Potential theory 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 .