Smoothness of the Beurling transform in Lipschitz domains

Let Ω⊂ℂ be a Lipschitz domain and consider the Beurling transform of χ Ω: Let 1<p<∞ and 0<α<1 with αp>1. In this paper we show that if the outward unit normal N on ∂Ω belongs to the Besov space B p,pα-1/p(∂Ω), then Bχ Ω is in the Sobolev space W α,p(Ω). This result is sharp. Further, together with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling transform is bounded in W α,p(Ω) if N belongs to B p,pα-1/p(∂Ω), assuming that αp>2. © 2012 Elsevier Inc.