Local regularity for parabolic nonlocal operators

TL;DR

This paper extends Moser's 1971 results by establishing local regularity estimates and a weak Harnack inequality for weak solutions of parabolic integro-differential operators, uniformly as the order approaches 2.

Contribution

It provides robust local a priori H"older estimates and a weak Harnack inequality for parabolic nonlocal operators, extending classical results to the nonlocal setting.

Findings

Established local H"older regularity estimates.

Proved a weak Harnack inequality.

Results are stable as the order approaches 2.

Abstract

Weak solutions to parabolic integro-differential operators of order $\alpha \in (\alpha_0, 2)$ are studied. Local a priori estimates of H\"older norms and a weak Harnack inequality are proved. These results are robust with respect to $\alpha \nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.