On relations between norms in weighted Lebesgue and weighted Hölder spaces for operators with local singularities

Abstract

The norms in weighted Holder spaces and in weighted Lebesgue spaces are different in their character and any direct connection between the norms of these spaces should not be expected. However, in this work, a special class of operators was found, for which we obtained an inequality that connects the norms of these operators acting on weighted Lebesgue spaces and acting on weighted Holder spaces. A description of such operators and a relation among parameters of these spaces are given. In particular, integral operators with local endpoint singularities belong to the considered class. These results can be used in the study of operators on weighted Holder spaces, based on known results for operators acting on weighted Lebesgue spaces.