Fractional Poincare and logarithmic Sobolev inequalities for measure spaces

TL;DR

This paper extends Poincare and logarithmic Sobolev inequalities to measure spaces with fractional derivatives, broadening their applicability to spaces with minimal geometric structure and providing new insights for various mathematical and physical models.

Contribution

It introduces generalized inequalities for fractional derivatives in measure spaces with minimal geometric assumptions, including spaces of homogeneous type and applications to graph Laplacians and Boltzmann operators.

Findings

Established fractional Poincare inequalities for measure spaces.

Provided applications to graph Laplacians and Boltzmann collision operators.

Connected inequalities to recent developments in kinetic theory.

Abstract

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].