Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings

TL;DR

This paper explores how the special extracancellation property of even Calderon-Zygmund kernels influences the regularity of quasiconformal mappings and solutions to elliptic equations, revealing new insights into their Lipschitz continuity.

Contribution

It demonstrates the impact of the extracancellation property of even Calderon-Zygmund kernels on the Lipschitz regularity of quasiconformal mappings and elliptic equations.

Findings

Establishes a connection between extracancellation property and bilipschitz quasiconformal mappings.

Shows that certain Beltrami coefficients lead to Lipschitz regularity.

Provides a new perspective on second order elliptic equations in divergence form.

Abstract

We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a density of class Lip epsilon on the domain. The crucial fact in the argument is the special extracancellation property of even Calderon-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far-reaching consequences. The main result may also be viewed as a Lipschitz regularity result for the Beltrami equation, and so for certain planar second order elliptic equations in divergence form.