Functional Analysis and Applications Seminar 

06/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Natasha G. Samko, Centro de Análise Funcional e Aplicações, Faro

Weighted boundedness of singular operators in Morrey spaces

We study the problem of weighted boundedness of the one-dimensional singular operator $S$ with Cauchy kernel in Morrey spaces on a curve. The weight function may be a product of a finite number of almost monotonic functions with nodes on the curve. The boundedness of the operator $S$ in case of such a weight is reduced to the boundedness of Hardy type operators in Morrey spaces with this weight. We prove the latter, which enables us to obtain sufficient conditions for the boundedness of the singular operator $S$ in terms of the Matuszewska-Orlicz indices of weights. A special attention is paid to the case of power weights where we prove that the conditions on the weight are also necessary for the boundedness. We also discuss an application to the study of Fredholmness of singular integral equations in weighted Morrey spaces, the interest to this investigation being caused by the non-separability of Morrey spaces.