Modular and norm inequalities for operators on the cone of decreasing functions in Orlicz space

Modular and norm inequalities are considered on the cone of all nonnegative functions as well as on the cone $\Omega$ of all nonnegative decreasing functions in the weighted Orlicz space. Reduction theorems are proved for the norm of positively homogeneous operator on the cone $\Omega$. We show that it is equivalent to the norm of a certain modified operator on the cone of all nonnegative functions in this space. Analogous results are established for modular inequalities.