Consider a Lipschitz domain [katex]\Omega[/katex] and the Beurling transform of its characteristic function [katex] \mathcal{B} \chi_\Omega(z)= – {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) [/katex]. It is shown that if the outward unit normal vector [katex] N [/katex] of the boundary of the domain is in the trace space of [katex] W^{n,p}(\Omega) [/katex] (i.e., the Besov space [katex] B^{n-1/p}_{p,p}(\partial\Omega) [/katex]) then [katex]\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega) [/katex]. Moreover, when [katex] p>2 [/katex] the boundedness of the Beurling transform on [katex] W^{n,p}(\Omega) [/katex] follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.