Let D be a planar Lipschitz domain and consider the Beurling transform of the
characteristic function of D, B(1_D). Let 1<p<\infty and 0<a<1 with ap>1. In
this paper we show that if the outward unit normal N on bD, the boundary of D,
belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of
1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together
with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling
transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD),
assuming that ap>2.