Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

Piotr Graczyk (1) , Todd Kemp (2) , Jean-Jacques Loeb (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 MATH - UC San Diego - Department of Mathematics [Univ California San Diego]

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 strong logarithmic Sobolev inequality for such log-subharmonic functions.

Domaines: Probabilités [math.PR]

Piotr Graczyk, Todd Kemp, Jean-Jacques Loeb. Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions. 2013. ⟨hal-00906171⟩