Maximal Inequalities in Bilateral Grand Lebesque Spaces Over Unbounded Measure

TL;DR

This paper develops precise non-asymptotic norm estimates for maximum distributions in rearrangement invariant spaces over unbounded measures, with applications to martingale theory and Fourier series.

Contribution

It introduces exact rearrangement invariant norm estimates over unbounded measures using entropy and generic chaining techniques, advancing the understanding of maximal inequalities.

Findings

Derived non-asymptotic exact norm estimates for maximum distributions

Applied results to martingale theory and Fourier series analysis

Enhanced the theoretical framework for rearrangement invariant spaces

Abstract

In this paper non-asymptotic exact rearrangement invariant norm estimates are derived for the maximum distribution of the family elements of some rearrangement invariant (r.i.) space over unbounded measure in the entropy terms and in the terms of generic chaining. We consider some applications in the martingale theory and in the theory of Fourier series.