# The Schauder estimate for kinetic integral equations

## TL;DR

This paper proves interior Schauder estimates for kinetic equations involving integro-differential diffusion operators, providing a priori regularity bounds under ellipticity and continuity conditions.

## Contribution

It establishes the first Schauder estimates for kinetic equations with integro-differential diffusion, extending regularity theory to this class of equations.

## Key findings

- Derived a priori Schauder estimates for kinetic integro-differential equations.
- Established regularity results under ellipticity and H"older continuity assumptions.
- Extended classical Schauder theory to kinetic equations with nonlocal diffusion.

## Abstract

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 H\"older continuity conditions on the kernel of $\mathcal L_v$, we obtain an a priori estimate for $f$ in a properly scaled H\"older space.