New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces

TL;DR

This paper introduces new formulas for decreasing rearrangements using nonlinear Hardy-Littlewood inequalities, and explores properties of new Orlicz-Lorentz related spaces, including their duals and Banach envelopes.

Contribution

It derives novel formulas for rearrangements and characterizes the duals and biduals of generalized Orlicz-Lorentz spaces, extending their analysis to non-linear and quasi-Banach settings.

Findings

New formulas for decreasing rearrangements derived

Identification of K"othe duals with classical Orlicz-Lorentz spaces

Introduction of a new class of rearrangement invariant Banach spaces

Abstract

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K\"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope.