Regularity of [formula omitted] and Lipschitz domains in terms of the Beurling transform

Abstract

Let Ω⊂C be a bounded C1 domain, or a Lipschitz domain “flat enough”, and consider the Beurling transform of χΩ:BχΩ(z)=limε→0−1π∫w∈Ω,|z−w|>ε1(z−w)2dm(w). Using a priori estimates, in this paper we solve the following free boundary problem: if BχΩ belongs to the Sobolev space Wα,p(Ω) for 0<α⩽1, 1<p<∞ such that αp>1, then the outward unit normal N on ∂Ω is in the Besov space Bp,pα−1/p(∂Ω). The converse statement, proved previously by Cruz and Tolsa, also holds. So we haveB(χΩ)∈Wα,p(Ω)⟺N∈Bp,pα−1/p(∂Ω). Together with recent results by Cruz, Mateu and Orobitg, from the preceding equivalence one infers that the Beurling transform is bounded in Wα,p(Ω) if and only if the outward unit normal N belongs to Bp,pα−1/p(∂Ω), assuming that αp>2.