Regularity estimates for diffusion semigroups on weighted Sobolev spaces

TL;DR

This paper establishes regularity estimates for diffusion semigroups generated by second order elliptic operators on weighted Sobolev spaces, using bounds on derivatives of fundamental solutions to parabolic equations.

Contribution

It provides new norm estimates for diffusion semigroups on weighted Sobolev spaces under specific weight conditions, extending classical derivative bounds.

Findings

Derived bounds for semigroup norms on weighted Sobolev spaces

Extended classical derivative bounds to weighted settings

Applicable to elliptic operators with smooth bounded coefficients

Abstract

In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces, where the weight satisfies some specific conditions. Our proof relies on a classical bound for the derivatives of fundamental solution to parabolic equations.