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

Abstract

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 = L₁(R₊; g) gives exact descriptions of associated norms for Λ_Φ,ν.