H\"older estimates for non-local parabolic equations with critical drift

TL;DR

This paper extends regularity results for non-local parabolic equations with non-symmetric kernels, providing key lemmas and inequalities to establish higher regularity of solutions.

Contribution

It introduces new oscillation and Harnack inequalities for non-symmetric kernels, advancing the understanding of regularity in non-local parabolic equations.

Findings

Established oscillation lemma for non-symmetric kernels

Proved Harnack inequality for solutions

Achieved higher regularity estimates for solutions

Abstract

In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide an Oscillation Lemma and a Harnack Inequality which can be used to prove higher regularity estimates.