Smoothness of the Beurling transform in Lipschitz domains

Let Ω⊂CΩ⊂C be a Lipschitz domain and consider the Beurling transform of χΩχΩ:BχΩ(z)=limε→0−1π∫w∈Ω,|z−w|>ε1(z−w)2dm(w). Let 11αp>1. In this paper we show that if the outward unit normal N on ∂Ω   belongs to the Besov space Bp,pα−1/p(∂Ω), then BχΩBχΩ is in the Sobolev space Wα,p(Ω)Wα,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 Wα,p(Ω)Wα,p(Ω) if N   belongs to Bp,pα−1/p(∂Ω), assuming that αp>2αp>2.