Hardy type inequalities in the category of 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. In this paper, we try to adjust the scheme used by Bradley for Hardy’s inequalities with general weights to the Hausdorff setting. It is not surprising that the obtained necessary conditions differ from the sufficient conditions as well as that both depend not only on weights but also on the kernel that generate the Hausdorff operator. For the Hardy and Bellman operators, the obtained necessary and sufficient conditions coincide and reduce to the classical ones. © Springer Nature Switzerland AG 2019.