Poincar\'e Inequalities in Bilateral Grand Lebesgue Spaces

TL;DR

This paper derives non-asymptotic Poincaré inequalities within Bilateral Grand Lebesgue Spaces, providing sharpness examples to demonstrate the bounds' precision.

Contribution

It introduces new non-asymptotic Poincaré inequalities in Bilateral Grand Lebesgue Spaces, expanding the understanding of functional inequalities in these spaces.

Findings

Derived sharp non-asymptotic Poincaré inequalities

Provided examples demonstrating the inequalities' sharpness

Extended classical inequalities to Bilateral Grand Lebesgue Spaces

Abstract

In this paper we obtain the non - asymptotic estimations of Poincare type between function and its gradient in the so - called Bilateral Grand Lebesgue Spaces. We also give some examples to show the sharpness of these inequalities.