Estimates and structure of α-harmonic functions

By Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwaśnicki. 2006

118 citations recorded by the source corpus.

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on Dc∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0c6eeaed674d1a1bb05ef5c29378f2c00b3a8db2

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “The approach to BHI via approximate factorization was applied to isotropic stable processes in [26], to stable-like subordinate diffusion on the Sierpiński gasket in [55], and to a wide class of isotropic Lévy processes in [65].”
Boundary Harnack principle for subordinate Brownian motions (2007, background): “In a recent paper [6], Bogdan, Kulczycki and Kwasnicki proved a version of the boundary Harnack inequality for nonnegative harmonic functions of rotationally invariant stable processes in arbitrary open sets.”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “The first one [2] is the boundary Harnack principle for nonnegative s-harmonic functions, which reads as follows: assume that u and v are two nonnegative functions in a Lipschitz domain Ω, which satisfy (−∆)su ≡ 0 and (−∆)sv ≡ 0 in Ω ∩ Br(x0) for some ball Br(x0) centered at x0 ∈ ∂Ω. Assume also…”
Ten equivalent definitions of the fractional laplace operator (2015, background): “Various results are also known for the restriction of L to a domain: detailed properties of L in a domain with Dirichlet condition can be found in [8, 9, 10, 11, 33, 38]; see also [6] for the study of L in a domain with certain reflection.”
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators (2007, methodology): “We refer the reader to [ 7 ] for a general perspective, more references and a definitive version of the boundary Harnack principle for � α/2 . We would also like to mention a forthcoming paper [25] on gradient perturbations of � α/2 in terms of the Green functions.”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “We remark that the existence of ratios and Martin representation of nonnegative harmonic functions of L may likely be obtained with the results and toolbox presented in this paper and [13].”