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, a) whose decreasing rearrangements with respect to the Lebesgue measure on R (+) belong to the weighted Orlicz space L (I broken vertical bar,nu). Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P: I >(I broken vertical bar,nu) -> Y to those of the norm of its restriction to the cone of nonnegative step functions in L (I broken vertical bar,nu). The application of these results to the identity operator from I >(I broken vertical bar,nu) to the weighted Lebesgue space Y = L (1)(R (+); g) gives exact descriptions of associated norms for I >(I broken vertical bar,nu).