Intrinsic Hölder Continuity of Harmonic Functions

Abstract

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

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

“…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.…”

Scaling invariant Harnack inequalities in a general setting

“…In Section 4, we discuss properties of an associated "Green function" which allow us to prove a Krylov-Safonov estimate for the corresponding capacity. This leads to Theorem 4.10 and, using recent results on Hölder continuity from [12], to Theorem 4.12 on (Hölder) continuity of harmonic functions.…”

“…In fact, assuming that the constant function 1 is harmonic on X, [12, Corollary 3.2] implies even the Hölder continuity of all functions in H + b (U), U ∈ U(X 0 ). To see this we only have to verify property (J 0 ) in [12], that is, the following. Proof.…”

Schauder Estimates for Equations Associated with Lévy Generators

“…Let us mention that the questions, which we discuss in this paper, are also related to the regularity of harmonic functions: If g = 0 and = 0 in (1), i. e. Af = 0, then f is harmonic for A, and there is an extensive literature on the regularity of functions which are harmonic for a Lévy generator, cf. [13,16,26,38] and the references therein. The regularity of solutions to elliptic integro-differential equations Af = g has been studied, more generally, for classes of Lévy-type operators, see e. g. [1,3,11,17,22], and for non-linear integro-differential operators, greatly influenced by the works of Barles et.…”

“…Moreover, there are numerous results on the regularity of functions which are harmonic with respect to a Lévy generator, see e.g. [7,8,16,18,30]. Let us mention that the regularity of solutions integrodifferential equations Af = g has been studied, more generally, for classes of Lévy-type operators, see e.g.…”