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arxiv: 2402.03571 · v2 · submitted 2024-02-05 · 🧮 math.PR · math.AP

Boundary Harnack principle on uniform domains

Aobo Chen This is my paper

Pith reviewed 2026-05-24 04:06 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords boundary Harnack principleuniform domainselliptic Harnack inequalityGreen functionsharmonic functionsmetric measure spacespotential theory
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The pith

Scale-invariant boundary Harnack principle holds for uniform domains in spaces satisfying scale-invariant elliptic Harnack inequality, without geodesic assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a proof of the scale-invariant boundary Harnack principle that applies to uniform domains whenever the ambient space obeys a scale-invariant elliptic Harnack inequality. The argument removes the usual geodesic hypothesis on the space and obtains Green-function existence from a cited external result instead of assuming it at the outset. A reader would care because the boundary Harnack principle supplies uniform control on the ratios of positive harmonic functions that vanish on the same portion of the boundary; extending it to non-geodesic settings widens the class of metric measure spaces in which such control is known to hold.

Core claim

We present a proof of scale-invariant boundary Harnack principle for uniform domains when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach does not assume the underlying space to be geodesic. Additionally, the existence of Green functions is also not assumed beforehand and is ensured by a recent result from M. T. Barlow, Z.-Q. Chen and M. Murugan.

What carries the argument

The scale-invariant boundary Harnack principle, which gives uniform comparability of positive harmonic functions vanishing on a common boundary set inside a uniform domain.

If this is right

  • Positive harmonic functions vanishing on the same boundary set remain comparable at interior points whose distances to the boundary are comparable.
  • The result applies directly to non-geodesic metric measure spaces that satisfy the elliptic Harnack inequality.
  • Green functions are available in these spaces via the cited existence theorem.
  • Boundary regularity and Harnack-type estimates extend to the same class of domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The argument may apply to certain random-walk models on graphs or fractals that fail to be geodesic.
  • It opens the possibility of deriving similar boundary principles from Dirichlet-form assumptions alone.
  • Concrete verification could begin with explicit non-geodesic uniform domains constructed from the cited Green-function result.

Load-bearing premise

The underlying space satisfies a scale-invariant elliptic Harnack inequality.

What would settle it

Exhibit a uniform domain inside a space obeying scale-invariant elliptic Harnack inequality together with a pair of positive harmonic functions vanishing on the same boundary portion whose ratio is unbounded near the boundary.

Figures

Figures reproduced from arXiv: 2402.03571 by Aobo Chen.

Figure 4.1
Figure 4.1. Figure 4.1: Five specified points x ⋆ ξ , y⋆ ξ , z⋆ ξ , ξr, ξ′ r . Define A4 = A3 + 2AA3 + (1 + 16K)A −1 . The following lemma, which compares Green functions on some specified points is useful to give estimates on a region. Note that [BM19, Lemma 5.10 and Lemma 5.11] also use such estimates without proof, since in the context of [BM19], such estimates naturally hold as the space is assumed to be geodesic, allowing … view at source ↗
read the original abstract

We present a proof of scale-invariant boundary Harnack principle for uniform domains when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach does not assume the underlying space to be geodesic. Additionally, the existence of Green functions is also not assumed beforehand and is ensured by a recent result from M. T. Barlow, Z.-Q. Chen and M. Murugan.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes the scale-invariant boundary Harnack principle for uniform domains in metric measure spaces satisfying a scale-invariant elliptic Harnack inequality. The proof constructs suitable barriers, applies the given Harnack inequality to control ratios of positive harmonic functions vanishing on the boundary, works directly with the uniform-domain chain condition to avoid any geodesic assumption, and invokes the Barlow-Chen-Murugan theorem solely to guarantee the existence of Green functions.

Significance. If the result holds, the removal of the geodesic assumption is a meaningful advance for potential theory on non-geodesic spaces such as certain fractals or irregular metric spaces. The direct use of the elliptic Harnack inequality together with the uniform chain condition supplies a clean, self-contained argument once the external Green-function result is granted.

minor comments (2)
  1. [§1] §1, line 3: the phrase 'scale-invariant elliptic Harnack inequality' is used before its precise statement; a forward reference to the exact formulation in §2 would improve readability.
  2. [Introduction] The citation to Barlow-Chen-Murugan is given only in the abstract and introduction; adding the full bibliographic details at first use in the main text would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive report. We are pleased that the removal of the geodesic assumption is viewed as a meaningful advance and that the argument is considered clean and self-contained once the Barlow-Chen-Murugan result is invoked.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation assumes a scale-invariant elliptic Harnack inequality as input and invokes an external result (Barlow-Chen-Murugan) solely for Green-function existence; the boundary Harnack principle is then obtained by constructing barriers and applying the given Harnack inequality together with the uniform-domain chain condition. No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a load-bearing uniqueness theorem, and the non-geodesic setting is handled directly from the stated assumptions without circular redefinition. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the scale-invariant elliptic Harnack inequality as a domain assumption and on the external theorem for Green functions; no free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption The underlying space satisfies a scale-invariant elliptic Harnack inequality
    Stated as the hypothesis under which the boundary Harnack principle holds (abstract).
  • domain assumption Green functions exist by the result of Barlow, Chen and Murugan
    Invoked to avoid assuming Green functions a priori (abstract).

pith-pipeline@v0.9.0 · 5568 in / 1236 out tokens · 26570 ms · 2026-05-24T04:06:38.689375+00:00 · methodology

discussion (0)

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Reference graph

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