  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.
