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Sobolev regularity of the Beurling transform on planar domains. (English) Zbl 1375.30026

Summary: 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.

MSC:

30C62 Quasiconformal mappings in the complex plane
42B37 Harmonic analysis and PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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