HARDY TYPE INEQUALITIES WITH POWER WEIGHTS FOR HAUSDORFF OPERATORS

Classical Hardy’s inequalities are concerned with the Hardy operator and its adjoint, the Bellman operator. Hausdorff operators in their various forms are natural generalizations of these two operators. Recently, the scheme used by Bradley for Hardy’s inequalities with general weights has been adjusted to the Hausdorff setting. For the Hardy and Bellman operators, the obtained necessary and sufficient conditions coincide and reduce to the classical ones. However, in this paper, we show that sufficient conditions for Hausdorff operators can be simplified. These are used for purely power weights, while for more general weights, only general necessary conditions are applied. The corresponding results are supplemented by examples.