Potential theoretic approach to Schauder estimates for the fractional Laplacian

Bucur, Claudia; Karakhanyan, Aram

doi:10.1090/proc/13227

Potential theoretic approach to Schauder estimates for the fractional Laplacian
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by Claudia Bucur and Aram L. Karakhanyan PDF

Proc. Amer. Math. Soc. 145 (2017), 637-651 Request permission

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 $\omega _f$.

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