Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain $\mathrm{\Omega }$ and the Beurling transform of its characteristic function $$\mathcal{B}$ \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) $\mathrm{.}Itisshownthatiftheoutwardunitnormalvector$N$oftheboundaryofthedomainisinthetracespaceof$W^{n,p}(\Omega)$ (i.e., the Besov space $B_{p,p}^{n-1\mathrm{/}p}(\mathrm{\partial }\mathrm{\Omega })$) then $$\mathcal{B}$ \chi_\Omega \in W^{n,p}(\Omega)$\mathrm{.}Moreover,when$p>2$ the boundedness of the Beurling transform on $W^{n,p}(\mathrm{\Omega })$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.