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Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators | | Zhen-Qing Chen
; Yan-Xia Ren
; Ting Yang
; | | Date: |
9 Jan 2015 | | Abstract: | Suppose $dge 2$ and $0<eta<alpha<2$. We consider the non-local operator
$mathcal{L}^{b}=Delta^{alpha/2}+mathcal{S}^{b}$, where
$$mathcal{S}^{b}f(x):=lim_{varepsilon o
0}mathcal{A}(d,-eta)int_{|z|>varepsilon}left(f(x+z)-f(x)
ight)frac{b(x,z)}{|z|^{d+eta}},dy.$$
Here $b(x,z)$ is a bounded measurable function on
$mathbb{R}^{d} imesmathbb{R}^{d}$ that is symmetric in $z$, and
$mathcal{A}(d,-eta)$ is a normalizing constant so that when $b(x, z)equiv
1$, $mathcal{S}^{b}$ becomes the fractional Laplacian
$Delta^{eta/2}:=-(-Delta)^{eta/2}$. In other words,
$$mathcal{L}^{b}f(x):=lim_{varepsilon o
0}mathcal{A}(d,-eta)int_{|z|>varepsilon}left(f(x+z)-f(x)
ight) j^b(x,
z),dz,$$ where $j^b(x, z):= mathcal{A}(d,-alpha) |z|^{-(d+alpha)}+
mathcal{A}(d,-eta) b(x, z)|z|^{-(d+eta)}$. It is recently established in
Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)geq 0$ on
$mathbb{R}^d imes mathbb{R}^d$, 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. | | Source: | arXiv, 1501.2023 | | Services: | Forum | Review | PDF | Favorites | |
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