On improvement of summability properties in nonautonomous Kolmogorov equations

TL;DR

This paper characterizes supercontractivity, ultraboundedness, and ultracontractivity of evolution operators for nonautonomous second-order parabolic equations with unbounded coefficients, using Harnack estimates and logarithmic Sobolev inequalities.

Contribution

It provides new characterizations and sufficient conditions for these properties in nonautonomous Kolmogorov equations, extending previous results.

Findings

Established Harnack type estimates for the evolution operator

Derived logarithmic Sobolev inequalities related to the evolution system of measures

Provided sufficient conditions for supercontractivity, ultraboundedness, and ultracontractivity

Abstract

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided.