Estimates for the norms of monotone operators on weighted Orlicz–Lorentz classes

A monotone operator P mapping the Orlicz–Lorentz class to an ideal space is considered. The Orlicz–Lorentz class is the cone of measurable functions on R+ =(0, ∞) whose decreasing rearrangements with respect to the Lebesgue measure on R+ belong to the weighted Orlicz space LΦ,ν. Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P: ΛΦ,ν →Y to those of the norm of its restriction to the cone of nonnegative step functions in LΦ,ν. The application of these results to the identity operator from ΛΦ,ν to the weighted Lebesgue space Y = L1(R+; g) gives exact descriptions of associated norms for ΛΦ,ν.