Abnormal boundary decay for stable operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914302614568960 |
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| author | Cho, Soobin Song, Renming |
| author_facet | Cho, Soobin Song, Renming |
| contents | Assume $α\in (0, 2)$ and $d\ge 2$. Let $\mathcal L^α$ be the generator of a symmetric, but not necessarily isotropic, $α$-stable process $X$ in $\mathbb R^d$ whose Lévy density is comparable with that of an isotropic $α$-stable process. In this paper, we show that the $C^{1, \rm Dini}$ regularity assumption on an open set $D\subset \mathbb R^d$ is optimal for the standard boundary decay property for nonnegative $\mathcal L^α$-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 $C^{1, \rm Dini}$ open set and $h$ is a nonnegative function which is $\mathcal L^α$-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 ${\rm dist} (x, D^c)^{α/2}$. (ii) If $D$ is a $C^{1, \rm Dini}$ open set, then, as $x\to \partial D$, the rate at which $p^D(t,x,y)$ tends to 0 is ${\rm dist} (x, D^c)^{α/2}$. (iii) For any non-Dini modulus of continuity $\ell$, there exist non-$C^{1, \rm Dini}$ 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 $D$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abnormal boundary decay for stable operators Cho, Soobin Song, Renming Analysis of PDEs Probability 60J45, 60J50, 35K08, 47G20 Assume $α\in (0, 2)$ and $d\ge 2$. Let $\mathcal L^α$ be the generator of a symmetric, but not necessarily isotropic, $α$-stable process $X$ in $\mathbb R^d$ whose Lévy density is comparable with that of an isotropic $α$-stable process. In this paper, we show that the $C^{1, \rm Dini}$ regularity assumption on an open set $D\subset \mathbb R^d$ is optimal for the standard boundary decay property for nonnegative $\mathcal L^α$-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 $C^{1, \rm Dini}$ open set and $h$ is a nonnegative function which is $\mathcal L^α$-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 ${\rm dist} (x, D^c)^{α/2}$. (ii) If $D$ is a $C^{1, \rm Dini}$ open set, then, as $x\to \partial D$, the rate at which $p^D(t,x,y)$ tends to 0 is ${\rm dist} (x, D^c)^{α/2}$. (iii) For any non-Dini modulus of continuity $\ell$, there exist non-$C^{1, \rm Dini}$ 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 $D$. |
| title | Abnormal boundary decay for stable operators |
| topic | Analysis of PDEs Probability 60J45, 60J50, 35K08, 47G20 |
| url | https://arxiv.org/abs/2510.03961 |