Counterexamples to some pointwise estimates of the maximal Cauchy transform in terms of the Cauchy transform

Daniel Girela-Sarrión

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

Motivated by the work of Mateu, Orobitg, Pérez and Verdera, who proved inequalities of the form T*f ≲ M(Tf) or T*f ≲ M2(Tf) for certain singular integral operators T, such as the Hilbert or the Beurling transforms, we study the possibility of establishing this type of control when T is the Cauchy transform along a Lipschitz graph. We show that this is not possible in general, and we give a partial positive result when the graph is substituted by a Jordan curve.
Original languageEnglish
Pages (from-to)657-675
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume38
Issue number1
DOIs
Publication statusPublished - 1 Jan 2013

Keywords

  • Calderón-Zygmund theory
  • Cauchy transform
  • Cotlar's inequality

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