Sobolev regularity of the Beurling transform on planar domains

Abstract

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B}{\chi }_{\Omega }(z)=-p.v.\frac{1}{\pi z^2}\ast {\chi }_{\Omega }(z)$. 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 $B_{p,p}^{n-1/p}(\partial \Omega )$) 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.

33 pages, 8 figures. arXiv admin note: text overlap with arXiv:1507.04332

References in corpus (4)

• Beltrami equation with coefficient in Sobolev and Besov spaces
• A T(P) theorem for Sobolev spaces on domains
• On Sobolev extension domains in $R^n$
• A geometric characterization of planar Sobolev extension domains

Cited by in corpus (3)

• Weighted Sobolev estimates of the truncated Beurling operator
• Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
• Harmonic and Trace Inequalities in Lipschitz Domains