On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian

Abstract

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^μ in Rⁿ, n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev.

As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)^μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there is an interval [A_n, 1/2n−A_n], where A_n→1, as n→∞, with μ-values for which the property does not hold. This interval is non-empty for n≥8.