Abnormal boundary decay for stable operators

Abstract

Assume $\hat{I}\pm \in (0,2)$ and $d\ge 2$. Let ${\mathcal{L}}^{\hat{I}}\pm$ be the generator of a symmetric, but not necessarily isotropic, $\hat{I}\pm$-stable process $X$ in ${\mathbb{R}}^d$ whose Lévy density is comparable with that of an isotropic $\hat{I}\pm$-stable process. In this paper, we show that the $C^{1,\mathrm{Dini}}$ regularity assumption on an open set $D\subset {\mathbb{R}}^d$ is optimal for the standard boundary decay property for nonnegative ${\mathcal{L}}^{\hat{I}}\pm$-harmonic functions in $D$, and for the standard boundary decay property of the heat kernel $p^D(t,x,y)$ of the part process $X^D$ of $X$ on $D$ by proving the following: (i) If $D$ is a $C^{1,\mathrm{Dini}}$ open set and $h$ is a nonnegative function which is ${\mathcal{L}}^{\hat{I}}\pm$-harmonic in $D$ and vanishes near a portion of $\partial D$, then the rate at which $h(x)$ decays to 0 near that portion of $\partial D$ is $\mathrm{dist}(x,D^c{)}^{\hat{I}\pm /2}$. (ii) If $D$ is a $C^{1,\mathrm{Dini}}$ open set, then, as $x\to \partial D$, the rate at which $p^D(t,x,y)$ tends to 0 is $\mathrm{dist}(x,D^c{)}^{\hat{I}\pm /2}$. (iii) For any non-Dini modulus of continuity $\ell$, there exist non-$C^{1,\mathrm{Dini}}$ open sets $D$, with $\partial D$ locally being the graph of a $C^{1,\ell }$ function, such that the standard boundary decay properties above do not hold for $D$.

46 pages. Revised version; accepted for publication in the Journal of Differential Equations