

sesses this property (as proved in l7j). Recently there has been great interest

in boundary behavior of solutions of (1.1) on strongly pseudo-convex domains

(i.e. (1.2) positive definite) and quite precise results have been obtained in terms

of continuity and Holder norms (see [6], [3] and [4]).

When the Levi-form is semidefinite but not strictly positive definite then

further invariants control the question of local regularity (see [8]). If the Levi-

form is identically zero in a neighborhood then local regularity does not hold

(see l8j). On the other hand, here we obtain the following result, which is a spe-

cial case of our main theorem (3.19).

(1.4) Theorem. // M CC C™ is pseudo-convex with smooth boundary, cl . e

C°°(/V)), / = 1, • • • , tz, and if the cl . satisfy the compatibility conditions (1.3)

then for each integer m there exists a function u, , £ CmitA) satisfying (1.1).

The question immediately arises whether, under the assumptions of the above

theorem, there exists a solution u £ C°°ÍM). This remains an open problem; in

particular, it would be interesting to know whether the solution u which is orthog-

onal to holomorphic functions is smooth.

It is convenient to use the standard notation:

du = Z u- dz ., cl = 2-f cl. dz.

(1.5)

z. JI

da= Z (a _ - cl _)dz . t\ dz,<ze i*k kz . '


GLOBAL REGULARITY FOR d 275

The system (1.1) then becomes du = a and the compatibility condition (1.3) be-

comes da = 0. The questions posed here are studied via the r}-Neumann problem

with respect to the type of weights introduced by Hörmander in [5l\ The essen-

tial tools are the techniques developed in [7] and [9]. In C" this means consid-

ering the "weighted" hermitian form:

(1.6) Rticß, </» = foci, fdlfr) + {Ocß, ûiflfj))

where

ftiz) = expfo V2t\z\2), <t>='E<p.dz., 0cß = -'£cß)z,

and ( , ) is the sum of the L -inner products of the components. Given a we

then show there exists a solution cß to the variational problem

(1.7) Rticßt, iß) = {a, ftiß), for all t/z,

under the boundary conditions

(L8) Zr </>.= Zt- t/>. = 0 on bM.Z .' I Z . I

l Î

Then a solution ot (1.1) is given by

(1.9) "t-L&ft*?

and we show that given m then zz e Cm{M) if t is sufficiently large.

In case Al is a manifold the above outline has to be somewhat modified.

First, to define / , we must assume that there exists a function with properties

analogoustto |z| (i.e. which is strongly plurisubharmonic in a neighborhood of

bM). Second, in general, the d-cohomology of Al is different from zero and this

is related to the fact that then R is not positive definite. We then consider the

positive definite form Q {eß, \fi) = R.ieß, iß) + (</>, /Ai) and solve the variational

problem (1.7) for this form. Using this solution we then obtain the desired solu-

tion of (1.1) under the additional necessary condition that a is orthogonal to the

"harmonic space". Throughout this paper we follow the type of program set up

in [8]; for more detailed explanations we refer to [lj.

2. The f3-Neumann problem with weights. Let Al be a complex hermitian

manifold of dimension n, and let M CC M be an open submanifold of M whose

closure, M, is compact. We denote by bM the boundary of Al and we assume

that there exists a neighborhood U of bM and a real-valued function r £ C°°fo/)

such that dr 4 0 and r{P) = 0 if and only if P £ bM. We normalize r so that

(2.1) \dr\ = 1 on bM,


276 J. J. KOHN

where | | denotes the length defined by the hermitian metric. The sign of r is

chosen so that r < 0 in M and r> 0 outside of M.

By Clt''q we denote the space of forms of type (/>, a) on M which are C°°

up to and including the boundary, i.e. they are restrictions to M of C forms on

M'. We set

(2.2) Q = Z(îp-q.

In terms of local holomorphic coordinates z •• • , z on a coordinate neighbor-

hood V we can express (f> £ u by

(2-3) (p = Z(p,,dz' A dz1,> J

where (p.. £ C°°ÍV O M); / = (z,, • ••, í ) with 1 < i, < • ■ • < i < n, 1 < p < n;

/ = (/,,•••,/) with 1 <;',<••••< ; < n, 1 < q < n; dz1 = dz . A .. . A fife.' M' ''9 — 1 Z?_>_2_> ;l

and ¿F' = dz~. /\ • ■ ■ A dz . .71 ~ z? zV?

The operator <?: Cl * Cl is expressed in terms of local coordinates by

(2-4) f(p = Z -^ dz, A¿F'A^ = Z(- l)l'l -IL ¿« ziz' A ¿F<«\z, dz,

Ze Ze

where |/| = p, (kj) is the ordered (a + l)-tuple consisting of k, and the elements

of / and

ÍO Ü0 ilsign

f A 4B,'2-5) e = / 0 if A has repeated elements

of the permutation A —» B otherwise.

Then

(2-6) cXCt^)cfi^ + 1 and d2 = 0.

The hermitian structure induces an inner product on Cl for each P £ Ni.

Thus if <p, 1/7 £ (3 we denote by (<£, <A)p their inner product at P and by (0, yj)

the function whose value at P is (ci, <A)p. In terms of coordinates we have

(2-7> <^>=Z^KLg';'KL,

where

g'J-KL = (ziz'A ¿*J. z7zK A cEL)

M |gf»*t„.,lW*Wi'»,»...î,l/l,W if |f| - |X| and l/l « |L|;

((J if either |/| ¿ |K| or |/| 4 \L\,

and glk = {dz., dz ), g'! = (dz ., dz f.



Further if cß, iß £ (1 the L -inner product and norm are defined by

(2-9) {cß, iß) = f {cß, iß)dV and \\cß\\2 = (0, cß),•7 M

where dV is the volume element induced by the hermitian metric. We denote by

(1 the inner hilbert space obtained by completing (1 under the norm || ||.

The formal adjoint of d is denoted by d: Cl ' <J and defined by the require-

ment that

(2.10) {aeß,iß) = {cß,diß)

fot all iß with compact support in Al. Then

(2.11) 0(0*'«) CO*-«-1 and tf2 = 0.

If P £ Al we denote by u the space of forms at P, by Tp (and T*) the

real tangent (and cotangent) space to Al at P. Then for each cotangent rj £ Tp

with ï] 4 0 the symbol of d is a linear map a„(d, rj): (íp —' (l„ given by

(2-12) Op{d, 77)0 = (np77) A 6,

where II : Up' © Up' 'UJ is the projection. The symbol of t>, denoted

by crp(ô, rj): ä p —' (l , is then the adjoint of o „id, rj) with respect to the inner

product { , )p Integration by parts gives us

(2.13) ideß, y) = {cß, diß) + ¡bM{o{0, dr)cß, iß)dS,

for all cß, \ß £ il, where dS is the volume element on bM. We define the space

SCO by

(2.14) -3)= \cß £ä\apio, idr)p)eßp = 0 for all P £ bM\.

Thus we have

(2.15) («></,, </» = {cß, diß)

whenever cß £ A and iß £ Cl. On il we define the hermitian form Q: ii x JJ ' C

by

(2.16) Qicß, iß) = {dé, diß) + (!></>, dip) + icß, iß).

Let ¡1' be the hilbert space obtained by completing U under the norm Qieß, cß)/2.

Since we have

(2.17) Ofo>, c4)> ||cS||2,

there is a natural imbedding of 1< in (l. Then by a well-known theorem in

hilbert space theory there exists a unique operator F with Dom(F) C J such that


278 J. J. KOHN

(2.18) Q(cp, ¿) = (Fcp, if,)

fot all 0 £ Dom(F) and yj £ A), and AÍF) = il, where J\ÍF) denotes the range of

F. F is called the Friedrichs representative of Q; it is selfadjoint and its in-

verse is bounded. For smooth forms in Dom(p) we have

(2.19) Fçh = G0 + cp,

where D is the complex laplacian defined by

(2.20) G = dô + tWL

In [71 (see also in L9j and llj) the smooth elements of Dom (F) are character-

ized by the following boundary conditions:

(2.21) Dom ÍF) n CÍ = Í0 e 3)| ¡30 e 3)1.

The space of harmonic forms K is defined by

(2.22) K = |0 e3)|O(0, 0) = ||ci|| 2 |.

It then follows that X can also be characterized by the following:

(2.23) K = 10 e 25| ¿V> = 0 and d*0 = 0| = |0 e Dom (p)| F0 = 0S;

here d denotes the L2-closure of d and d* its L -adjoint (noting that Dom id*)

C D). Now we formulate the L --d-Neumann problem as follows:

(2.24) Problem. Find a bounded operator Nv: Cl —» Cl such that

(I) ÎUzV)CDom(F),

(II) JlÍN) = K, where TUN) denotes the null space of N,

(III) K(rV) 1 K,

(IV) if a e Cl and a 1 K, then

(2.25) QÍNCL, i/>) = (a, ¡/» + (Net, i/>) for all y> e 33.

It is clear that if such an operator N exists, satisfying (I) through (IV), then it

is unique. Furthermore, N also satisfies the following:

(A) If H: Cl —'K denotes the orthogonal projection then whenever a e Cl,

we have the orthogonal decomposition

(2.26) a = dôNa + údNa + Ha.

(B) dN = Nd, d*N = Nd* and FN = NF.

(C) If et e Cl and dcL = 0 (i.e. a is in the domain of the L closure of d)

and if et 1 H then

(2-27) a=dôNa,

and 0 = dNcL is the unique solution of the equation cl = deb which is orthogonal

to H.



Let À be a C nonnegative function on zVI . With A fixed and for each / >

0 we will define the ¿-Neumann problem of weight t; the above problem will then

correspond to / = 0. For cß £ Cl

(2.28) {cß,iß)u)îcß,e-tXiß), \\cß\\2U) = {cß,cß)(t).

Observe that the norms || ||. . are equivalent to the norm || ||n «= || || , hence a

function is in the completion under any of these norms if and only if it is square

integrable. So that for each t the inner product (2.28) can be extended to u. We

define d : (l * (1 by

(2.29) oeß=eUoie~Xteß)= oeß - toiô, dk)cß

so we see that

(2.30) o-idt,rj) = <JÍO,-n).

Further, we have as in (2.13),

(2.31) {*t<t>, iß)(t) = {cß, diß)(t)+fhM{o{d, dr)ó, e~Xtiß)dS,

so that, when cß £ j), the boundary term vanishes.

We define Q : fj x £ —» C by

(2.32) Qt{cß, iß) = {dcß, diß){i) + {dtcß, uiß){t) + (c6, iß)(t).

Again, the norm Q ÍÓ, cß) 2 is equivalent to Qieß, cß) 2 for each t and hence,

for each t, the completion of D under 0 may be identified with £ and, for

each t, the inner product (2.32) extends to J).

As in (2.18), we have for each t and each a eu a unique cß £ D such that

(2.33) Qt(-4>t,ifj) = ia,ifj)U) for all iß £ ®.

Then F , the Friedrich's representative of Q , is defined by F ef> = a and

Dom(F ) = Dom(F) is independent of t. We have

(2.34) FiçS = D/0 + ef>,

where

(2.35) a, = <?>>, + */•

Analogously to (2.22) we define the space of weighted harmonic forms K by

(2.36) Kt = \cß£%Qticß, cß) =11011^1.


280 J. J. KOHN

We can now formulate the d-Neumann problem of weight t with respect to the

function À as follows: Find a bounded operator N : Cl > Cl which satisfies prop-

erties I, II, III and IV of 2.24 and hence also (A), (B) and (C) with N, F, K, Q, H

and i> replaced by N , F , K , Q , /7 and t) respectively, and 1 and adjoint

are understood with respect to the inner product ( , ) .

Our interest centers principally on property (C) which we can reformulate as

follows:

(2.37) Proposition. // a £ Cl and if cl is in the domain of the L -closure of

d with dcL = 0 and if

(2.38) (a,i/;) = 0,

wherever t>0 = 0 and i/V £ h and if for some t the d-Neumann problem of weight

I is solvable in the above sense, then there exists u such that du - cl and the

unique u satisfying this equation which is orthogonal to X (ztt the ( , ). . inner

product) is given by u- ON cl.

Proof. We simply observe that if d 6 = 0, i.e. t/Kc~Xi(9) = 0, then (2.38)

implies

(a, 6) = 0, hence alH„ under ( , ), ,,u> t uz

and since da = 0 we have the desired result by property (C).

3. The main theorem. The concepts of plurisubharmonic and pseudo-convex

are well-known in the theory of several complex variables, we recall them below.

(3-1) Definition. If P £ Al' we denote by Tp' the vectors of type (l, 0) at

P; it L e Tp then in local coordinates we have

(3.2) L » Z ** J-, dz.

If A £ C (M ) we define the complex Hessian of A at P to be the hermitian form

Tl,ü . ,on / p' given by

(3.3) L ^((dd\)p, L A I).

The function A is call plurisubharmonic at P it the complex Hessian is positive

semidefinite and it is called strongly plurisubharmonic at P if the complex Hes-

sian is positive definite.

In terms of local coordinates the complex Hessian at P is represented by

the matrix' ((r52A/dz dz Ap).

(3.4) Definition. M is called pseudo-convex if for each P £ bM the restric-

tion of the complex Hessian of r to T„' Cl CTpibM) is semidefinite and M is



called strongly pseudo-convex if this restriction is positive definite. Here

CTpfoAl) denotes the complexified tangent space of bM at P.

In terms of local coordinates, we are looking at the hermitian form,

(3-5) fo1, ...,«") ^ Z l-~--\ cCäk

where

j.k \dzjdzk/p

«3.6, Z J^-0.

(3.7) Theorem. // AI is pseudo-convex and if there exists a function À on

M which is strongly plurisubharmonic in some neighborhood of bM, then there

exist a nonnegative function p £ C°°(zVl) and a compact set K C Al such that p is

strongly plurisubharmonic outside of K, and, if for each c > 0, Al is defined by

(3.8) M = IP £ M\ p{P) < c\,

then Al CC Al is compact.c r

Proof. We will show that the function p £ C°°(Al) which outside of a suffi-

ciently large compact set K is given by

(3.9) fz. = - log \r\ + C\,

where C is a large constant, satisfies the properties required above

Let z • • ■ , z be a holomorphic coordinate system with origin on bM and

(3.10) L = rp(L) + vAh),

where

Iamrz (P)

(3.11) Vo{L) =--T—_Zr_ (P)

domain U. Suppose P £ U and L £ Tp is given by (3.2). We write

Pv

iP)\2 z, à z .

Then the vector rp{L) is in the tangent space at P of the surface r = r{P). The

pseudo-convexity of Al implies that there exist f > 0 and A > 0 such that

(3-12) ^rZz W-M^/kptU)* >.- A|L|2|r(P)|k

whenever |r(P)| < e and P £ U, where

(3.13) |L|2 = XV |B»|2.

=1


282 J. J. KOHN

To see this choose e so small that the points P with ¡r(P)| < e ate in a smooth

product neighborhood V of bM. Denote by 77: V —, bM the projection in this pro-

duct. Let kiP) denote the left side of (3.12); since k is smooth we have

kiP) - kiniP)) = Oi\L\2\riP)\)

when |r(P)| < e and P e V. This combined with the pseudo-convexity of M yields

(3.12).

From (3-10) we obtain

z,. 1

Z Z L; ze

aja k +- Zr + c.Z-A _ a '«*

(3.14) Zr _ rÍL).AD, +L \Lir)\2+cZ\ - a'akZ .Z . I * 2 7 z.z -zl.

; zez z.

; fe

\\r(L)\\viL)\ + |tz(L)|

The strong plurisubharmonicity of p. for small r is then a consequence of (3.12),

(3.14), the fact that |L(r)| = \fL)\, the estimate

(l/|r|)|KL)||v(L)| < large const |r(L)|2 + small const (l/r2)|zy(L)| 2,

and the fact that A is strongly plurisubharmonic near bM. The remaining prop-

erties that are needed are immediate consequences of the definition of p.

Observe that whenever bM has a neighborhood U which is Stein then there

exists a function A which is strongly plurisubharmonic on a neighborhood V CC

U of bM. Namely, let /. , • • • , fN be a set of holomorphic functions which sep-

arate points of V and such that the \df \ span the whole «-dimensional space of

(1, 0)-forms at each point of V. Then the function

A(3.15)

is strongly plurisubharmonic on V since

N

z U27=1

(3.16) Za a'äkz z,

1 k

Z f <*'t-" ' mz .

In [2] Grauert gives examples of pseudo-convex manifolds on which there

does not exist a function p as described in the above theorem.

To formulate the main theorem we use Sobolev norms on the space Cl. For

each nonnegative integer s we define the norm || \\s on Cl by

(3.17) - Z Z ZfnnM |Da0\a\ss ¡J

(v)|21; '

dV,



where Uv is a finite covering of M by holomorphic coordinate systems, ep.A ate

the components of </> (as in (2.3)) with respect to the coordinates in L/ • as usual

Da = d\a\/dxal ...ax"2", with x.= Re(z) * =Im(z), j = I, . . . , u, and1 ¿n 7 11T" 7

a = a j. . . . + a The topology defined by this norm is independent of the1 ' I ¿77 7i

choice of coordinate cover; we denote by Cl the completion of Cl under

Here we recall the well-known relation between differentiability and the Sobolev

spaces. For each integer r we denote by Cl the space of forms whose compo-

nents in each coordinate neighborhood U are in C{U C\ Al), and set (l = Cf.

(3.18) Lemma. Cl C Cl whenever s > n. In particular, if cß £ il for alls s—n — r ' ' ~ s '

s then eß £ Ct.

The principal result of this paper is given in the following.

(3.19) Main theorem. // Al CC Al is a pseudo-convex manifold with a smooth

boundary and if there exists a nonnegative function À on M which is strongly

plurisubharmonic in a neighborhood of bM, then there exists a number T„ such

that the d-Neumann problem of weight t is solvable {in the sense of §2) for each

t > Tn. Furthermore, for each s there exists a number T such that

(3.20) N (CÎ ) C Ö , H iä )c& , H c öt s s t s s t s

whenever t > T . Finally, whenever t > Tn then Kt''q, with q > 1, is finite

dimensional, its dimension is independent of t and it represents the d-cohomology

of Al.

As in Proposition (2.37), we obtain the following consequence of the main

theorem by applying Lemma (3.18).

(3-21) Corollary. Under the same hypotheses as above, the operators N

and H map (1 into Ct provided t > T ; in particular they map Ct intot r s s — n r — s' r j f

CÎ . Furthermore if a £ U , da = 0 and (2.38) holds then there exists u £ US —77 ' s' _ x _ s

(and hence u £ Ct ) such that du = a. In particular if a e Cl with da = 0 and

(2.38) holding then, for each m, there exists a u £ il such that du = a

4. /\ priori estimates. Our starting point is the following estimate which is

proved by Hörmander in L5J-

(4.1) Proposition. // AI C Al is pseudo-convex and if there exists a nonnega-

tive function X on M which is strongly plurisubharmonic in a neighborhood of

bM then there exists a function f £ C^(zVl) and constants T, C and for each t

a constant C such thatt

(4.2) tU\\2 <CQtió, 0) + C/||/ç4||2,


284 J. J. KOHN

whenever 0 £ £p'q with q > 1 and t >T, where || ||. . and || || are the norms

defined by (2.28) and (3.17) respectively.

First we have the following lemma which is an immediate consequence of the

ellipticity of Q.

(4.3) Lemma. If g £ CfÍM) and g' £ C™iM) so that g = 1 o7Z the support

of g and if a £ Cl, 0 £ X with

(4.4) 0/0, t/z) = (a, ,/7)(0

/or a// yj £ £ and some fixed t, then, for each s > 0, there exists a constant

C > 0 so that

(4.5) lls0lls+2 <c\\g'a\\s-

(4.6) Theorem. Under the same hypotheses as Proposition (4.1), for each

nonnegative integer s there exist constants T and C such that if a £ (A4',q

with a > 1 and if (4.3) holds with 0 e33p'? for all yb £ 1)p'q and t > T then we

have

(4-7) \\i>\\s<Cj4s,

Proof. The argument is along the lines of the proof of Theorem 2 in [9J. We

choose U to be a suitably small neighborhood of P £ bM on which there is de-

fined a boundary coordinate system as well as a boundary complex frame, which

we describe below. A boundary coordinate system is a system of real coordinates

f' ' ' ' ' f -\'r> wnere r IS tne function defining the boundary described in §2.

A boundary complex frame is a set of orthonormal vector fields, L • • • , L on

U of type (1, 0), such that L (r) = 0 if / = 1, • • • , 2tz - 1 and L ir) = 1. We let1

;

<y , oj" be the bases of (1, 0)-forms on U which is dual to L .,•••, L

Any form 0 on- U Cl M can be expressed in terms of the oo as follows:

(4-8) é=ZàuofAof,

where co = a> A • ■ • A co p anc[ fJ = oj A • • • A co q. The restrictions of

forms in 3) to U C\ M ate then characterized by

(4.9) ^¡jhtA =0 whenever n £ J.

If a is a (2?r)-tuple of integers we denote by D the operator given by

(4.10) Da = i-i)\a\d\a\/dt,1 ■.■du2n-fdr°~2n1 ¿n— 1

and if a- is a (2n - l)-tuple we define D, by



(4.11) D¡=i-tHd\a\/dtai1...dt^Z-l1.

These operators will be interpreted as acting on forms on U Ci M by applying

them to each component in (4.8). Thus if eß is the restriction to U D Al of a form

in 3) then so is Dêß. Now if £,, C e C™{U Ci Al) and C = 1 on the support of C,

we wish to prove the estimate

(4.12) t Z KDaeß\\2(t)<Cs Z K'D^\\2(t) + Csjf,|cx||2

\a\<s la|ss

for at cß £ S)p,q satisfying (4.3), where C is independent of /.

Before proving (4.12) we will show how the desired estimate (4.7) follows

from (4.12) and Lemma (4.2). Let U. , • • • , UN be a covering of ¿>zVl by neighbor-

hoods such as described above, and let 40, C, , • • • , fo be functions in C (Al)

such that suppfo0) CC Al and supp (O CC U . C\ M fot f = 1, • • • , N and such that

<4-13> Ifol.

It follows from (4.12) that

=i

(4.14) N<csZ Z i||^î)Da0||(2i)+||^DarA||(2oi+cSi(|fo||;

y=i \a\<s

where supp(z^J) CC Al, suppfo') CC U . O M fot f = 1, • • • , N and Ç. = 1 on the

support of C■ for / «= 0, • • • , N.

Now since SN=1 C'\Cq - 1| < const Sé;=1 C- then for t sufficiently large the

first term on the right is smaller than the left-hand side of (4.14) and hence can be

"absorbed". The second term on the right is less that ||g<p||s for a suitable

g £ C^(Al) and hence can be estimated by (4.5) which yields the desired (4.7).

Now we will prove (4.12) by induction on 5. We observe that (4.3) implies

(4.15) Qtio,o) = ia,cß)t<\\cß\\2u)+\\a\\2U);

this combined with (4.2) and (4.5) yields the desired result for s = 0 if t is suf-

ficiently large. Suppose that (4.7) has been proven for s — 1. Since cß £ Ju'''q

then Cuja e j)^'q and substituting this in (4.2) we have

(4.16) t\\ÇDlé\\2u) < CQtiCDlcß, (Djip) + C^fiDjcßW]

if \y\ = s, the last term can be estimated by ||g$||s+I with a suitable g £ C™{M)

and hence by const ||a.|| using (4.5). Integration by parts gives


286 J. J. KOHN

Qti&l<t>, &l<p)

= e/0,D^2D^0)

+ o{i\\[d,CDl](ß\\U) + \\[Ot,CDl]o\\(t)+\\CDl,Ht)'

(4.17)+ Ct Z ÍQ^'D^,CDjcb)fQt(CDl^,CDl^

\ß\<s

+ ídéUDyb, dl, <Dp0){í) + iôè, [[01, *1, £D¡](p)(t)

+ \\KDyh,dW2(z) - »l >muyBy (2.29) we have

(4.18) [ôt, 0?] = [*, £d£] - ítod», ¿A), 0J].

Note that the first term on the right is an operator of order s and is independent

of t, and that the second term on the right is of order s — 1. Thus the error term

in (4.17) can be estimated by

(4.19)

small const Q {0y^, 0y<f) + large const Z \\CD

cit)u'm_v

Xrt

where the first two constants are independent of t and C has compact support

in ii H M with C = 1 on the support of C- By (4.3) and integration by parts,

we have

Q(cp,DlC2Dl<p) = iCDla, 0¡&.(4.20)

Choosing a covering and a partition of unity as above, summing over y with |y|

< s and over the partition we obtain, using the inductive hypothesis,

(4.21) t Z Kpyb4>\\2{t)< const Z H^'DÎI^ + CWIIall2,7,|y|s.s i,\ßfs

where the first constant on the right is independent of /. To conclude the proof

of the desired estimate (4.7) we must control the normal derivatives. Since Q is

elliptic the boundary bM is noncharacteristic and so we have

(4.22)dr

< const

(i)

k(¿0,£0)+ Z H'DfrwÙ

for all 0 £ Jb, the constant being independent of t. Replacing 0 with D ,0,

where |ff | = s - 1, applying (4.17), (4.19) summing over the partitions of unity

we obtain


(4.23)r- 1

ífoo

GLOBAL REGULARITY FOR d

12

287

•v const KX<to

\l) + cMMl-vi. \y\<s

again the first constant is independent of t. The condition (4.3) implies that

(4.24) cx= D +

this is a determined elliptic system in which the second order terms are indepen-

dent of t, hence in 1/ n Al, we can solve for d cß/dr obtaining

(4.25) -~- = Aa + Z Bt(9=2

Z C'Q — D cß + first order terms,

foUi ^

where A, Bq and C$ are matrices independent of the parameter t. Applying D

with |ff| = s — 2 to (4.2 5), taking the norm || ||, , and combining with (4.23) we

obtain

(4.26) zi,\cr\<S-2 or (z)

where the dots stand for the right-hand side of (4.23). By successive differentia-

tions of (4.25) and proceeding similarly we obtain

12(4.27) z

j.\cr\<s-l<&;<*

<

(0

for k = 1, • • • , s, where again the dots represent the right side of (4.26). The

desired inequality (4.17) is finally obtained by using the inequalities (4.27) to

replace the first term on the right of (4.21) by the first term on the right of (4.23);

then / is chosen large enough to absorb this term in the left-hand side. This

gives the desired estimate for the tangential derivatives of order s and hence,

by (4.27), also of all other derivatives of order 5.

5. Proof of the main theorem. We begin by "regularizing" the a priori esti-

mates of the previous section. That is, we want to show that the derivatives for

which we obtained a priori bounds actually exist.

Throughout this section we will assume that M CC Al is pseudo-convex and

that there exists A, £ C°°(M ) which is strongly plurisubharmonic in a neighborhood

of bM.

(5-1) Proposition. For each nonnegative integer s there exists a number

T such that for any fixed t > T and a £ Cp.9 with

that

> 1 if cß £ 3)p-q such

(5.2) Qtießt, iß) = ia,iß){t(z)


288 J. J. KOHN

for all 1/7 e 3) then 0 £ Cl . Furthermore 0 satisfies the estimate (4.17).

Prool. The proof follows exactly the same lines as §4 in l9J. We outline it

here. For 8 >0 let Q be the quadratic form defined by

(5.3) qJ(0, ^) = 2,(0, <A) + S(0, l/,)i

tot 0, 0 e X, where ( , ), denotes the inner product associated with the Sobolev

norm || || . Let J^s be the completion of J) under Q ; this is independent of t

and for 8 >0 it is also independent of 8 and 3yg C £. If a £ (JP.9 then, by the

Friedrichs' representation theorem, there exists a unique 0 £ £p,q such that

(5.4) ef(0(S, </» = (a, <A)(/)

for all </7 e l^-*. Furthermore, if 8 > 0 then 0, is a solution of a coercive bound-

ary value problem and hence 0* £ a*'« if a e ÔM then 0(S e (?£•«,. By the

same method as in Theorem (4.6) it can be shown that if t > T and ¿> > 0 then

(5.5) ltëJll,<coost||a||a.

Then we can choose a sequence i<5v| with lim 8V = 0 such that the sequence

\v~ (0 1 + • • • + (p( )\ converges in || ]| ; it is easy

unique 0 satisfying (5.2) and hence 0 fc (lt>,q C if).

(5-6) Lemma. For any integer s, (AP,q Cl Dom(F ) is dense in Dom (F ) Ocv si t

A)p-q in the norm 0 if a > 1 and t > T .— t ' ' — — s

Proof. If 0 e Dom(F ) Ci W-* let a = F{(p, let ay e CÎ"-« such that a =

lim av in 3"-«, let 0V be such that Q(i<f>v, >l>) = (<*■„, ̂)(() for all uV e 3>'«.

By the above 0„ fc U.P'9 and we have

(0 ! + • • • + cb( )\ converges in || || ; it is easy to see that the limit is the

QMv - 07 0, - 0) < KII(/)II02, - 0ll(i) < \K\\U)\fQt(<pv -4»4>v- 4)

which proves that 0 = lim (pv in the norm Q .

(5.7) Lemma. If q > 1 and t sufficiently large then sip,q is finite dimen-

sional and there exists C > 0 such that for all 0 e l/p,q with 0 1 }ip'q we

have

(5-8) ll0ll(2o<C(||d0||(2;)+||^0||(2o).

Proof. From (4.1) and Lemma (4.2) we have

(5-9) H0ll(2o<C(||d"0!|(2i)+||^0||2i)+ llgF^H^),



where g £ C™{M). If cß £ Kp-q then the above reduces to \cß\ 2() < C\\gcß\\ 2_ ,.

This implies that unit sphere in i\p,q is compact and hence Ap,q is finite dimen-

sional. Further, we have

(5.10) \\gFti<f>)\\2_1 < const i\\dcß\\2 + ||tf0||2 + ügcSII2.,).

So (5.8) is deduced by the usual contradiction argument, i.e. take a sequence

év lKf.« with ||<M(()=1 and

(5.11) uvfU)>-wv\\2U) + ii*m2,,>.

then combining this with (5.9) and (5.10) we have ||0„|fo) < const ||g<£„||_i fot

v large. This shows that a subsequence of the \epv\ converges in L, anc^ be-

cause of (5.11) also in Q to an element eß but cß 1 KP,q, ||<£ll(,) = 1 and, by

(5-11), eß £ sip,q which is a contradiction proving that (5.8) must hold.

(5.2) Proposition. For each s there exists a number T such that if q >

1 and t > T then Kp,q C Cl and if a 1 M0,9 then there exists a unique eß e— S t S ] t IT

3>-« n 3 such that cß 1 K"-q ands T t

(5.13) Q\/cß,iß)-{cß,iß)t = {a,ifj)t

for all iß £ ®.

ProoL The proof is by the same method as is (5.4) of L9J. Let 6^, • • •, 0N

be a basis of Kp'q, q > I. For each positive v and eß, iß £ JÖp'q we define the

quadratic form P by

1 N(5.14) p^(0, iß) = Qt{cß, iß) - (cß, iß){t) + - Z (cß, e)(t)ie., iß){t).

7 = 1

We write

(5.15) eß = cß'+cß",

where cß' eH^ and eß" 1 Kp-q; then

(5.16) PVtieß, cß) = Qt{cß", cß") - (cS", cß"){t) + v- X \\eß'\\t),

hence by Lemma (5.7) we obtain

(5.17) U\\t)< ""st PVtieß,cß);

note that this constant depends on v. Proceeding as in Proposition (5-1), we

conclude that for sufficiently large t, if a £ (lp'q, q >l, then there exists a

unique <j>v £ Dp'^ such that


290 J. J. KOHN

(5.18) Pvf(bv,yj) = (a,^){t)

for all é £ 3>"z and c?v e (IP-«. Furthermore

(5-19) ¡|0j|s< const ||a||s>

where the constant again depends on u. It is easy to see from the derivation of

(5-19) that there exists C > 0 such that

(5-20) H0vlls<C(||a||s+||0j),

for all V.

Now we shall prove that np,q C Cl . If dim np'q = 0 there is nothing to

prove, otherwise set 0Q = 0 and assume 6. £ Cl for /' = 0, • • •, k with k < N;

we shall construct 0 6 Kp-q H fi with ||0|| = 1 and id, 0),, = 0 for / < k. Int s " " ' ; (t) ft ■

this way we will construct a basis of sip,q which is contained in Cls. Let a £

(1 such that a is orthogonal to 0 . for j' < k but so that et is not orthogonal to

0 . . Let 0j, satisfy (5.18), then the sequence [||0j|S is unbounded, for if it

were bounded we could by (5.20) find a subsequence converging to a 0 in || || _j

and then 0 would satisfy

(5-21) Qticp, 0) - (0, xf,)it) = (a, 0)(O

for all <A e3)i','ï. Setting 0 = 0., the left-hand side is zero for all / and the

right-hand side is different from zero for ] = k + 1, which is a contradiction.

Thus the sequence i!|0„||i is unbounded and hence we can find a subsequence

I0V S such that lim.|l0v || = °°. Setting /3. = 0 /||0V ||, it follows thati l i ' z i

(5.22) pV,'(ß ., 0) = (a, 0)/||0v II-

From (5.20) we see that ||/3.|| is bounded; now choose a subsequence of the ß.

such that its arithmetic means converge to 0 in || || and such that the subse-

quence converges to 6 in || \s_v Then 0 £ U^, ||öj| = 1 and Qid, f) = id, f)

so that 6 £ sip,q. Further setting 4> = 0 in (5-18) we have,/ ° m

(5.23) lz-1|(0 0 )l2 = (a,(5 )1 " V 772 ' 772

and hence (bv is orthogonal to 6 . for / = 1, ■ ■ • , k so that 0 is also orthogonal

to d. for / = 1, - • •. k. Therefore Kp-q C fl .; ; ' ' Ajj t s

If a 1 }ip'q) a e Cl , we proceed as above to find fv satisfying (5.18).

Now, however, the sequence iy0„||! is bounded, for if it were unbounded then

we could construct ß as above but now ß 1 Xp,q, ||/3j| = 1 and ß £5\p'q which

is a contradiction. So that by (5.20) we see that there is a subsequence of the (pv

which converges in || ¡| _ , to 0 and whose arithmetic means converge in || ||

to 0 so that 0 £$p'q C 3 , co i KP,Q and 0 satisfies (5.13) as required.



(5-24) Lemma. Let L = F — 1, then the range of L is closed if t is suf-

ficiently large.

Proot. It suffices to prove that there exists a constant C > 0 such that for

every eß £ Dom (F ) with cß 1 A we have

(5-25) U\\u)<C\\Lcß\\uy

We write Dom foO = 0 Dom fofo O äK« and denote by Lp-q the restriction of

L( to Dom fofo H 35?.?. Then L¡ is the direct norm of the Lp-q and (5.25) fol-

lows by proving it for each Lp,q separately. The case q > 1 follows from the

construction given above, or we can deduce (5-25) directly from (5.8) by noting

that

(5.26) \\dH2n+\\*tH2t)^LPAq<P>t\ir

To show that the range of Lp' is closed we proceed as in [7j. Namely,

we first observe that if 6 £ DomiLP' ) then dd £ JJP' , thus we can apply (5.8)

to dd and obtain

(5.27) il^||20<C¡|Lf.°o||20.

Furthermore, since Lpl is selfadjoint its range is dense in the orthogonal com-

plement to Kp- . Hence, if cß i Kfo° and f>0 there exists 0 £ Dom(Lp-°) so

that ¡10 - Lp-°d\\ < f, so if eß £ Domfofo0) we have

II0II(O < K07 L?-°0)(t)| + 4<p\\{1) < \\dep\\U)\\de\\u) + e||0||(()

< ¡|Lf'°0ll(ol|Lfo°o||{() + mu) < llLf'Vll(/) + AU\U) + Ifof'VlD

which concludes the proof.

We can now define theoperator N required in the main theorem (3-19).

Namely, we set N (H ) = 0 and N a = cß if a = L eß w¡th eß I H . Proposition

(5.12) and Lemma (5.24) show that N is well defined and satisfies the required

regularity as well as the properties (I) through (IV) of (2.24). The further prop-

erties (A), (B) and (C) of (2.24) then follow easily from the properties (I) through

(IV) and the regularity (cf. [7] of [l]).

REFERENCES

1. G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann

complex, Arm. of Math. Studies, no. 75, Princeton Univ. Press, Princeton, N. J., 1972.

2. H. Grauert, Remerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81

(1963), 377-391. MR 29 #6054.


292 J. J. KÖHN

3. H. Grauert and I. Lieb, Das Ramirezchc Integral und die Lösung der Gleichung

âf = a. im Reriech der beschrenktan Formen, Proc. Conf. Complex Analysis (Rice Univ.,

Houston, Tex., 1969), Rice Univ. Studies 56 (1970), no. 2, 29-50. MR 42 #7938.

4. G. M. Henkin, Integral representations of functions holomorphic in strictly pseudo-

convex domains and some applications, Mat. Sb. 78 (120) (1969), 611—632 = Math. USSR

Sb. 7 (1969), 597-616. MR 40 #2902.

5. L. Hormander, L estimates and existence theorems for the d operator, Acta.

Math. 113 (1965), 89-152. MR 31 #3691.

6. N. Kerzman, Holder and Lp estimates for solutions of du = f in strongly pseudo-

convex domains, Comm. Pure Appl. Math. 24 (1971), 301-379. MR 43 #7658.

7. J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, II, Ann.

of Math. (2) 78 (1963), 112-148; ibid. (2) 79 (1964), 450-472. MR 27 #2999; MR 34 #8010.

8. -, Roundary behavior of d on weakly pseudo-convex manifolds of dimension

two, J. Differential Geometry 6 (1972), 523-542.

9. J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure

Appl. Math. 18 (1965), 443-492. MR 31 #6041.

DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY08540


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