Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions. (1st December 2015) - Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques
Abstract: We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ( LSH ) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.