Tail estimations for functions belonging to Grand Lebesgue Spaces builded on the set with infinite measure

TL;DR

This paper explores the precise relationship between tail behaviors of measurable functions and their norms in Grand Lebesgue and Orlicz spaces over infinite measure sets, providing exact estimates and illustrative examples.

Contribution

It establishes exact reciprocal relations between tail decay and norms in GLS and Orlicz spaces on infinite measure sets, with concrete examples.

Findings

Exact bilateral relations between tail behavior and norms.

Illustrative examples confirming the estimates.

Extensions to functions on infinite measure sets.

Abstract

We establish the bilateral exact reciprocal interrelations between a tail behavior of a measurable functions and its norm in the suitable Grand Lebesgue Space (GLS) as well as Orlicz one, builded over the set with infinite measure. We bring also some examples in order to illustrate the exactness of offered estimates.