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An optimal fractional Hardy inequality on the discrete half-line

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Pith's one-line read For every σ ∈ (0,1], the paper identifies an explicit weight W_σ^op such that (−∆_N)^σ − W_σ^op is nonnegative, critical, and null-critical on the discrete half-line, and derives the sharp constant for the classical n^{−2σ} Hardy…

desk verdict Solid optimal Hardy weight for the discrete half-line; the flagged local-finiteness worry is likely a presentational gap, not a mathematical one. read the letter →

arxiv 2507.06716 v2 pith:GYXKLDC2 submitted 2025-07-09 math.AP math-phmath.CAmath.FAmath.MPmath.SP

Ujjal Das , Rubén de la Fuente-Fernández This is my paper
classification math.APmath-phmath.CAmath.FAmath.MPmath.SP MSC 26D1526A33
keywords HardyinequalityfractionalLaplaciandiscretehalf-linecriticalityoptimalweightgroundstaterepresentationuniquecontinuation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the fractional Laplacian $(-\Delta_{\mathbb{N}})^\sigma$ on the discrete half-line $\mathbb{N}$ has an explicit optimal Hardy weight $W^\mathrm{op}_\sigma$ for every exponent $\sigma \in (0,1]$. The weight is a ratio of Gamma functions, and subtracting it leaves an operator that is nonnegative, critical, and null-critical, which is the strongest sense in which a Hardy weight can be 'as large as possible'. This answers an open question from prior work about the sharp constant in the Hardy inequality with the classical weight $n^{-2\sigma}$, giving the upper bound $C_\sigma = 4^\sigma \Gamma((3+2\sigma)/4)^2 / \Gamma((3-2\sigma)/4)^2$. For $\sigma = 1$ the new weight is pointwise larger than a previously known optimal weight near infinity, so optimal Hardy weights are not unique. As an application, the paper obtains a Landis-type unique continuation result at infinity for fractional Schrödinger equations on $\mathbb{N}$.

What carries the argument

The load-bearing object is the Riesz potential $I_\alpha(n)$ of (3.1), defined spectrally as the inner product of Chebyshev basis vectors with $2^{-\alpha}(1-x)^{-\alpha}$, and given explicitly in (3.2) as a ratio of Gamma functions with asymptotic $I_\alpha(n) \asymp n^{2\alpha-2}$. Its role is to make the identity $(-\Delta_{\mathbb{N}})^\sigma I_\alpha = I_{\alpha-\sigma}$ (Proposition 3.2) true, which turns the quotient $W_{\alpha,\sigma} = I_{\alpha-\sigma}/I_\alpha$ into a Hardy weight via the Agmon–Allegretto–Piepenbrink-type theorem. The ground-state representation (Proposition 3.4) then rewrites the quadratic form of $(-\Delta_{\mathbb{N}})^\sigma - W_{\alpha,\sigma}$ as a positive sum of squared differences weighted by $I_\alpha(n) I_\alpha(m)$, reducing criticality questions to asymptotics of $I_\alpha$ and the kernel $K^\sigma_{m,n}$. The threshold $\alpha = (3+2\sigma)/4$ is exactly where $\sum_n I_\alpha(n)^2 W_{\alpha,\sigma}(n)$ diverges, which is the null-criticality condition that upgrades criticality to optimality.

What would settle it

For a fixed $\sigma \in (0,1]$, evaluate the claimed identity $(-\Delta_{\mathbb{N}})^\sigma I_\alpha = I_{\alpha-\sigma}$ at several $n$ using the explicit kernel in (2.9) and the explicit Gamma formula in (3.2); if the equality fails at any $n$ for $\alpha = (3+2\sigma)/4$, the proposed optimal weight is not a Hardy weight.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Riesz potential $I_\alpha$ defined through Chebyshev polynomials satisfies the transfer identity $(-\Delta_{\mathbb{N}})^\sigma I_\alpha = I_{\alpha-\sigma}$ for $\sigma < \alpha < 1+\sigma$, so the quotient $W_{\alpha,\sigma} = I_{\alpha-\sigma}/I_\alpha$ is automatically a Hardy weight. The paper shows that $W_{\alpha,\sigma}$ is critical exactly for $\alpha \leq (3+2\sigma)/4$ and null-critical only at the endpoint $\alpha = (3+2\sigma)/4$; the endpoint weight $W^\mathrm{op}_\sigma(n) = 4^\sigma \frac{\Gamma((3+2\sigma)/4)^2}{\Gamma((3-2\sigma)/4)^2} \frac{\Gamma(n-(1+2\sigma)/4)\Gamma(n+(5-2\sigma)/4)}{\Gamma(n+(-1+2\sigma)/4)\Gamma(n+(5+2\sigma)/4)}$ is therefore optimal. Criticality uses an explicit logarithmic null-sequence for $\sigma<1$ and the Liouville comparison principle for $\sigma=1$, while null-criticality follows from the asymptotics $I_\alpha(n) \asymp n^{2\alpha-2}$ and $W_{\alpha,\sigma}(n) \asymp n^{-2\sigma}$. A corollary is that the best constant at infinity for the classical weight $n^{-2\sigma}$ equals $C_\sigma = 4^\sigma \Gamma((3+2\sigma)/4)^2 / \Gamma((3-2\sigma)/4)^2$.

Load-bearing premise

The load-bearing premise is that a theory built for graphs where each point connects to finitely many neighbors still applies to the fractional Laplacian, where every point connects to infinitely many others; the paper uses that theory to prove optimality without an explicit check of its hypotheses.

Editorial extensions

If this is right

  • For every $\sigma \in (0,1]$, the classical fractional Hardy inequality on $\mathbb{N}$ with weight $\gamma n^{-2\sigma}$ holds for all $0 < \gamma \leq C_\sigma$, and $C_\sigma = 4^\sigma \Gamma((3+2\sigma)/4)^2 / \Gamma((3-2\sigma)/4)^2$ is the best constant at infinity.
  • The optimal weight $W^\mathrm{op}_\sigma$ decays like $n^{-2\sigma}$ but is null-critical, so no pointwise larger weight can be inserted into the Hardy inequality without destroying it.
  • For $\sigma = 1$, $W^\mathrm{op}_1(n) = \frac{1}{4}(n^2 - \frac{9}{16})^{-1}$ exceeds the previously known optimal weight for the standard discrete Laplacian for all sufficiently large $n$, so optimal Hardy weights for a single operator are not unique.
  • Any solution $u$ of $(-\Delta_{\mathbb{N}})^\sigma u + V u = 0$ with $V \leq 0$ outside a finite set and $|u| = O(n^{\sigma-1/2})$ that satisfies $\liminf_{n\to\infty} |u| n^{2-2\sigma} = 0$ must vanish identically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Riesz-potential construction might extend to the subcritical range $\sigma \in (1, 3/2)$ if the signed kernel that blocks the graph-Laplacian representation can be handled by a generalized criticality theory; the paper explicitly leaves this as an open question.
  • Because $W^\mathrm{op}_\sigma$ is null-critical, the Rayleigh quotient $\langle(-\Delta_{\mathbb{N}})^\sigma f, f\rangle / \langle f, n^{-2\sigma} f\rangle$ over functions supported far out should approach $C_\sigma$, so the constant could be verified numerically from finite truncations.
  • The uniqueness (up to scaling) of the Agmon ground state for $(-\Delta_{\mathbb{N}})^\sigma - W^\mathrm{op}_\sigma$ suggests a rigidity statement: any positive supersolution of $(-\Delta_{\mathbb{N}})^\sigma$ decaying like a power must be a multiple of $I_{(3+2\sigma)/4}$.
  • The contrast with the continuum sharp constant highlights that the lattice changes the constant, not just the formulation; tracing where the difference enters (boundary conditions versus long-range jumps) could transfer the method to other discrete domains.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies the fractional Laplacian (-Δ_N)^σ on the discrete half-line N. Using the representation of this operator as a graph Laplacian for σ∈(0,1], the authors construct a family of positive Hardy weights W_{α,σ} = I_{α-σ}/I_α, where I_α is a Riesz potential defined in (3.1). They prove that W_{α,σ} is critical for (-Δ_N)^σ if and only if α ≤ (3+2σ)/4, and null-critical at α=(3+2σ)/4, yielding an explicit optimal Hardy weight W^op_σ with asymptotic n^{-2σ}; for σ=1 it is asymptotically larger than the Keller-Pinchover-Pogorzelski weight. A Landis-type unique continuation theorem for positive supersolutions is derived as an application.

Significance. The result is significant: it answers the optimal-weight question posed in [27] for the subcritical range σ∈(0,1], extends the integer-lattice construction of [34] to the half-line, determines the best Hardy constant at infinity for the weight n^{-2σ}, and gives a sharp unique continuation criterion. The proof is constructive and parameter-free: the weight is explicit in Gamma functions, the ground-state representation is derived directly from the kernel, and the criticality threshold is obtained from sharp summability estimates rather than from fitted parameters. These are concrete strengths. The main reservation is a missing verification of the structural hypotheses in the imported criticality theory.

major comments (1)
  1. [Section 2.2 and the proofs of Propositions 3.6, 3.7, 3.10 and Theorem 1.2] The criticality arguments rely on Propositions 2.5, 2.7, Remark 2.8, and Theorem 2.11 imported from [37], [36], and [11]. For σ∈(0,1), the graph {N, -eK^σ, R^σ} is not locally finite: by Remark 2.3(iv), -eK^σ_{m,n} ≍ |m-n|^{-1-2σ} is nonzero for every m≠n. The Introduction (Section 1, page 3) states that local finiteness was imposed in [36] 'in order to avoid certain technical difficulties in finding optimal Hardy-weights for graph Laplacian,' and Remark 2.8 cites [36, Theorem 5.3] for the uniqueness of the Agmon ground state. The manuscript does not verify that the theorems of [37] and [11] used here remain valid for infinite-range, summable edge weights, nor that the fractional graph satisfies the standing assumptions of those papers. Since the null-sequence criterion (Proposition 2.7) drives Proposition 3.6 and the Liouville comparison principle (Theorem 2.11) drives Propositions 3.7 and Theorem 1.2, the optimality claim in Theorem 1.1 is conditional on an unstated structural hypothesis. The authors should either state explicitly that the cited criticality theory does not require local finiteness, with precise references, or prove the needed extension to infinite-range summable edge weights.
minor comments (5)
  1. [Definition 2.1] The definition of the domain F_X contains a misprint: the summability condition should be Σ_{m∈X} b_{n,m}|f(m)| < ∞ for every n, not Σ_{m∈X} b_{n,m}|f(n)|.
  2. [Section 2.1, Eq. (2.11)] The displayed formula for R^σ_n has a 0·∞ ambiguity at σ=1: the expression involving sin(πσ) is not literally well-defined there. The value R^1_n = δ_{1,n} should be obtained by a limiting argument or by a direct computation for σ=1.
  3. [Section 3.2, Eq. (3.13)] The simplified energy functional Q^σ_α is defined with sums over n,m∈Z, but I_α and K^σ are defined on N, and the proof of Proposition 3.6 sums over 1≤n<m. Please correct the index set or explicitly define the extensions to Z.
  4. [Section 3.4, Lemma 3.9] The notation 'Qσ−α' in the proof of Lemma 3.9 is confusing and should be typeset as Q^σ_α (the subscript is α, not −α).
  5. [Proposition 3.6, proof of (3.15)] In the case analysis for (3.15), situation (i) writes '(2 − 1)²' where '(1 − 0)²' is meant, and situation (ii) contains the typo '=≤'. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal Hardy-weight is derived from an explicit Riesz-potential identity and external criticality theory, not from a fitted parameter or a self-citation chain.

full rationale

The derivation is self-contained relative to its stated inputs. The central object W_{\alpha,\sigma} is not fitted: it is the explicit quotient I_{\alpha-\sigma}/I_\alpha, with I_\alpha defined by (3.1) and the identity (-\Delta_{\mathbb{N}})^{\sigma} I_\alpha = I_{\alpha-\sigma} proved in Proposition 3.2 via the spectral representation (2.7), the kernel estimate (2.13), and the growth bounds (3.6) taken from [27]. Criticality is established by explicit null-sequences (Proposition 3.6) or by the Liouville comparison principle from [11], whose stated assumptions do not include the half-line optimal weight; null-criticality is a direct asymptotic computation (Proposition 3.8). The necessity threshold \alpha \le (3+2\sigma)/4 (Proposition 3.10) uses Gamma-function monotonicity and the earlier null-criticality computation, not a parameter fitted to the claimed conclusion. The main review concern visible in the text, namely whether the criticality theory of [37] applies to the non-locally-finite graph \{\mathbb{N}, -\tilde K^{\sigma}, R^{\sigma}\} despite the local-finiteness remark in the introduction, is a hypothesis-verification or correctness issue, not a circular reduction: no equation of the conclusion is assumed among the inputs, and no fitted constant is renamed as a prediction. The self-citations to [11] and [12] supply general theorems with independent content and do not load-bear by already containing Theorem 1.1. Therefore no circular step is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the spectral representation of the fractional Laplacian, the non-positivity of the kernel (proved in the paper), and the applicability of graph criticality theory. There are no fitted constants or new physical entities.

assumptions (5)
  • domain assumption The unitary spectral representation (-Δ_N)^σ = U^{-1} M_{2^σ(1-x)^σ} U from prior work defines the operator.
    Used in Section 2.1 and Proposition 3.2; accepted without proof.
  • domain assumption The criticality theory of [37], including the null-sequence criterion, Agmon ground state uniqueness, and the ground-state representation identity, applies to the graph {N,-eK^σ,R^σ}.
    Load-bearing for Propositions 3.4 and 3.6; the graph is not locally finite in the combinatorial sense, and the paper does not explicitly justify that [37] covers this case.
  • domain assumption The kernel and Riesz-potential estimates from prior work: |K^σ_{m,n}| bounded by |m-n|^{-2σ-1} and the growth law for J_β(m).
    Used to justify dominated convergence in Proposition 3.2 and to control the null-sequence energy in Proposition 3.6.
  • domain assumption Liouville comparison principle and the decay-to-triviality criteria for positive Schrödinger operators on graphs from prior work.
    Used for the σ=1 criticality proof and for the unique-continuation application.
  • standard math Standard gamma-function identities, including the reflection formula and digamma monotonicity.
    Background facts; the relevant computations are given in Appendix A.

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Cite this review

Pith. "Pith review of An optimal fractional Hardy inequality on the discrete half-line." pith.science (2026). https://pith.science/paper/GYXKLDC2

@misc{pith2026250706716,
  author       = {Pith},
  title        = {Pith review of: An optimal fractional Hardy inequality on the discrete half-line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYXKLDC2}},
  note         = {Machine review of arXiv:2507.06716}
}
abstract

In the context of Hardy inequalities for the fractional Laplacian $(-\Delta_{\mathbb{N}})^{\sigma}$ on the discrete half-line $\mathbb{N}$, we provide an optimal Hardy-weight $W^{\mathrm{op}}_{\sigma}$ for exponents $\sigma\in\left(0,1\right]$. As a consequence, we provide the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $n^{-2\sigma}$ on $\mathbb{N}$. It turns out that for $\sigma =1$ the Hardy-weight $W^{\mathrm{op}}_{1}$ is pointwise larger than the optimal Hardy-weight obtained by Keller--Pinchover--Pogorzelski near infinity. As an application of our main result, we obtain unique continuation results at infinity for the solutions of some fractional Schr\"odinger equation.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.