Mei-Chi Shaw

Estimates for the -Neumann problem and nonexistence of C^2 Levi-flat hypersurfaces in CP^n

Abstract.Let Ω be a pseudoconvex domain with C2 boundary in , n ≥ 2. We prove that the -Neumann operator N exists for square-integrable forms on Ω. Furthermore, there exists a number ε0>0 such that the operators and the Bergman projection are regular in the Sobolev space Wε ( Ω) for ε<ε0. The -Neumann operator is used to construct -closed extension on Ω for forms on the boundary bΩ. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L2-holomorphic (p, 0)-forms on any domain with C2 pseudoconcave boundary in with p > 0 and n ≥ 2. As a consequence, we prove the nonexistence of C2 Levi-flat hypersurfaces in .

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 Let D 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

A decomposition problem on weakly pseudoconvex domains

Let Ω ⊂ Cn be a bounded pseudoconvex domain with defining function ρ of class CK , 2 ≤ K ≤ ∞. Let ν ≥ 1 be a real number such that ρ = −(−ρ)1/ν is a strictly plurisubharmonic exhaustion function on Ω. For a bounded open set Ω0 ⊃ Ω we obtain for S = Ω \ Ω the following theorem. Theorem 1. Let k be a nonnegative integer and tk = [2νmax(4+3k, n−1 5 ) +1]. Then there exists a Ck map W k = (W k 1 , · · · , W k n ) : Ω × S −→ Cn with the following properties: (i) W k(·, ζ) is holomorphic for all ζ ∈ S and for (z, ζ) ∈ Ω × S one has

The range of the tangential Cauchy-Riemann operator over a small ball

In this paPer we study the closed range property of tangential Cauchy-Riemann equations of an abstract CR structure over a small ball. Let D be the set {x E R*“’ 1 1x1 4, Kuranishi [ 141 and Akahori [ 11 have studied the L* theory

\( \bar{\partial } \) and \( {\bar{\partial }_b} \) Problems on Nonsmooth Domains

In this survey article we want to describe our method for constructing barriers on weakly pseudoconvex domains with smooth boundaries and give some applications to nonsmooth pseudoconvex domains. We also consider annuli of a very general type which are difference sets of a large piecewise smooth pseudoconvex domain with a union of piecewise smooth pseudoconvex domains in the interior. We do not impose any further conditions on the Levi form. For more details on the proofs, see the papers [24], [27], [28], [29].

Non-closed Range Property for the Cauchy-Riemann Operator

In this paper we study the non-closed range of the Cauchy-Riemann operator for relatively compact domains in \(\mathbb C^{n}\) or in a complex manifold. We give necessary and sufficient conditions for the \(L^{2}\) closed range property for \(\overline{\partial }\) on bounded Lipschitz domains in \(\mathbb C^{2}\) with connected complement. It is proved for the Hartogs triangle that \(\overline{\partial }\) does not have closed range for (0, 1)-forms smooth up to the boundary, even though it has closed range in the weak \(L^{2}\) sense. An example is given to show that \(\overline{\partial }\) might not have closed range in \(L^{2}\) on a Stein domain in complex manifold.

On the

For certain annuli in

L^2

\mathbb{C}^n

-Dolbeault cohomology of annuli

,

Subelliptic estimates for the

n\geq 2