{"paper":{"title":"Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ting Yang, Yan-Xia Ren, Zhen-Qing Chen","submitted_at":"2015-01-09T03:05:32Z","abstract_excerpt":"Suppose $d\\ge 2$ and $0<\\beta<\\alpha<2$. We consider the non-local operator $\\mathcal{L}^{b}=\\Delta^{\\alpha/2}+\\mathcal{S}^{b}$, where $$\\mathcal{S}^{b}f(x):=\\lim_{\\varepsilon\\to 0}\\mathcal{A}(d,-\\beta)\\int_{|z|>\\varepsilon}\\left(f(x+z)-f(x)\\right)\\frac{b(x,z)}{|z|^{d+\\beta}}\\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\\mathbb{R}^{d}\\times\\mathbb{R}^{d}$ 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, $$\\mathc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02023","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}