Potential theoretic approach to Schauder estimates for the fractional Laplacian

We present an elementary approach for the proof of Schauder estimates for the equation (−Δ)su(x) = f(x), 0 < s < 1, with f having a modulus of continuity ωf, based on the Poisson representation formula and dyadic ball approximation argument. We give the explicit modulus of continuity of u in balls Br(x) ⊂ Rn in terms of ωf.