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.