The Schauder estimate for kinetic integral equations

We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. study of the form $f_t + v \cdot \nabla_x f = \mathcal L_v c$, where $\mathcal L_v$ is an diffusion operator order $2s$ acting in $v$-variable. Under suitable ellipticity and H\"older continuity conditions on kernel L_v$, we obtain a priori estimate $f$ properly scaled space.