A T(P) theorem for Sobolev spaces on domains

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Resum

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 0 < s 1, 1 p ∞ with sp 2 and a Lipschitz domain Ω ⊂ C, the Beurling transform Bf = −p.v. 1 πz2 ∗ f is bounded in the Sobolev space Ws,p(Ω) if and only if BχΩ ∈ Ws,p(Ω). In this paper we obtain a generalized version of the former result valid for any s ∈ N and for a larger family of Calderón–Zygmund operators in any ambient space Rd as long as p 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 wefind a sufficient condition in terms of Carleson measures for p d. In the particular case s = 1, this condition is in fact necessary, which yields a complete characterization.
Idioma originalAnglès
Pàgines (de-a)2946-2989
Nombre de pàgines44
RevistaJournal of Functional Analysis
Volum268
Número10
DOIs
Estat de la publicacióPublicada - 1 de gen. 2015

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