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Sobolev Regularity for the Bergman Projection on Relatively Compact Domains in Hermitian manifolds

Phillip S. Harrington

Abstract

Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for $L^2$ Sobolev regularity of the Bergman projection acting on $L^2$ sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.

Sobolev Regularity for the Bergman Projection on Relatively Compact Domains in Hermitian manifolds

Abstract

Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for Sobolev regularity of the Bergman projection acting on sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.

Paper Structure

This paper contains 10 sections, 14 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Sobolev Spaces in Manifolds
  3. Hermitian Geometry
  4. The Basic Estimate
  5. A Solution Operator
  6. Estimates for the Bergman Projection
  7. Proof of Theorem \ref{['thm:DF_psh']}
  8. Examples
  9. Hopf Manifolds
  10. An Application of Theorem \ref{['thm:DF_psh']}

Key Result

Theorem 1.3

Let $M$ be a Hermitian manifold of dimension $n\geq 2$ and let $\Omega\subset M$ be a relatively compact domain with Lipschitz boundary. Let $\psi\in C^2(M)$. Suppose that for some $0\leq a<b\leq 1$, there exists a Lipschitz defining function $\rho$ for $\Omega$ such that $\rho|_\Omega\in C^2(\Omega on $\Omega$ for $\eta=a$ and $\eta=b$. Then the Bergman projection $P_\psi$ is continuous in $W^s(\

Theorems & Definitions (26)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 16 more