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Boundary Harnack Principle and Gradient Estimates for Fractional Laplacian Perturbed by Non-local Operators
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Suppose d ≥ 2 and 0 < β < α < 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)≡1, \(\mathcal {S}^{b}\) becomes the fractional Laplacian Δβ/2:=−(−Δ)β/2. In other words,$$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta){\int}_{|z|>\varepsilon}\left( f(x+z)-f(x)\right) j^{b}(x, z)\,dz, $$ 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 [11] that, when jb(x, z)≥0 on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\), there is a conservative Feller process Xb 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 Xb (or equivalently, of \(\mathcal {L}^{b}\)) in any κ-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of Xb in open sets.KeywordsHarmonic functionBoundary Harnack principleGradient estimateNon-local operatorGreen functionPoisson kernelMathematics Subject Classification (2010)Primary 60J45Secondary 31B0531B25References1.Blumenthal, R.A., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554 (1961)MathSciNetMATHGoogle Scholar2.Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. 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Ph.D thesis, Peking University (2012)Copyright information© Springer Science+Business Media Dordrecht 2016Authors and AffiliationsZhen-Qing Chen1Yan-Xia Ren2Ting Yang34Email author1.Department of MathematicsUniversity of WashingtonSeattleUSA2.LMAM School of Mathematical Sciences and Center for Statistical SciencePeking UniversityBeijingChina3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina4.Beijing Key Laboratory on MCAACIBeijingChina About this article CrossMark Print ISSN 0926-2601 Online ISSN 1572-929X Publisher Name Springer Netherlands About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s11118-016-9554-1_Boundary Harnack Principle and Gra", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s11118-016-9554-1_Boundary Harnack Principle and Gra", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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