Potential theoretic approach to Schauder estimates for the 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 $\omega_f$.