Jean-Jacques Loeb

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

We prove an intrinsic equivalence between strong hypercontractivity (sHC) and a strong logarithmic Sobolev inequality (sLSI) 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 throughLSH functions, and use it to prove the equivalence of (sHC) and (sLSI) for such log-subharmonic functions.

On open analytic and subanalytic mappings

This paper is a completion of some results of Ronga–Gamboa and Hirsch: namely, we characterize those subanalytic (or definable in some o-minimal structure) mappings of class that are open. The results give raise to some natural questions about the relations between the openness and the properness of a real analytic map. We present some results in two variables and give several explanatory examples. This is partly motivated by the fact that apart from the Pinchuk example, there are apparently no known examples of polynomial open but non-proper self-maps of the plane.