Harnack inequalities for non-local operators of variable order

By Richard F. Bass, M. Kassmann. 2004

156 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6f30d9911fa8f16b4aeb34678ec76174fefafeba

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian (2005, background): “The class of considered operators gradually extended, see [47], [45], [4], [18], [5], and the references given there.”
Regularity theory for fully nonlinear integro‐differential equations (2007, result): “Although there are some known results about Harnack inequalities and Hölder estimates for integro-differential equations with either analytical proofs [10] or probabilistic proofs [3, 4, 5, 12], the estimates in all these previous results blow up as the order of the equation approaches 2.”
Potential Theory of Subordinate Brownian Motion (2009, background): “A version of a weak Harnack inequality appeared in [3], and our proof follows the arguments there.”
Non-local dirichlet forms and symmetric jump processes (2006, methodology): “An example in [BK05a] shows that the large jumps, although in many ways less interesting, can cause the Harnack inequality to fail.”
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].”
Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order (2005, methodology): “However, as assumption (d) in Example 2 shows, some continuity in x is required. In [ BK04 ], no continuity in x appeared.”
Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part (2009, background): “Let Y λ be the process associated with the following Dirichlet form: EY λ(f, f) = 1 2 ∫ Rd d ∑ i,j=1 aij(x) ∂f(x) ∂xi ∂f(x) ∂xj dx + ∫∫ |x−y|≤λ (f(y) − f(x))2J(x, y)dxdy, (3.3) so that Y λ has jumps of size less than λ only.”
Nonlocal Diffusion and Applications (2015, methodology): “We refer to [7, 86, 64] and to the references therein for a detailed introduction to the fractional Harnack Inequality, and to [38] for general estimates of this type.”
Regularity theory for parabolic nonlinear integral operators (2011, unknown): “Along these lines, see Bass and Kassmann [3, 2] (see also [4]).”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “nd Bernstein-type regularity results (see [14]). Though conceptually diﬀerent, the problems related to mixed order equations are closely related in spirit to the ones of variable order equations (see [34, 7, 8, 41]), which have themselves concrete signiﬁcance in applied sciences (see e.g. [42, 31]).”
Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions (2006, background): “In a recent paper [2], Bass and Kassmann proved the Harnack inequality for a class of processes corresponding to non-local operators of variable order.”