Regularity for parabolic integro-differential equations with very irregular kernels

TL;DR

This paper establishes H"older regularity for a broad class of parabolic integro-differential equations with highly irregular kernels, extending previous results and introducing new proof techniques suited for fractional order problems.

Contribution

It introduces a novel proof method avoiding convex envelopes and a new covering argument, applicable to kernels with singular measures, asymmetry, and vanishing points.

Findings

Proves H"older regularity for general integro-differential equations

Includes kernels with singular measures and asymmetry

Provides potential applications to Boltzmann equation regularization

Abstract

We prove H\"older regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering argument which is better suited to the fractional order setting. Our main result involves a class of kernels which may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the Boltzmann equation.