Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces

We prove generalizations of the Poincaré 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 Poincaré inequalities for graph Laplacians, fractional Poincaré inequalities of Mouhot, Russ, and Sire (2011) [17], and implications for recent work of the author and R.M. Strain on the Boltzmann collision operator (Gressman and Strain, 2010, 2011, 2011 [9], [10] and [11]).