Singular and fractional integral operators on weighted local Morrey spaces

TL;DR

This paper characterizes weighted inequalities for Riesz transforms and Calderón-Zygmund operators on weighted local Morrey spaces, providing conditions for boundedness and extending results to fractional operators and power weights.

Contribution

It offers new characterizations of inequalities for singular and fractional integral operators on weighted local Morrey spaces, including sharp results for power weights.

Findings

Characterization of weighted inequalities for Riesz transforms

Boundedness conditions for Calderón-Zygmund operators

Sharp results for fractional operators with power weights

Abstract

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calder\'on-Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results in a larger range of exponents for the usual versions of weighted Morrey spaces.