The Schauder estimate for kinetic integral equations

Abstract

We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion. We study equations of the form $f_t+v\cdot {\nabla }_{x}f={\mathcal{ℒ}}_{v}f+c$, where ${\mathcal{ℒ}}_v$ is an integrodifferential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of ${\mathcal{ℒ}}_v$, we obtain an a priori estimate for $f$ in a properly scaled Hölder space.