Sobolev regularity of the Beurling transform on planar domains

TL;DR

This paper investigates the Sobolev regularity of the Beurling transform applied to characteristic functions of planar domains, establishing conditions on boundary normals that imply the transform's Sobolev regularity, with implications for quasiconformal mappings.

Contribution

It proves that boundary normal regularity in Besov spaces ensures the Beurling transform of the characteristic function lies in the same Sobolev space, and shows boundedness for p>2.

Findings

Beurling transform of characteristic functions is in W^{n,p} under boundary regularity conditions.

Normal vector in trace space implies Sobolev regularity of the transform.

Boundedness of the Beurling transform on W^{n,p} for p>2.

Abstract

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) $. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(\Omega)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partial\Omega)$) then $\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(\Omega)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.