Regularity for parabolic integro-differential equations with very irregular kernels

Abstract

We prove Hölder regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof that avoids the use of a convex envelope as well as give a new covering argument that is better suited to the fractional order setting. Our main result involves a class of kernels that 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.