The ∂̄-neumann Operator on Lipschitz Pseudoconvex Domains with Plurisubharmonic Defining Functions

On a bounded pseudoconvex domain in C with a plurisubharmonic Lipschitz defining function, we prove that the ∂̄-Neumann operator is bounded on Sobolev (1/2)-spaces. 0. Introduction LetD be a bounded pseudoconvex domain in C with the standard Hermitian metric. The ∂̄-Neumann operator N for (p, q)-forms is the inverse of the complex Laplacian = ∂̄ ∂̄∗ + ∂̄∗∂̄ , where ∂̄ is the maximal extension of the Cauchy-Riemann operator on (p, q)-forms with L2-coefficients and ∂̄∗ is its Hilbert space adjoint. The existence of the ∂̄-Neumann operator for any bounded pseudoconvex domain follows from Hörmander’s L2-existence theorems for ∂̄ . The ∂̄-Neumann problem serves as a prototype for boundary value problems that are noncoercive and is of fundamental importance in the theory of several complex variables and partial differential equations. The ∂̄-Neumann problem has been studied extensively when the domain D has smooth boundary (see J. Kohn [21], [22] or H. Boas and E. Straube [4], [6] and the references within). In this paper we study the ∂̄-Neumann operator on a Lipschitz domain D when D has a plurisubharmonic defining function (see Theorem 1). Let H (p,q)(D) denote Hilbert spaces of (p, q)-forms with H (D)-coefficients. Their norms are denoted by ‖ ‖s(D) for s ≥ 0. The principal result of this paper is the following theorem. theorem 1 Let D C be a bounded pseudoconvex Lipschitz domain with a defining function that is plurisubharmonic in D. The ∂̄-Neumann operator N is bounded from DUKE MATHEMATICAL JOURNAL Vol. 108, No. 3, c © 2001 Received 30 September 1999. Revision received 22 August 2000. 2000 Mathematics Subject Classification. Primary 35N15; Secondary 32W05. Shaw’s work partially supported by National Science Foundation grant number DMS 98-01091.