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.
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