Boundary estimates and a Wiener criterion for the fractional Laplacian

Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-\Delta)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the extended weighted equation in a subset of $\R^{n+1}$. As a consequence, we characterize regular boundary points for $(-\Delta)^su=0$ by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.