Some Recent Results on Weakly Pseudoconvex Domains Nessim Sibony CNRS URA D 0757, Université de Paris-Sud, Mathématiques, Bâtiment 425 F-91405 Orsay Cedex, France
We give here a survey of some recent results concerning analysis in weakly pseudoconvex domains in C" with smooth boundary. Let Q C (C" be a smoothly bounded domain with defining function r, more precisely r is a smooth function defined in a neighborhood U of Q such that Q = {z G U ; r(z) < 0} and dr ^ 0 on dQ. The domain Q is pseudoconvex iff for z G dQ and t G C"
(Lr(z)t, t) = Ys Ta^W* ^ ° whenever Ik
jüZk
]£ - Ä , = 0. ;=i
^
If the inequality is strict for t ^ 0 the domain is said to be strictly pseudoconvex. Pseudoconvex domains with smooth boundary are just the domains of holomorphy with smooth boundary. The analysis on strictly pseudoconvex domains received much attention in the late sixties and in the seventies, the explicit construction of kernels in order to solve 3-equation was one of the main tools to solve function theoretic questions in these domains, see [HeL]. The fact that a strictly pseudoconvex domain is locally biholomorphic to a strictly convex domain is crucial in this approach. Such a simple local model does not exist when the Levi form L is not positive definite. We give first few examples to show the type of difficulties we have to face.
1. Examples a) The Kohn-Nirenberg example [KN1]. Let Q = {(z,w) G (C2 ; Re w+ | z \2k +t | z |2 Re(z2/c~2) < 0}. If | t \< 2£ZT then Q is pseudoconvex. If | t |> 1 and k > 3 then there is no supporting analytic set to Q at the point 0. Hence Q is not biholomorphically equivalent in a neighborhood of 0 to a convex domain. b) Non embeddability into convex domains. There exists a smooth pseudoconvex domain ß € C 3 and p G dQ, such that for any N and any convex domain U C 0, into U, [Sil]. c) The "worm" domain. Diederich and Fornaess [DF1] have exhibited a pseudoconvex domain fì^C2 with smooth boundary such that Q does not have Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 © The Mathematical Society of Japan, 1991
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a Stein neighborhood basis. Moreover given e > 0 it is possible to construct a "worm" domain Qe such that there is no plurisubharmonic (p.s.h.) function Q in Qe satisfying — Ade < Q < —Bde with positive constants A, B. Here d denote the distance to the boundary of Qe, see [DF1] and [Ki]. This last property implies the non existence of a V1 plurisubharmonic defining function for Q8, if e < 1. Despite the smoothness of dQ, the structure of the set W(dQ) — {z G dQ ; there exists t G 2. Theorem 3.5. There exists a pseudoconvexdomain U 2. Observe that dU is not smooth. The example is based on the fact that the holomorphic functions in LP (U), p > 2, extend holomorphically to a domain Ü which is not contained in U. Such phenomenon cannot happen when U is Runge. If one tries to solve the d-equation in a Hartogs domain U = {(z,w) G (C2 ; z G Q cz C, | w \< exp(—cp(z))} where cp is subharmonic in an open set fiC(C, one is led to the problem in one complex variable : solve du/dz = / in Lp(Q,cp) with the estimate i/p
/
2 the equation du/dz = / has no solution in Lpf(A, cp). Using these one variable results, we prove the following theorem. Theorem 3.6. There exists Ö ^ C 4 pseudoconvex with smooth boundary, strictly pseudoconvex except at one point with the following property : for every p > 2 there exists a d-closed form ß G LPQ^(Q) such that the equation du = ß has no solution u G LP'(Q) if pf G ]y/i + 9 p - 3,p]. The result is probably not sharp. There is also a smooth pseudoconvex domain flc£3 such that d : LP(Q) -> LP0^(Q) does not have closed range. The following questions are then quite natural. i) Suppose Q C (Cw is pseudoconvex with smooth boundary. Let s > 0, for which sf < s does the operator d : AS'(Q) -> A^(Q) has closed range ? ii) Let 1 < p < oo, for which p1 < p, does the operator d : LP'(Q) -> LP0^(Q) has closed range ? The following partial answer due to Bonami-Sibony [BoSi] is a consequence of Kohn's result and of a Sobolev embedding theorem which basically says that if du G L*(Q) for t large enough then the dimension 2n in the usual Sobolev embedding theorem may be replaced by (n + 1). Theorem 3.7. Let Q ^ (ST be a pseudoconvex domain with smooth boundary. Let a G ifg),i) 0^) a à -closed form. a)IfO *^r and a G ^J~i) (^)> ^ e n 3w = a ftas a solution in Aa(Q), with
as-*?. Remarks, i) In ail cases the solution u belongs to HS(Q). ii) The examples mentioned in Theorem 3.4 show that there is necessarily a loss in the regularity of the solution.
Some Recent Results on Weakly Pseudoconvex Domains
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4. 8 -Neumann Problem Let Q C (C" be a pseudoconvex domain with smooth boundary. Let a be a 0 in a neighborhood ofpGdQ iff dQ is of finite type at p. The existence of a subelliptic estimate of order s > 0 near a point p implies that the canonical solution to du = g is smooth near p if g is smooth in a neighborhoof of p, [KN2]. The best e in the subelliptic estimate is known only for domains of finite type in C 2 , [FeK, Ca4, FSi2] and for convex domains of finite type in C" [FSi2]. Real progress has been made recently in the understanding of regularity properties of the canonical solution of the d and dt equation for domains of finite type in (C2, see the survey paper by Christ in these proceedings [Ch]. The understanding of the local geometry of domains of finite type in v = {cp ; cp p.s.h. continuous in Q,
p yv T+(cp) = hmsup , z-weß log || z -
