The $\overline {\partial }$ problem on domains with piecewise smooth boundaries with applications

Abstract

Abstract. Let Ω be a bounded domain in C n such that Ω has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equationwhere α is a smooth ∂-closed (p, q) form with coefficients C ∞ up to the bundary of Ω, 0 ≤ p ≤ n and 1 ≤ q ≤ n. In particular, Equation (0.1) is solvable with u smooth up to the boundary (for appropriate degree q) if Ω satisfies one of the following conditions: i) Ω is the transversal intersection of bounded smooth pseudoconvex domains. ii) Ω = Ω 1 \ Ω 2 where Ω 2 is the union of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex convex domain with a piecewise smooth boundary. iii) Ω = Ω 1 \ Ω 2 where Ω 2 is the intersection of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary. The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for ∂ b on domains with piecewise smooth boundaries in a pseudoconvex manifold.Let Ω be a bounded domain in C n such that Ω has a piecewise smooth boundary. In this paper we study the solvability of the Cauchy-Riemann equationwhere α is a smooth ∂-closed (p, q) form with coefficients C ∞ up to the boundary of Ω, 0 ≤ p ≤ n and 1 ≤ q ≤ n. In particular, we prove that Equation (0.1) is solvable with u smooth up to the boundary (for appropriate degree q) if Ω satisfies one of the following conditions: i) Ω is the transversal intersection of bounded smooth pseudoconvex domains. ii) Ω = Ω 1 \Ω 2 , where Ω 2 is the union of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary.iii) Ω = Ω 1 \ Ω 2 , where Ω 2 is the intersection of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary.The boundary regularity problem for ∂ has been studied extensively by two very different methods: by L 2 a priori estimates and by the integral kernel approach.

Function Theory and Holomorphic Maps on Symmetric Products of Planar Domains

“…Note that though the ∂-Neumann operator on Σ n U is non-compact, the ∂-problem is globally regular. The situation on Σ n U is therefore analogous to that on cartesian products (see [12,33]. )…”

The Cauchy–Riemann equations on product domains

“…Under the same assumption as in Corollary 6.1, if f ∈ C ∞ p,q (Ω), is a ∂-closed form, with q = 0, then there exists u ∈ C ∞ p,q−1 (Ω) such that ∂u = f . For domains which are the intersection of a finite number of smoothly bounded pseudoconvex domains, such that the boundaries meet transversely at each point of intersection, the existence of a solution to the ∂-equation smooth up to the boundary has been obtained before [25] using integral kernels. This includes the result of Corollary 6.2, but our method here is simpler and also leads to estimates in Sobolev spaces.…”

“…The regularity for the canonical solution of the ∂-equation and the ∂-Neumann operator on a polydisc have been studied extensively (see [12,13,14,4] and the references in these works.) There is also a considerable amount of work for the ∂-equation on domains with Lipschitz boundary or piecewise smooth domains (see [25]). Notice that a product domain is only piecewise smooth even if each factor domain has smooth boundary.…”

Product domains, multi-Cauchy transforms, and the ∂¯ equation

“…The bibliographies in [16,29,35] give references to more specialized results. The papers [11] and [31] are highlights of results obtained after the mid-80s.…”