A T(P) theorem for Sobolev spaces on domains

Martí Prats, Xavier Tolsa

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. 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.1πz2*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 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.
Original languageEnglish
Pages (from-to)2946-2989
JournalJournal of Functional Analysis
Volume268
Issue number10
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Calderón-Zygmund operators
  • Carleson measures
  • Harmonic analysis
  • Sobolev spaces

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