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.
|Journal||Annales Academiae Scientiarum Fennicae Mathematica|
|Publication status||Published - 1 Jan 2013|
- Calderón-Zygmund theory
- Cauchy transform
- Cotlar's inequality