On improvement of summability properties in nonautonomous Kolmogorov equations
Luciana Angiuli, Luca Lorenzi · Communications on Pure & Applied Analysis · 2013
Under suitable conditions, we obtain some characterizations of supercontractivity,ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class ofnonautonomous second order parabolic equations with unbounded coefficientsdefined in $I\times R^d$, where $I$ is a right-halfline. For this purpose, we establishan Harnack type estimate for $G(t,s)$ and a family of logarithmicSobolev 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 andultracontractivity to hold are also provided.