Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 38, 2013, 657-675

COUNTEREXAMPLES TO SOME POINTWISE ESTIMATES OF THE MAXIMAL CAUCHY TRANSFORM IN TERMS OF THE CAUCHY TRANSFORM

Daniel Girela-Sarrión

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193 Bellaterra (Barcelona), Spain; dgirela 'at' mat.uab.cat

Abstract. Motivated by the work of Mateu, Orobitg, Pérez and Verdera, who proved inequalities of the form T*f \lesssim M(Tf) or T*f \lesssim 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.

2010 Mathematics Subject Classification: Primary 42B20.

Key words: Calderón-Zygmund theory, Cauchy transform, Cotlar's inequality.

Reference to this article: D. Girela-Sarrión: Counterexamples to some pointwise estimates of the maximal Cauchy transform in terms of the Cauchy transform. Ann. Acad. Sci. Fenn. Math. 38 (2013), 657-675.

Full document as PDF file

doi:10.5186/aasfm.2013.3828

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