Potential theoretic approach to Schauder estimates for the fractional Laplacian

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$.