On the parabolic Adams theorem and its applications to diffusion processes

TL;DR

This paper explores the parabolic Adams theorem's applications in estimating solutions to divergence form parabolic operators and Itô equations with singular drift terms.

Contribution

It introduces new methods to estimate evolution families and derivatives of solutions using the parabolic Adams theorem and its corollaries.

Findings

Estimates for evolution families in Lp spaces for divergence form operators.

Bounds on moments of derivatives of solutions to Itô equations with singular drifts.

Demonstration of the theorem's utility in stochastic and PDE contexts.

Abstract

We show how the parabolic version of the Adams theorem and its corollary can be used to estimate in $L_{p}$ the evolution family associated to a divergence form second-order parabolic operator with parabolic Morrey lower-order terms and also how to estimate the moments of the derivatives of solutions of It\^o equations with respect to the initial data when the drift term has singularities.