A T(P) theorem for Sobolev spaces on domains

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given 02sp>2 and a Lipschitz domain Ω⊂CΩ⊂C, the Beurling transform Bf=−p.v.1πz2⁎f is bounded in the Sobolev space Ws,p(Ω)Ws,p(Ω) if and only if BχΩ∈Ws,p(Ω)BχΩ∈Ws,p(Ω).In this paper we obtain a generalized version of the former result valid for any s∈Ns∈N and for a larger family of Calderón–Zygmund operators in any ambient space RdRd as long as p>dp>d. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for p⩽dp⩽d. In the particular case s=1s=1, this condition is in fact necessary, which yields a complete characterization.