Necessary and sufficient conditions for the boundedness of the genuine singular integral operators in local Morrey-type spaces

Abstract

The problem of the boundedness of a Calderon-Zygmund singular integral operator T in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted L_p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness of  T  in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, for a genuine Calderon-Zygmund singular integral operator these sufficient conditions coincide with the necessary ones.