Potential theoretic approach to Schauder estimates for the fractional Laplacian

TL;DR

This paper introduces an elementary method using potential theory to establish Schauder estimates for the fractional Laplacian, providing explicit continuity bounds based on the source function's modulus of continuity.

Contribution

It offers a new, simplified proof technique for Schauder estimates for the fractional Laplacian utilizing Poisson representation and dyadic ball approximation.

Findings

Explicit modulus of continuity for solutions in terms of source function

Elementary proof approach for Schauder estimates

Applicable to equations with $0<s<1$ fractional Laplacian

Abstract

We present an elementary approach for the proof of Schauder estimates for the equation $(-\Delta)^s u(x)=f(x), \,0<s<1$, with $f$ having a modulus of continuity $\omega_f$, based on the Poisson representation formula and dyadic ball approximation argument. We give the explicit modulus of continuity of $u$ in balls $B_r(x)\subset \mathbb{R}^n$ in terms of $\omega_f$.