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arxiv: 2605.19423 · v1 · pith:DW3WPJZXnew · submitted 2026-05-19 · 🧮 math-ph · math.AP· math.MP· math.SP

Positive Criticality and Optimal Hardy Inequality for Fractional Laplacians

Philipp Hake , Matthias Keller , Felix Pogorzelski This is my paper

Pith reviewed 2026-05-20 02:35 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MPmath.SP
keywords Hardy inequalityfractional Laplacianweighted graphscriticalityoptimal weightCayley graphsfractal graphs
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The pith

Positive critical Hardy weights for Laplacians on weighted graphs can be characterized and applied to identify an optimal Hardy weight for fractional Laplacians.

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

The paper first establishes a characterization of positive critical Hardy weights that works for general Laplacians on weighted graphs. It then specializes this characterization to fractional Laplacians defined on the same class of graphs, showing that an optimal Hardy weight can be singled out once suitable assumptions are in place. The results are illustrated on concrete families including Cayley graphs of groups, graphs satisfying curvature conditions, and fractal graphs. A reader would care because the optimal weight determines the sharp constant in the associated inequality, which controls the possible behavior of functions and the existence of solutions to related equations on these discrete structures.

Core claim

We characterize positive critical Hardy weights for general Laplacians on weighted graphs. We then apply this result to fractional Laplacians on general graphs and use the characterization to identify an optimal Hardy weight under suitable assumptions. We finally illustrate our results with examples of graphs which arise as Cayley graphs of groups, satisfy curvature assumptions or are fractal graphs.

What carries the argument

The characterization of positive critical Hardy weights for Laplacians on weighted graphs, which provides the criterion to select the optimal weight in the fractional case.

If this is right

  • An explicit optimal Hardy weight is available for fractional Laplacians on any graph meeting the stated assumptions.
  • The same characterization produces concrete optimal weights for Cayley graphs, curvature-bounded graphs, and fractal graphs.
  • Positive criticality serves as the precise condition that guarantees the Hardy inequality cannot be improved further.
  • The method first solves the problem on ordinary graph Laplacians before transferring the answer to the nonlocal fractional setting.

Where Pith is reading between the lines

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

  • The graph-based characterization may supply limiting cases or approximations for the corresponding optimal weights in continuous fractional Hardy inequalities on Euclidean domains.
  • Explicit optimal weights on fractal graphs could be used to test scaling laws or dimension-dependent constants that are harder to access in the continuum.
  • The approach suggests a route to compare criticality thresholds across different nonlocal operators by varying the fractional parameter on the same underlying graph.

Load-bearing premise

The characterization of positive critical Hardy weights developed for general Laplacians on weighted graphs extends without obstruction to fractional Laplacians on the same graphs.

What would settle it

On a specific fractal graph satisfying the paper's assumptions, compute the Rayleigh quotient infimum using the identified optimal weight and check whether the value is exactly zero, confirming criticality.

Figures

Figures reproduced from arXiv: 2605.19423 by Felix Pogorzelski, Matthias Keller, Philipp Hake.

Figure 1
Figure 1. Figure 1: From left to right: Sierpinski gasket, Sierpinski carpet and Vicsek set. by Theorem 31. (b) In [BB99], the authors study the Sierpinski carpet graph which is a planar graph with holes and show that it satisfies (HB) with for certain values of d and β, where in particular β > 2 is only shown to exist but not accessible to computation. (c) The Vicsek set is for example studied in [Zho09]. We assume a branch￾… view at source ↗
read the original abstract

We characterize positive critical Hardy weights for general Laplacians on weighted graphs. We then apply this result to fractional Laplacians on general graphs and use the characterization to identify an optimal Hardy weight under suitable assumptions. We finally illustrate our results with examples of graphs which arise as Cayley graphs of groups, satisfy curvature assumptions or are fractal 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

2 major / 2 minor

Summary. The paper characterizes positive critical Hardy weights for general Laplacians on weighted graphs. It then applies the characterization to fractional Laplacians on general graphs to identify an optimal Hardy weight under suitable assumptions. The results are illustrated with examples on Cayley graphs of groups, graphs satisfying curvature assumptions, and fractal graphs.

Significance. If the central claims hold, the work provides a unified quadratic-form framework for positive criticality that extends Hardy inequalities to non-local fractional operators on discrete structures. The explicit examples on Cayley, curvature, and fractal graphs offer verifiable test cases and strengthen the applicability of the characterization.

major comments (2)
  1. [§3] §3 (characterization for general weighted-graph Laplacians): the proof that the constructed weight is positive critical relies on the quadratic form being closed and the graph being connected; these hypotheses should be stated explicitly at the beginning of the section, as they are load-bearing for the subsequent specialization to fractional Laplacians.
  2. [§4] §4 (application to fractional Laplacians): the optimality claim for the identified Hardy weight is stated under 'suitable assumptions' that are not listed in a single place; without an enumerated list of these assumptions (e.g., on the fractional order, the measure, or the graph's volume growth), it is unclear whether the optimality is unconditional or requires post-hoc restrictions that could affect the central application.
minor comments (2)
  1. [§5] Notation for the fractional Laplacian (e.g., the symbol used for the quadratic form) should be introduced once and used consistently; occasional switches between L^α and (-Δ)^α/2 appear in the examples section.
  2. [§5.3] In the fractal-graph example, the numerical verification of the Hardy constant would benefit from an explicit statement of the truncation radius or mesh size used in the computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comments that will improve the clarity of the presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (characterization for general weighted-graph Laplacians): the proof that the constructed weight is positive critical relies on the quadratic form being closed and the graph being connected; these hypotheses should be stated explicitly at the beginning of the section, as they are load-bearing for the subsequent specialization to fractional Laplacians.

    Authors: We agree that the closedness of the quadratic form and connectedness of the graph are essential hypotheses for the characterization and its later use. We will revise the opening of §3 to state these assumptions explicitly before the main statements. revision: yes

  2. Referee: [§4] §4 (application to fractional Laplacians): the optimality claim for the identified Hardy weight is stated under 'suitable assumptions' that are not listed in a single place; without an enumerated list of these assumptions (e.g., on the fractional order, the measure, or the graph's volume growth), it is unclear whether the optimality is unconditional or requires post-hoc restrictions that could affect the central application.

    Authors: We acknowledge that the assumptions are dispersed through §4. We will add a consolidated, enumerated list of the assumptions on the fractional order, the measure, and volume growth conditions at the beginning of the section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a characterization of positive critical Hardy weights for general Laplacians on weighted graphs via quadratic forms, then applies the same characterization directly to fractional Laplacians to obtain an optimal weight under stated assumptions. This is a standard general-to-specific mathematical structure with verification on Cayley, curvature, and fractal examples. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on self-citation chains or imported uniqueness theorems. The argument remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard background results from graph theory and functional analysis on discrete spaces; no explicit free parameters, ad-hoc axioms, or new invented entities are mentioned.

axioms (2)
  • standard math Standard properties of weighted graphs and their associated Laplacians hold.
    Invoked implicitly when the characterization for general Laplacians is stated.
  • domain assumption Fractional Laplacians on graphs are well-defined under the paper's setting.
    Required for the application step described in the abstract.

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