# Schauder estimates for equations associated with L\'evy generators

## TL;DR

This paper establishes Schauder estimates for solutions to integro-differential equations linked with Le9vy process generators, broadening understanding of regularity properties for a wide class of such stochastic processes.

## Contribution

It introduces a method to derive Schauder estimates for Le9vy generators using gradient estimates of transition densities, applicable to stable processes and subordinate Brownian motions.

## Key findings

- Schauder estimates are established for a broad class of Le9vy generators.
- Insights into the domain of the infinitesimal generator for processes with characteristic exponent a b |a|^{b}.
- Analysis of the optimality of results via the Cauchy process domain.

## Abstract

We study the regularity of solutions to the integro-differential equation $Af-\lambda f=g$ associated with the infinitesimal generator $A$ of a L\'evy process. We show that gradient estimates for the transition density can be used to derive Schauder estimates for $f$. Our main result allows us to establish Schauder estimates for a wide class of L\'evy generators, including generators of stable L\'evy processes and subordinate Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a L\'evy process whose characteristic exponent $\psi$ satisfies $\text{Re} \, \psi(\xi) \asymp |\xi|^{\alpha}$ for large $|\xi|$. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.