A smooth pseudoconvex domain in C 2 for which L -estimates for c5 do not hold

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A s m o o t h pseudoconvex domain in C 2 for which L -estimates for c5 do not hold

Bo Berndtsson(1)

Let ~ be a smoothly bounded domain in C n. It is well known (see [HL] and [0]) t h a t if :D is strictly pseudoconvex then we can solve the c%equation with estimates in L p for any l < p < c ~ . It has also been known for some time t h a t this is no longer true if l) is merely pseudoconvex. Namely, Sibony [$2] found an example of such a domain in C 3 where L~ do not hold. T h e reader should also consult the p a p e r [FS1] which contains a discussion of LP-estimates in general and m a n y counterexamples to this type of questions. However, all counterexamples known seem to treat the case n > 3 and LP-estimates for p > 2 .

In this p a p e r we shall prove

T h e o r e m 1. There is a smoothly bounded Hartogs domain in C 2, and a O- closed (0, 1)-form g in ~), which extends continuously to 9 , such that the equation Ou=g has no bounded solution.

Recall t h a t a Hartogs domain is a domain of the form

(1)

:D---- {(z, w); Iwl < e-r(*>}

where ~o is subharmonic. If e.g. ~o is smooth in the disk and

1 1

~o= ~ l o g 1_1zl2 near the b o u n d a r y of the disk, t h e n 0:D will be smooth.

There is a special reason why we are interested in the case n=2. T h e form g in T h e o r e m 1 extends continuously to 0l). So, the same example shows t h a t we d o n ' t have L~176 for 0b either. But in C 2 there is a duality between 0b in L ~ and in L 1. Therefore we get

(1) Supported by Naturvetenskapliga Forskningsrgdet

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210 Bo Berndtsson

T h e o r e m 2. There is a sequence of functions gn on 01? such that

(i) lignllL1 <1,

(ii) there is Un in L 1 such that

ObUn=gn,

(iii) if O v n = g , then Ilv, llnl --* c~.

In other words, we do not have L 1 estimates for C~b either. Another way of ex- pressing the conclusion of Theorem 2 is that the closed and densely defined operator 0b: L 1--*L 1 does not have closed range.

W h e t h e r one can solve the cb-equation i n / 9 with Ll-estimates is another question, which I do not know the answer to. In a recent paper by Bonneau and Diederich ([BD]), Ll-estimates with a logarithmic loss are proved. One should also compare the results by Feffermann Kohn, Christ and Nagel-Rosay-Stein-Wainger (see [FK], [C], [NRSW]), which contain sup-norm estimates, and even H61der estimates for c~b in domains of finite type in C 2 (thus in particular for domains with real-analytic boundary). Recently a more elementary proof of a slightly weaker result was obtained by Range ([R]).

Our construction is quite different from the one in [$2] (it is actually more similar to the earlier one in IS1]). It is based on the relation between estimates for the cb-equation in domains of the form (1), and estimates for the one dimensional c~-equation in the disk with weight e -n~' where n C N .

It is well known (see [FS1], [FS2] or [B]) that if ~ is an arbitrary subharmonic function in the disk, then one can in general not solve the equation

(2)

Ou

02 f in the disk with estimates

(3) sup lul e-~" _<_ c sup I/le -~.

A A

The analogous question for smooth ~'s is whether one can solve (2) with the estimate

(4) sup lul e-n~ < C s u p [fie - n ~ .

A A

where C is a constant that does not depend on n (nor f of course). It turns out (see Section 2) that if we have L ~ - e s t i m a t e s in a domain 7? of type (1), then one can solve the c~-equation in the disk with the estimate (4). Hence, all we need to do to prove Theorem 1, is to find a subharmonic function ~ C C ~ 1 7 6 for which this is impossible. This is the object of Section 1. In Section 2 we show how this implies Theorems 1 and 2.

This paper was finished when I visited the CIMAT in Guanajuato. I would like to thank Xavier Gdmez-Mont for his invitation and all his help, and the CIMAT for its support.

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S e c t i o n 1

In this section we shall study estimates of the form (1.1) sup I.le -~' _< C s u p I f i e - "

for solutions to the equation

Ov o2 f

in the unit disk, A. We are interested in for which functions ~ such an estimate holds for all f in, say, C ~ ( A ) , and also for which functions r (1.1) holds with a fixed constant for ~ - - n r n - l , 2, 3 ....

P r o p o s i t i o n 1.2.

We can solve the O-equation in A with the estimate

(1.1)

if and only if the following inequality holds

(1.3)

fA 'c~'e~ <--C/A 0_~ e~

V ~ E C ~ ( A ) .

The best constants in

(1.1)

and

(1.3)

are the same.

Proof.

The fact that

Ov/O2=f

is equivalent to f c ~ = -

/o~

v~-~ Vc~ E C ~ ( A ) . If v satisfies (1.1) we get

Taking the supremum over all i with I / l < e " we get (1.3).

If, on the other hand (1.3) holds then

I,: ,<su.,,../I -

(1.4) Let

Define a linear functional

r o a ~

}_

F = ~ e ; ~ C : ~ CLI(9)

T: F--~C

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212 Bo Berndtsson

by

(1.4) implies that T is well defined and that IITll < C s u p I fie -~.

We can extend T to a linear operator T: Ll(/k) ---+ C

with the same norm. Hence there is a function u C L ~ ( A ) such t h a t

ff =f as

^{U ~=z e}

and IluIl~= IITll. Letting v = - u e ~ we have a solution to O v = f which satisfies (1.1).

The proof is complete. []

Note that the question whether (1.3) holds depends only on A~. In other words, if h is harmonic and (1.3) holds for ~, then it holds with ~ replaced by ~ + h (just multiply a by e h+i[z where/z is the harmonic conjugate to h).

We also remark that if (1.3) holds for all a e g ~ ( A ) , then it actually holds for all a in L 1 with compact support, which are such that O a / 0 2 is a finite measure.

We will use this remark at several points.

P r o p o s i t i o n 1.5. Assume that (1.3) holds with a fixed constant for ~ = n r where n C N . Then r is subharmonic.

Proof. Let A ~ C C A be a disk and let a = X A , . Then (1.3) implies A' enr <- C fo~, e nr Idzl"

Hence, if r on 0 A ' then r in A'. Since we can change r to r where h is any harmonic polynomial, we see that if r on 0A', then r in A'. This means that r is subharmonic. []

A similar argument shows t h a t if the analog of (1.3) in L2-norm

(1.6) 1~12e ~ < C / 0 a 2e~

holds for ~ - - n r then r is also subharmonic. In this case, the converse is also true. This follows from the inequality used in the proof of H6rmander's theorem

(see [H]). We shall now see t h a t in L l - n o r m the situation is quite different.

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P r o p o s i t i o n 1.7. Let qo be any subharmonic function in A with the property that ~o=-oo on some set with an interior accumulation point. Then (1.3) does not hold with any constant C.

Proof. A s s u m e t h a t a E A a n d t h a t qo(a)=-oo. T h e n it follows f r o m (1.3) t h a t

e~ o

,z_~

To see this consider (1.3) w i t h a replaced b y a / ( z - a ) . T h e n

Oa O~

Oz ~(z-a) = ~ l ( z - - a ) + . ~ a ,

b u t the last t e r m gives no c o n t r i b u t i o n since

e ~~

I t e r a t i n g this o b s e r v a t i o n we see t h a t if ~(al)=qo(a2) . . . qO(an)=-oo t h e n

f

(1.8)

J I77 Iz-ar 1-I1 I z - a j l "

I n p a r t i c u l a r we can t a k e a = X A , , w h e r e A ' c c A is a disk containing a n a c c u m u l a - tion point, p, of t h e set where ~ o = - o o . T h e n (1.8) c a n clearly not hold in t h e limit as all aj t e n d to p. H e n c e (1.3) c a n n o t hold. []

T h e preceding p r o o f shows a curious fact. If q0 is given b y

3

1

~log Iz-asl

~<~=~

1

t h e n (1.3) c a n n o t hold w i t h a fixed c o n s t a n t as a=(al,a2,a3)-+O. O n t h e o t h e r h a n d , in t h e limit we get

~o = log Izl, which does satisfy (1.3) (just m u l t i p l y a b y z).

We are now r e a d y to prove our m a i n technical result.

P r o p o s i t i o n 1.9. There is a subharmonic function @ in C~176 such that if C~ denotes the best constant in

fA io~ienr < Cn fA 00~ nr

(1.10) ^_ ^~ e Vc~ e C ~ ,

then lim Cn = oo.

Proof. Let ~ b e a function s u b h a r m o n i c in a n e i g h b o r h o o d of A which is harmonic n e a r 0 A a n d which is such t h a t

(1.3)

is violated. T h e n t h e r e is a sequence

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214 Bo B e r n d t s s o n

of smooth subharmonic functions ~k in ~ , which are harmonic near OA such that

~kl!P. Let II~kllc~(~)=Ak. Take a sequence of integers nk such that

nk/Ak--ce,*

and put C k : ~ k / n k . Then we have a sequence of smooth subharmonic functions such that

(i) lim IlCkllCk = 0 and

(ii) if Bk is the best constant in

f lalen~r < Bk / ~ e n~r ^wec~

then lim

Bk =oo.

Moreover, all Ck's are harmonic near hA, so A C k C C ~ ( A ) . Now, if A ' = A ( a , r ) CC A

is a small disk in A, and if r is one of our Ck's we transport ~ to A ~ by the following definition.

where G is the Green-potential in A.

Then v = r ( A ' , r is smooth and

A v = A ( r

in A' and A v = 0 out- side A ~. Moreover

(1.11) II/Wllck-= _< r-kllar ~ _< r-kllr

Choose a disjoint sequence of disks

Ak=A(a~, rk)

in A which converges to an interior point. By taking a sparse subsequence, renumbering and letting Ck =T(Ak, ~bk) we get that

(iii)

E~kllCkflc~

< ~ and (iv) lim

rkBk =oc.

Then (iii) together with (1.11) shows that r Ck

e c ~ ( ~ ) .

Assume, to get a contradiction, that (1.10) holds with C~_< C. Take in particular a E C ~ ( A k ) . T h e n

- <_c f e

~ ^I~lenr ^{Yak I Oe}

since the other @ ' s are harmonic in Ak. The change of variables ~-=ak + r k r trans- ports this estimate to A. We then get

_ 0 0 z n k

since A C k = A ( ~ k ( a + r f f ) ) . Taking in particular

n=nk

we obtain

Bk <_C/rk,

which contradicts lim rk Bk----o0. []

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S e c t i o n 2

W e shall now see how we c a n use t h e f u n c t i o n r f r o m P r o p o s i t i o n 1.9 t o p r o v e T h e o r e m 1.

P r o p o s i t i o n 2 . 1 . Let :D be a domain of the f o r m z) =

{ (z, w);

^{Izl <} ^1,Iwl < e - ~ ( z ) },

where ~ zs a smooth function. A s s u m e that for any O-closed f o r m f which extends continuously to ~ we can solve the equation Ou= f with u bounded. Then there is a constant C such that for any

fEC~(A1/2)

and any h e n we can solve O u / O ~ = f in A1/2 with

(2.2) sup [uIe - ~ < C sup I fi e - ' ~

AZ/2 2'1/2

Proof. A s s u m e t h e conclusion is false. T h e n t h e r e is a sequence n k - * oc a n d a sequence of fk C C ~ ( A 1 / 2 ) such t h a t

s u p Ifkle -nk~' --: ak ~ 0 a n d if uk are a n y s o l u t i o n s for

T h e n

~ t k

02 - f k .

sup [uk le -nk~~ --* co.

B y choosing a s p a r s e s u b s e q u e n c e we c a n a s s u m e ~ ak < c~. L e t

O O

f = y ~ f k ( z ) w nkd2.

0

T h e s u m is a b s o l u t e l y a n d u n i f o r m l y c o n v e r g e n t in :D. So, f is c o n t i n u o u s o n :D a n d 0 f = 0 . B y h y p o t h e s i s we c a n find a f u n c t i o n uEL~ such t h a t O u = f . Since f has no d ~ c o m p o n e n t , u is h o l o m o r p h i c in w, a n d we c a n e x p a n d u in a p o w e r series

O4)

u(z,w) :Euo(z)wo

0 I d e n t i f y i n g coefficients of w n in O u = f we get

OUnk _ f k.

02

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216 Bo B e r n d t s s o n

B u t

so if r < e - ~ we get

s u p lttn(Z)l'F n ~_ IIltllL~.

A1/2

L e t t i n g

n=nk

a n d

rTe -~

we see t h a t

sup lu~le - - ~ ~ _< C A1/2

c o n t r a d i c t i n g t h e choice of

fk. []

T h e o r e m 1 is n o w a direct consequence. S t a r t i n g with our function r from P r o p o s i t i o n 1.9, we can scale it down to AU2. T h e n we can e x t e n d r to a s m o o t h s u b h a r m o n i c function ~ in A such t h a t ~ = 89 lo g ( i / ( 1 - I z 1 2 ) ) near 0 A . B y Proposi- tion 1.2 t h e conclusion of P r o p o s i t i o n 2.1 fails for ~. Hence, if we use ~ to define 79 we get a s m o o t h l y b o u n d e d d o m a i n in C 2 which satisfies the claim in T h e o r e m 1.

Let us now briefly discuss CSb on 079. Let

u(z, w)

be a b o u n d e d function on 079.

We can t h e n e x p a n d

u(z, w)

in a Fourier series

U(Z, e -9~+i0) ~'~ ~ Un(Z)C--Inl~e inO .

We can always e x t e n d u s m o o t h l y to 79. Let on t h e other h a n d f be a (0, 1) form in 79 which e x t e n d s continuously to :D. Finally let p be a n y s m o o t h defining function for 79. We say

Obu=f

if

OuAOQ=fAcSQ

on 079 in the sense of distributions.

Now, let in p a r t i c u l a r f be the form in T h e o r e m 1. B y extending u to 79 by

oc --1

U(Z, W)= ~ ~n(Z)Wn-~ ~ ~n(Z)W -n

0 --c~

we see t h a t if

Obu=f

^{t h e n}

Oun~ _ f k .

02

Again, by e s t i m a t i n g Fourier coefficients we see t h a t

Obu=f

has no b o u n d e d solution.

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We finally t u r n to the proof of T h e o r e m 2. This follows in principle from what we just said, b u t it is p r o b a b l y more instructive to give a direct construction.

We know from the construction of ~ t h a t there is a sequence of functions

ak

EC~(A1/2), and a sequence nk such t h a t

(a)

and

(b)

P u t for

(z,w)C0l)

Oak enk~=l O~

Oak w_nk d2 gk -~ -~Z

Since surface measure on 07) is equivalent to

idzAd2AdO

(a) means t h a t

]lgk IlL ~ (OD)

< C . Moreover

gk=Obt~kW -n~.

If now vk is any solution to

Obvk=gk

then

V k - - a k w - n k ~-

hk

has a holomorphic extension to 7). Hence

1 I S wei~176

2---~ vk(z, dO = a k w -n~.

7r

Therefore

IL k -n lLL1 < II'kllL1.

But (b) says precisely t h a t the left hand side here tends to infinity. Hence

IlVkI[LI

cannot be bounded, so we have proved T h e o r e m 2.

Remark added March

31, 1993. T h e construction in this p a p e r is based on the existence of a subharmonic function, ~, in the disk such t h a t sup-norm estimates for with the weight factor e - ~ fail. Shortly after the p a p e r was completed Fornaess and Sibony [FS2] noted t h a t their construction from [FS1] of a function with similar properties for LP-estimates, implies in the same way t h a t there is a s m o o t h l y bounded H a r t o g s domain in C 2 where LP-estimates fail for any p > 2 . Feeding the same function into our construction for 0b one gets a smoothly b o u n d e d domain in C 2 where C~b does not have closed range in any LP-space except for p = 2 . A more detailed explanation of this together with further examples of the same kind can be found in [B2].

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218 Bo Berndtsson: A smooth pseudoconvex domain R e f e r e n c e s

[ B 1 ] BERNDTSSON, B., Weighted estimates for 0 in domains in C, Duke Math. J. 66 (1992), 239-255.

[ B 2 ] BERNDTSSON, B., Some recent results on estimates for the cg-equation, in Pro- ceedings of conference in honor of P. Dolbeault, (to appear).

[BD] BONNEAU, P. and DIEDERICH, K., Integral solution operators for the Cauchy- Riemann equations on pseudoconvex domains, Math. Ann. 286 (1990), 77- 100.

[C] CHRIST, M., Precise analysis of 0 and cSb on domains of finite type in C 2, in Proceedings of the ICM Kyoto 1990, Springer-Verlag, Berlin-Heidelberg-New York, 1991.

[FK] FEEFERMAN, C. and KOHN, J. J., HSlder estimates on domains in two complex variables and on three-dimensional C R manifolds, Adv. in Math. 69 (1988), 233-303.

[FS1] FORNAESS, J. and SIBONY, N., LP-estimates for 0, in Several Complex Variables and Complex Geometry, Proc. Symp. Pure Math. 52:3, pp. 129-163, Amer.

Math. Soc., Providence, R. I., 1990.

[FS2] FORNAESS, J. and SIBONY, N., Pseudoconvex domains in C 2 where the Corona theorem and LP-estimates for c9 don't hold, Preprint.

[HL] HENKIN, G. M. and LEITERER, J., Theory of functions on complex manifolds, Birkh/~user, Boston, Mass., 1984.

[HI HORMANDER, L., L2-estimates and existence theorems for the cg-operator, Acta Math. 113 (1965), 89-152.

[NRSW] NAGEL, A., ROSAY, J. P., STEIN, E. M. and WAINGER, S., Estimates for the Bergman and Szeg5 kernels in C 2, Ann. of Math. 129 (1989), 113-149.

[0] OVRELID, N., Integral representations and LP-estimates for the cg-equation, Math. Scand. 29 (1971), 137-160.

[R] RANGE, R. M., Integral kernels and HSlder estimates for 0 on pseudoconvex domains of finite type in C 2, Math. Ann. 288 (1990), 63-74.

[81] SIBONY, N., Prolongement de fonctions holomorphes borndes et metrique de Carathdodory, Invent. Math. 29 (1975), 205-230.

[82] SIBONY, N., Un example de domaine pseudoconvexe regulier ou l'dquation O u = f n'admet pas de solution bornde pour f bornde, Invent. Math. 62 (1980), 235- 242.

Received August 15, 1991 B o Berndtsson

Department of Mathematics Chalmers University of Technology University of GSteborg

S-412 96 GSteborg S w e d e n

A smooth pseudoconvex domain in C 2 for which L -estimates for c5 do not hold