# $C^{2s}$ regularity for fully nonlinear nonlocal equations with bounded right hand side

## TL;DR

This paper proves sharp interior regularity estimates of class $C^{2s}$ for solutions to fully nonlinear nonlocal equations with bounded right hand side, extending previous linear and nonlinear results.

## Contribution

It establishes the first sharp $C^{2s}$ regularity results for fully nonlinear nonlocal equations with bounded data, generalizing prior linear and nonlinear theories.

## Key findings

- Solutions are in $C^{2s}$ when $Iu=f$ with $f$ bounded.
- Extends regularity results from linear to fully nonlinear nonlocal equations.
- Provides a regularity estimate for the nonlocal two membranes problem.

## Abstract

We establish sharp $C^{2s}$ interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right hand side. More precisely, we show that if $I$ is a fully nonlinear nonlocal concave or convex elliptic operator and $f\in L^\infty(B_1)$ then \[ Iu=f\quad\textrm{ in }\quad B_1 \quad \Rightarrow\quad u\in C^{2s}(B_{1/2}). \] This result generalizes the linear counterpart proved by Ros-Oton and Serra and extends previous available results for fully nonlinear nonlocal operators. As an application, we get a basic regularity estimate for the nonlocal two membranes problem.