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The Schauder estimate for kinetic integral equations

We establish interior Schauder estimates for kinetic equations with integro-differential 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 integro-differential 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.