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

Motivated by the work of Mateu, Orobitg, Perez and Verdera, who proved inequalities of the form T(*)f less than or similar to M(Tf) or T(*)f less than or similar to 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 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.