Lorentz-Shimogaki and Boyd Theorems for weighted Lorentz spaces

Abstract

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces Λpu(w). As a consequence, we give the complete characterization of the strong boundedness of H on these spaces in terms of some geometric conditions on the weights u and w, whenever p > 1. For these values of p, we also give the complete solution of the weak-type boundedness of the Hardy Littlewood operator on Λpu(w).