Fractional Poincare and logarithmic Sobolev inequalities for measure spaces

2012/05/25 by Gressman, Philip T.
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1205.5773

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].

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