On a pointwise inequality for even Legendre polynomials in high dimensional spheres

TL;DR

This paper establishes a pointwise inequality for even Legendre polynomials on high-dimensional spheres, incorporating spectral gap effects, which enhances previous results and aids in analyzing Fisher information in the Boltzmann equation.

Contribution

It introduces a new pointwise inequality for even Legendre polynomials in high dimensions, considering spectral gaps, improving upon recent prior results.

Findings

Derived a sharper inequality for Legendre polynomials

Incorporated spectral gap effects into the inequality

Enabled better analysis of Fisher information in kinetic theory

Abstract

We present a pointwise inequality for adjacent even Legendre polynomials in high dimensional spheres featuring the effect of spectral gaps. This improves a recent result of Imbert, Silvestre and Villani that is crucially used in their study of the Fisher information for the Boltzmann equation.