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

Motivated by the work of Mateu, Orobitg, Pérez and Verdera, who proved inequalities of the form $T_*f\lesssim M(Tf)$ or $T_*f\lesssim M^2(Tf)$ for certain singular integral operators $T$, such as the Hilbert or the Beurling transforms, we study the possibility of establishing this type of control for 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.

Motivats per la feina de Mateu, Orobitg, Pérez i Verdera, que van provar desigualtats del tipus $T_*f\lesssim M(Tf)$ o $T_*f\lesssim M^2(Tf)$ per alguns operadors integrals singulars $T$, com les transformades de Beurling o de Hilbert, estudiem la possibilitat d'establir aquest tipus de control per la transformada de Cauchy al llarg d'un graf Lipschitz. Provem que això no es possible en general, i donem un resultat parcial positiu quan el graf se substitueix per una corba de Jordan.