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New Properties of Holomorphic Sobolev-Hardy Spaces

William Gryc, Loredana Lanzani, Jue Xiong, Yuan Zhang

Abstract

We give new characterizations of the optimal data space for the $L^p(bD,σ)$-Neumann boundary value problem for the $\bar{\partial}$ operator associated to a bounded, Lipschitz domain $D\subset\mathbb{C}$. We show that the solution space is embedded (as a Banach space) in the Dirichlet space and that for $p=2$, the solution space is a reproducing kernel Hilbert space.

New Properties of Holomorphic Sobolev-Hardy Spaces

Abstract

We give new characterizations of the optimal data space for the -Neumann boundary value problem for the operator associated to a bounded, Lipschitz domain . We show that the solution space is embedded (as a Banach space) in the Dirichlet space and that for , the solution space is a reproducing kernel Hilbert space.
Paper Structure (8 sections, 12 theorems, 64 equations)

This paper contains 8 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz domains
  4. Properties of $\mathcal{H}_\alpha^{1,2} (D)$ for simply connected $D$
  5. $\mathcal{H}_\alpha^{1, 2}(D)$ is a reproducing kernel Hilbert space
  6. $\mathcal{H}_\alpha^{1,p}(D)$ is embedded in the Dirichlet Space
  7. Characterizations of $\mathfrak{n}^{p}(bD)$ for simply connected $D$
  8. A characterization of $\mathfrak{n}^p(bD)$ for multiply connected $D$

Key Result

Lemma 2.3

NecasV Let $D$ be a bounded Lipschitz domain. There exists a family $\{D_k\}_{k=1}^\infty$ of smooth domains with $D_k$ compactly contained in $D$ that satisfy the following:

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 16 more