Abstract:
We establish criteria for the validity of modular inequalities
for the Hardy operator
on the cone $\Omega$
of nonnegative decreasing functions from weighted Orlicz spaces
with general weight.
The result
is based
on the theorem on the reduction of modular inequalities
for positively homogeneous operators
on the cone $\Omega$,
which enables passing to modular inequalities
for modified operators
on the cone
of all nonnegative functions from an Orlicz space.
It is shown that, for the Hardy operator, the modified operator
is a generalized Hardy operator.
This
enables us to establish explicit criteria for the validity of modular inequalities.