Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain Ω and the Beurling transform of its characteristic function BχΩ(z) = −p.v. πz12 * χΩ(z). It is shown that if the outward unit normal vector N of the boundary of the domain is in the trace space of Wn,p(Ω) (i.e., the Besov space Bp,pn−1/p(∂Ω)) then BχΩ ∈ Wn,p(Ω). Moreover, when p > 2 the boundedness of the Beurling transform on Wn,p(Ω) follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.