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- W2953338897 abstract "Suppose $dge 2$ and $0<beta<alpha<2$. We consider the non-local operator $mathcal{L}^{b}=Delta^{alpha/2}+mathcal{S}^{b}$, where $$mathcal{S}^{b}f(x):=lim_{varepsilonto 0}mathcal{A}(d,-beta)int_{|z|>varepsilon}left(f(x+z)-f(x)right)frac{b(x,z)}{|z|^{d+beta}},dy.$$ Here $b(x,z)$ is a bounded measurable function on $mathbb{R}^{d}timesmathbb{R}^{d}$ that is symmetric in $z$, and $mathcal{A}(d,-beta)$ is a normalizing constant so that when $b(x, z)equiv 1$, $mathcal{S}^{b}$ becomes the fractional Laplacian $Delta^{beta/2}:=-(-Delta)^{beta/2}$. In other words, $$mathcal{L}^{b}f(x):=lim_{varepsilonto 0}mathcal{A}(d,-beta)int_{|z|>varepsilon}left(f(x+z)-f(x)right) j^b(x, z),dz,$$ where $j^b(x, z):= mathcal{A}(d,-alpha) |z|^{-(d+alpha)}+ mathcal{A}(d,-beta) b(x, z)|z|^{-(d+beta)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)geq 0$ on $mathbb{R}^dtimes mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $mathcal{L}^b$) in any $kappa$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets." @default.
- W2953338897 created "2019-06-27" @default.
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- W2953338897 date "2015-01-08" @default.
- W2953338897 modified "2023-09-26" @default.
- W2953338897 title "Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators" @default.
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- W2953338897 doi "https://doi.org/10.48550/arxiv.1501.02023" @default.
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