Hardy's operator and normability of generalized Lorentz-Marcinkiewicz spaces, with sharp or weakly sharp constant estimation

TL;DR

This paper introduces a generalized Banach quasinorm based on Hardy's operator for Lorentz-Marcinkiewicz spaces, providing an equivalent norm expression and analyzing sharp constant estimates.

Contribution

It extends classical Lorentz-Marcinkiewicz norms by defining a new Banach quasinorm using Hardy's operator and derives its equivalent norm expression.

Findings

Established a new Banach quasinorm for Lorentz-Marcinkiewicz spaces.

Derived an equivalent norm expression for the introduced quasinorm.

Provided estimates for sharp or weakly sharp constants in the norm.

Abstract

We introduce a Banach rearrangement invariant (tail) quasy-norm by means of Hardy's (Cesaro) average on the (measurable) functions defined on some measurable space which is a slight generalization of classical Lorentz-Marcinkiewicz norm and find for it an equivalent norm expression.