Rearrangement transformations on general measure spaces

TL;DR

This paper introduces a unified framework for rearrangement transformations on measure spaces, encompassing classical, Steiner's symmetrization, multidimensional, and discrete tree cases, through a general set transformation.

Contribution

It defines a general rearrangement via the Layer's cake formula and studies Lorentz spaces associated with these transformations, unifying various classical and modern approaches.

Findings

Unified approach to rearrangement transformations

Functional properties of Lorentz spaces studied

Applicable to classical, Steiner's, multidimensional, and discrete cases

Abstract

For a general set transformation ${\cal R}$ between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of ${\cal R}$, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.