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A T(P) theorem for Sobolev spaces on domains

Author:
Prats, Martí   Tolsa, Xavier  


Journal:
Journal of Functional Analysis


Issue Date:
2015


Abstract(summary):

Abstract

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 View the MathML source is bounded in the Sobolev space Ws,p(Ω) if and only if Ω∈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 we find 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.



Page:
2946-2989


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