Smoothness of the Beurling transform in Lipschitz domains

摘要

Let Ω?? be a Lipschitz domain and consider the Beurling transform of χ _Ω: Let 1<∞ 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,p) ~(α-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,p) ~(α-1)/p(?Ω), assuming that αp>2.