Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian

By Krzysztof Bogdan, Paweł Sztonyk. 2005

76 citations recorded by the source corpus.

We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack’s inequality. We also estimate the corresponding semigroup and the potential kernel.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02083d453669ef576f22601ac636065f2bbbfb40

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Density and tails of unimodal convolution semigroups (2013, background): “Estimates and references to anisotropic ν with prescribed radial decay and rough spherical marginals may be found in [28] (see also [7, 31] for more details in the case of homogeneous anisotropic ν).”
Estimates of transition densities and their derivatives for jump Lévy processes (2013, background): “More recent asymptotic results for stable Lévy processes are given in papers [30] and [4].”
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators (2007, methodology): “In particular, in the forthcoming paper [5] we give the estimates for the transition density of time-dependent Schrodinger perturbations of � α/2 . In passing, we refer the interested reader to [ 9 , 41] for information on the various asymptotics of the unperturbed semigroup generated by more genera”
The best constant in a fractional Hardy inequality (2008, background): “In particular, we expect that the results of [16], [17] may be used to obtain, for the anisotropic stable [9], [10] censored processes, the ruin probabilities generalizing [7, Theorem 5.”
Regularity theory for general stable operators (2014, background): “1), or related ones, has been widely investigated; see the works by Bass, Kassmann, Schwab, Silvestre, Sztonyk, and Bogdan, among others [1, 23, 22, 3, 39, 21, 2, 37, 5, 6, 7, 9, 24].”
Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, background): “Regarding other recent estimates [3, 17, 20, 23, 36] for the transition density and potential kernel of jumptype processes, we need to point out that generally these only concern processes without killing.”
Boundary regularity for fully nonlinear integro-differential equations (2014, background): “This is important when considering non translation invariant equations for which a comparison principle for viscosity solutions is not available.”
Harnack Inequalities: An Introduction (2007, background): “In Section 2, Harnack inequalities for nonlocal operators are mentioned only briefly although they attract much attention at present; see [121, 122].”
Transition density estimates for jump Lévy processes (2010, background): “More recent asymptotic results for stable Lévy processes are given in papers [36] and [4].”
Barriers, exit time and survival probability for unimodal Lévy processes (2013, methodology): “We test super- and subharmonicity by means of Dynkin’s generator of X in a way suggested by [13].”
Estimates and structure of α-harmonic functions (2006, unknown): “We also refer the reader to [ 16 ] to see the limitations of (53) in defining harmonic functions for other operators as opposed to (51).”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “3; see also [28] and the references therein for related estimates in the symmetric (but anisotropic) case.”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “(12) In view of (10) and the fact that each pt is a radial function, X is called the isotropic α-stable Lévy process (see [42], [15] for a discussion of general stable Lévy processes).”