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Weighted Sobolev estimates of the truncated Beurling operator

Yifei Pan, Yuan Zhang

Abstract

Given a bounded planar domain $D$ with $W^{k+1, \infty}$ boundary, $ k\in \mathbb Z^+$, and a weight $μ\in A_p, 1<p<\infty$, we show that the corresponding truncated Beurling transform is a bounded operator sending $W^{k, p}(D, μ)$ into itself. Weighted Sobolev estimates for other Cauchy-type integrals are also obtained.

Weighted Sobolev estimates of the truncated Beurling operator

Abstract

Given a bounded planar domain with boundary, , and a weight , we show that the corresponding truncated Beurling transform is a bounded operator sending into itself. Weighted Sobolev estimates for other Cauchy-type integrals are also obtained.
Paper Structure (3 sections, 5 theorems, 33 equations)

This paper contains 3 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Weighted Sobolev estimates

Key Result

Theorem 1.1

Let $D\subset \mathbb C$ be a bounded domain with $W^{k+1, \infty}$ boundary, $k\in \mathbb Z^+\cup\{0\}$. Assume $\mu\in A_p, 1<p<\infty$. There exists a constant $C$ dependent only on $D, k$, $p$ and $\mu$, such that for all $f\in W^{k, p}(D, \mu)$, and

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['mainS']}:
  • proof : Proof of Theorem \ref{['mainT']}: