Open Access
2017 Sobolev Regularity of the Beurling Transform on Planar Domains
Martí Prats
Publ. Mat. 61(2): 291-336 (2017). DOI: 10.5565/PUBLMAT6121701


Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function ${\mathcal B} \chi_\Omega(z)= - \text{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.


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Martí Prats. "Sobolev Regularity of the Beurling Transform on Planar Domains." Publ. Mat. 61 (2) 291 - 336, 2017.


Received: 31 July 2015; Revised: 27 April 2016; Published: 2017
First available in Project Euclid: 29 June 2017

zbMATH: 1375.30026
MathSciNet: MR3677864
Digital Object Identifier: 10.5565/PUBLMAT6121701

Primary: 30C62 , 42B37 , 46E35

Keywords: Beurling transform , David-Semmes betas , Lipschitz domains , Peter Jones' betas , quasiconformal mappings , Sobolev Spaces

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.61 • No. 2 • 2017
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