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Fractional and non-fractional Hardy inequality on a lattice $\Z^d$

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every 0<p<∞ and every dimension d, this paper pins down exactly which power weights make the discrete Hardy inequality on Z^d true, and proves those exponents are optimal.

desk verdict A solid, self-contained completion of the power-weight discrete Hardy picture, with a minor summation slip in Lemma 2.1 that is easy to fix. read the letter →

arxiv 2506.08273 v2 pith:GWWNYCJH submitted 2025-06-09 math.FA

Bart{l}omiej Dyda This is my paper
classification math.FA MSC 26D1526A3335J05
keywords Hardyinequalitydiscretefractionalintegerlatticepowerweightsoptimalexponentsnearest-neighbourdifferencesdyadicannuli
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 establishes a complete picture, within power weights, for the discrete Hardy inequality on the lattice $\mathbb{Z}^d$: sums of $|u(j)|^p$ divided by a power of $\|j\|_\infty$ are controlled by sums of $p$-th powers of nearest-neighbour differences, for every $0d$, and an arbitrarily small correction $\varepsilon>0$ in the critical case $d=p$ (and in dimension $1$ for $00$ and $p>0$ with $sp\ne d$. If the completeness claim is right, there is no remaining gap in the exponent range for power weights in the local case.

What carries the argument

The engine is Lemma 2.1, a dyadic-annulus estimate. With $A_n=\{j\in\mathbb{Z}^d_+: 2^{n-1}\le\|j\|_\infty\le 2^n-1\}$, the lemma bounds the full weighted sum $\sum_{j\ne0}|u(j)|^p/\|j\|_\infty^{sp}$ by a constant times $\sum_n\sum_{j\in A_n}\sum_{m\in A_{n+K}}|u(j)-u(m)|^p\,2^{-(n+K)(d+sp)}$, provided $K$ is chosen so that the contraction factor $2^{sp+1}(2^{p-1}\vee1)2^{-K|sp-d|}\le1$. This is the 'primary form' of the Hardy inequality. Proposition 2.3 converts these annulus pair sums into nearest-neighbour edge sums via a path-counting argument: any two points in nearby annuli are joined by a coordinate-wise shortest path, and each edge of such a path belongs to at most a bounded number of pairs; averaging over the $d$ cyclic permutations of the coordinate order removes the dependence on the starting coordinate. In the fractional case the same annulus lemma is applied directly, because the kernel $\|j-m\|_\infty^{-(sp+d)}$ matches the annulus denominators.

What would settle it

In the critical case $d=p=2$, take $u_n(j)=\min(\|j\|_\infty/n,\, n/\|j\|_\infty)$ and compute the ratio of $\sum_{j\ne0}|u_n(j)|^2/\|j\|_\infty^2$ to $\sum_{j\sim k}|u_n(j)-u_n(k)|^2$ as $n\to\infty$; if this ratio stays bounded, then the claimed failure at $\varepsilon=0$ is false.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: on $\mathbb{Z}^d_+$ (and by extension on $\mathbb{Z}^d$) the non-fractional Hardy inequality holds for every $0<p<\infty$, with weight exponent $t=1$ for $0<p\le 1<d$, $t=p$ for $1\le p<d$ and for $p>d$, $t=p+\varepsilon$ for $d=p$, and $t=1+\varepsilon$ for $d=1$, $0<p<1$. Theorem 1.1(6) adds that the exponents $1,p,p$ are optimal and that the $\varepsilon=0$ borderline versions fail. Theorem 1.3 gives the fractional counterpart: for $s>0$ and $sp\ne d$, the inequality with differences over all pairs and kernel $\|j-m\|_\infty^{-(sp+d)}$ holds with weight $\|j\|_\infty^{-sp}$ when $sp<d$ and when $sp>d$, and with weight $\|j\|_\infty^{-(sp+\varepsilon)}$ when $sp=d$; optimality of these exponents is not claimed. All results transfer from the positive orthant to the whole lattice with possibly larger constants.

Load-bearing premise

The completeness claim rests on the optimality arguments using the test function $u=1-v_n$; this function has infinite weighted sum for exponents $t\le d$, so the written proof does not exclude smaller exponents in that range.

Editorial extensions

If this is right

  • For power weights on $\mathbb{Z}^d_+$ and $\mathbb{Z}^d$, the nearest-neighbour Hardy inequality holds precisely on the exponent ranges named in Theorem 1.1, so the previously open low-exponent and critical-dimensional cases are settled.
  • At the critical dimension $d=p$, the weight $\|j\|_\infty^{-p}$ is forbidden, but $\|j\|_\infty^{-(p+\varepsilon)}$ is allowed for every $\varepsilon>0$, with the same $\varepsilon$ appearing as a weight on the right-hand side.
  • The fractional Hardy inequality is valid for all $s>0$ and all $p>0$ with $sp\ne d$, not only for $p=2$ and $0<s<1/2$; the known one-dimensional fractional results are special cases.
  • All statements pass from $\mathbb{Z}^d_+$ to the full lattice $\mathbb{Z}^d$ with possibly larger constants.
  • The constants are explicit and depend only on $d,p,s$ and the gap $\delta>0$ (or on $\varepsilon$), so the inequalities are uniform in the gap parameter.

Reading between the lines

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

  • The paper does not pursue it, but the same dyadic-annulus mechanism, since it uses only the shell count $\#A_n\asymp 2^{nd}$ and coordinate-wise paths, should transfer to other lattices of polynomial volume growth.
  • A natural test left implicit is whether logarithmic weights, rather than the $\varepsilon$ power correction, are the true borderline weights in the critical case $d=p$; the paper does not address such weights.
  • For the fractional theorem no optimality is proved; testing truncated-cone functions of the same type should reveal whether the weight exponent $sp$ in the supercritical range is necessary.
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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 / 4 minor

Summary. The paper proves discrete Hardy-type inequalities for functions on the positive lattice and on the full lattice Z^d, both in the local (finite-difference) and fractional (nonlocal) settings. The main positive results cover all exponents 0<p<∞ and all dimensions d for power weights, with explicit but not optimal constants: Theorem 1.1 treats the local inequalities on Z^d_+ and Z^d, including the critical cases d=p and d=1, and Theorem 1.3 treats the fractional inequalities for all s>0 with sp≠d. The proofs are organized around a common technical lemma, Lemma 2.1, which is then used with a path-counting proposition, Proposition 2.3, to derive the local and nonlocal inequalities. The paper also contains lower-bound constructions aimed at showing optimality of the power exponents in Theorem 1.1.

Significance. If the proof is made fully correct, the paper would give a fairly complete power-weight picture for discrete Hardy inequalities on Z^d_+ and Z^d, including cases that the paper identifies as new. The approach is self-contained and has a pleasant structure: the same annulus lemma yields both local and nonlocal inequalities, and the constants are explicitly tracked. The optimality constructions are concrete and do not rely on fitted parameters. The main caveat is that the central lemma contains a summation step that is not justified as written, and since both main theorems rely on that lemma, the correctness claim is currently load-bearing on an unproved step.

major comments (1)
  1. [§2, proof of Lemma 2.1, case sp<d] After summing (2.5) over n≥1 in the case sp<d, the second term on the right-hand side is (1/2) Σ_{m: ||m||∞≥2^K} |u(m)|^p/||m||^sp, not (1/2) Σ_{m∈Z^d_+\{0}} |u(m)|^p/||m||^sp. The annuli A_1,...,A_K are missing from this summed contribution, so the displayed inequality containing (1/2) times the full left-hand side is not an algebraic consequence of the preceding line. Since this absorption is used to derive (2.3), Lemma 2.1 is not proved as written. This is load-bearing because Theorems 1.1(1)-(2) and 1.3(1) are deduced from (2.3). The gap appears repairable by unrolling the recursion L_n ≤ B_n + (1/2)L_{n+K} and using finiteness of the total left-hand side to let the remainder vanish, but the manuscript needs to supply this argument and check what statement and constants are actually obtained.
minor comments (4)
  1. [Abstract and Introduction] The first sentence of the abstract contains a punctuation error: 'inequality, Our constants' should read 'inequality; our constants'.
  2. [§2, proof of Lemma 2.1] There is a typo 'rearraging' in the paragraph after (2.6); it should be 'rearranging'.
  3. [§3, Proposition 3.2(3.6)] The lower bound in (3.6) is stated only for t>d and t=d, but the divergence also occurs for t<d by the same estimate. Since the optimality argument for d≤p may need t<p with t≤d, it would be clearer to state explicitly that the left-hand side diverges for all t≤d. This is not a substantive issue: in the cases d≤p the theorem is asserted for all functions with u(0)=0, so an infinite left-hand side combined with a finite right-hand side is a valid contradiction.
  4. [§4, proof of Theorem 1.1(7)] The extension from Z^d_+ to Z^d is stated without proof details for Theorem 1.1, while Theorem 1.3(4) gives an indication. A short explanation analogous to that for Theorem 1.3(4) would make the completeness claim easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's inequalities are derived from its own Lemma 2.1 and explicit test functions, with no fitted parameters and no load-bearing self-citations.

full rationale

The derivation chain is self-contained. The positive inequalities in Theorems 1.1 and 1.3 are deduced from Lemma 2.1 together with Proposition 2.3 and Lemma 4.1; the optimality arguments in Theorem 1.1(6) use explicit test functions u_n and v_n from Propositions 3.1 and 3.2. No parameter is fitted to a subset of data and then renamed a prediction, no conclusion is assumed in the hypotheses, and no load-bearing claim rests on a citation to the author's own prior work. The references to prior results are contextual and not used to justify the new inequalities. A reviewer concern that the test function u = 1 - v_n has infinite left-hand side is not circularity: for the cases d <= p where it is used, Theorem 1.1(3)-(4) are stated for all functions with u(0)=0, so an infinite left side is a legitimate contradiction rather than an inadmissible input. Separately, the proof of Lemma 2.1 in the case sp < d appears to contain an algebraic gap when summing inequality (2.5): the second term runs over annuli A_{n+K}, so the displayed summed inequality omits the first K annuli; this is a potential non-circular proof-error, possibly repairable by iteration, but it does not make the argument circular and therefore does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The proofs are essentially self-contained; the only inputs are the dyadic shell partition, standard counting on integer lattice paths, and geometric series estimates. No new physical or mathematical objects are introduced, and the only hand-chosen parameter is the shell shift K.

free parameters (1)
  • K (annulus shift) = any integer satisfying (2.1)
    Chosen by hand to make the self-term absorption coefficient at most 1/2 in Lemma 2.1. The constants depend on K, but the truth of the inequalities does not depend on its specific value.
assumptions (4)
  • standard math The dyadic annuli A_n partition Z^d_+ without the origin.
    Each nonzero lattice point has sup norm in exactly one interval [2^{n-1}, 2^n - 1], used throughout the paper.
  • standard math The coordinate-ordered path construction and its counting bounds (2.10) and (2.14) are valid.
    These counts produce the factor d^{p∨1-2} in Proposition 2.3; an error here would change the local inequality constants.
  • domain assumption Infinite sums over annuli may be rearranged and limits interchanged for functions with finite left-hand side.
    Lemma 2.1 sums estimates over infinitely many annuli and passes to limits; monotone or dominated convergence is needed.
  • standard math The finite-box Poincare inequality (Lemma 4.1) holds for functions vanishing at 0.
    Proven in the paper; used to control the small-ball term in Theorem 1.1(3)-(4).

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Cite this review

Pith. "Pith review of Fractional and non-fractional Hardy inequality on a lattice $\Z^d$." pith.science (2026). https://pith.science/paper/GWWNYCJH

@misc{pith2026250608273,
  author       = {Pith},
  title        = {Pith review of: Fractional and non-fractional Hardy inequality on a lattice $\Z^d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWWNYCJH}},
  note         = {Machine review of arXiv:2506.08273}
}
abstract

We present simple proofs of a discrete fractional and non-fractional Hardy inequality, Our constants are explicit, but not optimal. In the class of power weights, we get a complete picture of when the non-fractional Hardy inequality holds, for any dimension $d$ of the lattice $\Z^d$ and exponent $0<p<\infty$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An optimal fractional Hardy inequality on the discrete half-line

    math.AP 2025-07 conditional novelty 6.0 of 10

    For σ in (0,1], the paper constructs an explicit optimal Hardy weight W^op_σ for (-Δ_N)^σ, with W^op_σ(n) approximately n^{-2σ} and an upper bound C_σ for the best constant in the n^{-2σ} Hardy inequality.

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