Poincare type inequalities for two different Bilateral Grand Lebesgue Spaces

TL;DR

This paper establishes non-asymptotic Poincare inequalities for functions and their weak gradients within Bilateral Grand Lebesgue Spaces on general metric spaces, demonstrating their sharpness.

Contribution

It introduces new Poincare inequalities in Bilateral Grand Lebesgue Spaces and proves their sharpness, extending classical results to these spaces.

Findings

Derived non-asymptotic Poincare inequalities in Bilateral Grand Lebesgue Spaces.

Proved the sharpness of these inequalities.

Extended classical Poincare inequalities to a broader functional setting.

Abstract

In this paper we obtain the non-asymptotic inequalities of Poincare type between function and its weak gradient belonging the so-called Bilateral Grand Lebesgue Spaces over general metric measurable space. We also prove the sharpness of these inequalities.