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arxiv: 2603.28358 · v2 · submitted 2026-03-30 · 🧮 math.AP

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A Wiener criterion at infinity for p-massiveness on weighted graphs

Lu Hao
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:40 UTC · model grok-4.3

classification 🧮 math.AP
keywords Wiener criterionp-massivenessweighted graphsgraph p-Laplacianvolume doublingPoincaré inequalityrelative capacityexterior Dirichlet problem
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The pith

A capacitary condition at infinity characterizes p-massive sets on weighted graphs.

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

The paper establishes a nonlinear Wiener criterion for p-massiveness at the point at infinity on infinite connected weighted graphs. Assuming volume doubling and a weak (1,p)-Poincaré inequality, every infinite connected p-massive set satisfies a dyadic condition on relative p-capacities between nested balls; the converse holds under an additional (p0) condition. Without those assumptions, p-massiveness is equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems of the graph p-Laplacian. This yields a characterization of bounded nonconstant p-harmonic functions via the existence of two disjoint massive sets. A reader would care because the criterion supplies an explicit geometric test for when boundary-value problems at infinity admit multiple solutions, extending classical potential theory to a discrete nonlinear setting.

Core claim

The central claim is that p-massiveness of an infinite connected set is equivalent to a dyadic capacitary condition at infinity, expressed through relative p-capacities in nested balls. Under volume doubling and a weak (1,p)-Poincaré inequality, every such massive set satisfies the condition; when the graph also satisfies the (p0) condition, the converse holds. Without the geometric assumptions, p-massiveness remains equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems. Consequently, bounded nonconstant p-harmonic functions exist if and only if there are two disjoint p-massive sets.

What carries the argument

The dyadic capacitary condition on relative p-capacities of annuli formed by nested balls; this serves as the test for whether a set is thick enough at infinity to produce non-trivial p-harmonic behavior.

If this is right

  • Every infinite connected p-massive set satisfies the dyadic capacitary condition on relative p-capacities under volume doubling and weak (1,p)-Poincaré inequality.
  • Under the additional (p0) condition the capacitary condition implies p-massiveness.
  • p-massiveness is equivalent to strengthened nonuniqueness for exterior Dirichlet problems even without volume doubling or Poincaré assumptions.
  • Bounded nonconstant p-harmonic functions exist precisely when two disjoint p-massive sets exist.
  • The Wiener criterion fits inside a broader equivalence between exterior boundary behavior and Liouville-type phenomena for the graph p-Laplacian.

Where Pith is reading between the lines

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

  • The equivalence to nonuniqueness suggests that finite-graph approximations could numerically certify massiveness by checking whether exterior Dirichlet problems admit multiple bounded solutions.
  • The capacitary test may extend to other nonlinear discrete operators, such as discrete mean-curvature or infinity-Laplacian equations on the same graphs.
  • On graphs where the (p0) condition fails, the one-way implication from massiveness to the capacitary condition could still be used to rule out Liouville-type theorems by exhibiting thick sets at infinity.

Load-bearing premise

The weighted graph satisfies volume doubling and a weak (1,p)-Poincaré inequality.

What would settle it

A concrete weighted graph obeying volume doubling and the weak (1,p)-Poincaré inequality, together with an infinite connected set that is p-massive yet fails the dyadic relative p-capacity condition (or satisfies the condition but is not massive when (p0) also holds).

Figures

Figures reproduced from arXiv: 2603.28358 by Lu Hao.

Figure 1
Figure 1. Figure 1: Region An and Bn. Theorem 1.7. Let (V, µ) be an infinite, connected, and locally finite graph. Then the following statements are equivalent: (1) Ω ⊂ V is a p-massive set. (2) For some (equivalently, every) f ∈ B(∂Ω), the Dirichlet problem (1.1) admits two bounded solutions u and v such that supΩ u ̸= supΩ v. The next theorem characterizes the failure of the p-Liouville property through disjoint massive set… view at source ↗
read the original abstract

We study boundary value problems at infinity for the graph $p$-Laplacian on infinite, connected, locally finite weighted graphs. Our main result is a Wiener criterion for $p$-massiveness. Assuming volume doubling and a weak $(1,p)$-Poincar\'e inequality, we show that every infinite connected $p$-massive set satisfies a dyadic capacitary condition expressed through relative $p$-capacities in nested balls; under the additional $(p_0)$ condition, the converse also holds. This yields a nonlinear criterion at the point at infinity in a rough weighted-graph setting and extends the Wiener viewpoint to a nonlinear discrete framework. We also prove, without these geometric assumptions, that $p$-massiveness is equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems. As a further consequence, bounded nonconstant $p$-harmonic functions are characterized by the existence of two disjoint massive sets. In this way, the Wiener criterion is placed in a broader and more flexible picture of exterior boundary behavior and Liouville-type phenomena on weighted graphs.

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 / 3 minor

Summary. The manuscript establishes a Wiener-type criterion for p-massiveness at infinity on infinite connected locally finite weighted graphs. Under volume doubling and a weak (1,p)-Poincaré inequality, every infinite connected p-massive set satisfies a dyadic capacitary condition expressed through relative p-capacities in nested balls; the converse holds under the additional (p0) condition. Without these geometric assumptions, p-massiveness is shown equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems. As a consequence, bounded nonconstant p-harmonic functions are characterized by the existence of two disjoint massive sets.

Significance. If the results hold, the work supplies a nonlinear discrete analogue of the classical Wiener criterion in a rough weighted-graph setting, cleanly separating the geometric hypotheses needed for the capacitary characterization from the equivalence to nonuniqueness for exterior problems. This yields a flexible framework for exterior boundary behavior and Liouville-type phenomena on graphs. The explicit separation of assumptions is a methodological strength that broadens applicability beyond the volume-doubling/Poincaré regime.

minor comments (3)
  1. Abstract: the phrase 'dyadic capacitary condition' is introduced without a one-sentence gloss; a brief parenthetical description of the nested-ball relative-capacity test would improve immediate readability for readers outside the immediate subfield.
  2. Introduction (likely §1): the (p0) condition is referenced as 'additional' but its precise statement (e.g., a lower bound on the measure of balls or a comparison for p-capacity) is not recalled; a short reminder of its form would help readers track when the converse direction applies.
  3. Notation: the distinction between 'p-massive' and 'infinite connected p-massive set' is used repeatedly; a single clarifying sentence early in the paper would eliminate any ambiguity about whether connectedness is part of the definition or an extra hypothesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and insightful summary of our manuscript on the Wiener criterion for p-massiveness at infinity on weighted graphs. We appreciate the recommendation for minor revision and the recognition of the separation between geometric assumptions and the equivalence to nonuniqueness for exterior problems. Since the report contains no specific major comments, we have no point-by-point responses to provide and will proceed with minor revisions as suggested.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central Wiener criterion is derived from the stated geometric hypotheses (volume doubling + weak (1,p)-Poincaré inequality) via capacity estimates and Harnack controls that are standard under those assumptions; the converse uses the additional (p0) condition but remains a direct implication rather than a redefinition. The separate equivalence between p-massiveness and non-uniqueness for exterior Dirichlet problems is proved without the geometric assumptions, confirming independent content. No load-bearing self-citation, fitted parameter renamed as prediction, or ansatz smuggled via prior work appears in the abstract or described theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on two standard geometric assumptions for metric measure spaces that are not derived in the paper: volume doubling controls ball volumes, and the weak Poincaré inequality controls function oscillations by gradients. No free parameters or new entities are introduced.

axioms (3)
  • domain assumption Volume doubling property on the weighted graph
    Controls the growth of measure of balls and is invoked to obtain the dyadic capacitary estimates.
  • domain assumption Weak (1,p)-Poincaré inequality
    Relates the p-energy of a function to its oscillation and is required for the capacitary condition to characterize massiveness.
  • domain assumption Additional (p0) condition
    Needed only for the converse implication in the Wiener criterion.

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Reference graph

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