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.


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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