Lorentz spaces for decreasing rearrangements of functions on trees

Abstract

We characterize the existence of a norm in the Lorentz spaces defined on trees, for non-linear decreasing rearrangements, as well as other functional properties (quasinormability, relationship with rearrangement invariant spaces, etc.). The main tool is the characterization of the saturation in Hardy-Littlewood's inequality for linearly decreasing functions.