A priori Hölder and Lipschitz regularity for generalizedp-harmonious functions in metric measure spaces

Abstract

Let (X, d, µ) be a proper metric measure space and let Ω ⊂ X be a bounded domain. For each x ∈ Ω, we choose a radius 0 < ̺(x) ≤ dist(x, ∂Ω) and let Bx be the closed ball centered at x with radius ̺(x). If α ∈ R, consider the following operator in C(Ω),Under appropriate assumptions on α, X, µ and the radius function ̺ we show that solutions u ∈ C(Ω) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.

$p$-harmonic functions by way of intrinsic mean value properties

“…We want to emphasize that, while the constant radius hypothesis is convenient for game-theoretic applications, the variable radius setting is natural for at least two reasons: it is closely related to the classical theory (p = 2) and it is intrinsic, in the sense that no extension of the domain is needed (this explains the term intrinsic in the title of the paper). In [AL16-1] and [AL18] we proved a number of results about existence, uniqueness and regularity of solutions of (1.4) with variable radius under different hypothesis on ρ and Ω. In this paper we substantially improve the existence results of [AL16-1] and provide also an approximation result.…”

“…Equicontinuity results. At this point we refer to [AL18,Theorem 4.5] for the equicontinuity of the sequence {T k ρ } k at interior points of Ω, where u ∈ K f (see also [AL16-1, Proposition 2.6]).…”

“…A significative difference between Lebesgue's setting and ours is that we actually use the convergence of iterates to prove existence of solution to the Dirichlet problem while in Lebesgue's note, existence is taken for granted and convergence is obtained as a consequence. Interior equicontinuity is more straightforward and was obtained, in different contexts, in [AL16-1] and [AL18]. However, boundary equicontinuity turns out to be a much more delicate matter.…”

Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure

“…Moreover, the Harnack inequality and the strong maximum principle hold for strongly harmonic functions as well as the local Hölder continuity and even local Lipschitz continuity under more involved assumptions, see [1]. It is important to mention here that similar type of problems were studied for a more general, nonlinear mean value property by Manfredi-Parvainen-Rossi and Arroyo-Llorente, see [3,4,19,20].…”

On a class of singular measures satisfying a strong annular decay condition

“…We say that (X, d, µ) satisfies a δ-annular decay condition (δ-ADC) if there exists a constant C > 0 such that The δ-ADC was apparently introduced by Colding-Minicozzi in manifolds ( [9]) and, independently, by Buckley ([7]) in metric spaces. In the last years the δ-ADC has been successfully used in several problems of Harmonic Analysis and Geometric Function Theory: when studying reverse Hölder inequalities and characterizations of A ∞ in metric spaces (see [13,14,15]), Hardy inequalities and T (b) theorems ( [4]), capacity estimates ( [6]) and also in connection to the regularity of functions satisfying certain mean value properties (see [1,3]). Remark 1.1.…”