Optimal functional inequalities for fractional operators on the sphere and applications

TL;DR

This paper establishes optimal fractional inequalities on the sphere, explores their implications for fractional heat flows, and derives weighted inequalities in Euclidean space, advancing the understanding of fractional operators and their applications.

Contribution

It introduces new optimal inequalities for fractional Laplacians on the sphere, including interpolations and remainder terms, with applications to heat flows and Euclidean weighted inequalities.

Findings

Optimal constants determined by spectral gap.

Inequalities interpolate between Sobolev and logarithmic Sobolev.

Weighted inequalities derived via stereographic projection.

Abstract

This paper is devoted to optimal functional inequalities for fractional Laplace operators on the sphere. Based on spectral properties, subcritical inequalities are established. Their consequences for fractional heat flows are considered. These subcritical inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities. Their optimal constants are determined by a spectral gap. In the subcritical range, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. We also consider inequalities which interpolate between fractional logarithmic Sobolev and fractional Poincar{\'e} inequalities. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, using a stereographic projection and scaling properties.