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. We 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.