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 L Φ,ν . Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P : Λ Φ,ν → Y to those of the norm of its restriction to the cone of nonnegative step functions in L Φ,ν . The application of these results to the identity operator from Λ Φ,ν to the weighted Lebesgue space Y = L 1 ( R + ; g ) gives exact descriptions of associated norms for Λ Φ,ν .