Smoothness of the Beurling transform in Lipschitz domains ✩

Let Ω ⊂ C be a Lipschitz domain and consider the Beurling transform of χΩ: BχΩ(z) = lim ε→0 −1 π w∈Ω,|z−w|>ε 1 (z − w)2 dm(w). Let 1 1. In this paper we show that if the outward unit normal N on ∂Ω belongs to the Besov space B α−1/p p,p (∂Ω), then BχΩ is in the Sobolev space 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) if N belongs to B α−1/p p,p (∂Ω), assuming that αp > 2. © 2012 Elsevier Inc. All rights reserved.