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 ${\omega }_f$.

14 pages

References in corpus (2)

• Nonlocal diffusion and applications
• Some observations on the Green function for the ball in the fractional Laplace framework

Cited by in corpus (1)

• Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions