Scaling invariant Harnack inequalities in a general setting

Hansen, Wolfhard; Netuka, Ivan

doi:10.1016/j.jmaa.2016.06.069

Cited by 3 publications

(1 citation statement)

References 23 publications

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“…To address this problem, some versions of elliptic Harnack inequalities that imply EHR are considered in some literatures such as [CKP1,CKP2,K1], in connection with the Moser's iteration method. We note that there are now many related work on EHI and EHR for harmonic functions of non-local operators; in addition to the papers mentioned above; see, for instance, [BS,CK1,CK2,CS,DK,GHH,Ha,HN,K2,Sil] and references therein. This is only a partial list of the vast literature on the subject.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

Chen

Kumagai

Wang

2019

Journal de Mathématiques Pures et Appliquées

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We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

Chen

Kumagai

Wang

2019

Journal de Mathématiques Pures et Appliquées

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Intrinsic Hölder Continuity of Harmonic Functions

Hansen

2017

Potential Anal

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In a setting, where only "exit measures" are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant Hölder continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling property ensures that Harnack inequalities imply Hölder continuity. Our approach covers recent results by M. Kassmann and A. Mimica as well as cases, where a Green function leads to an intrinsic metric.Keywords: Harmonic function; Hölder continuity; right process; balayage space; Lévy process.MSC: 31D05, 60J25, 60J45, 60J75. Harmonic functions in a general settingDuring the last years, Hölder continuity of bounded harmonic functions has been studied for various classes of Markov processes (see [14,8] and the references therein). The aim of this paper is to offer not only a unified (and perhaps more transparent) approach to results obtained until now, but also the possibility for applications in new cases. 1Let X be a topological space such that finite measures µ on its σ-algebra B(X) of Borel subsets satisfyThis holds if X is a separable metric space (on its completion every finite measure is tight). Let M(X) denote the set of all finite measures on (X, B(X)) (which we also consider as measures on the σ-algebra B * (X) of all universally measurable sets). Given a set F of numerical functions on X, let F b , F + be the set of all functions in F which are bounded, positive respectively.For great flexibility in applications, we consider an open neighborhood X 0 of a point x 0 ∈ X and suppose that we have a continuous real function ρ 0 ≥ 0 on X 0 with ρ 0 (x 0 ) = 0 and 0 < R 0 ≤ ∞ such that, for every 0 < r < R 0 , the closure of U r := {x ∈ X : ρ 0 (x) < r} is contained in X 0 . Let U 0 denote the set of all open sets V in X with V ⊂ U r for some 0 < r < R 0 .1 The final publication is available at Springer via http://dx

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