{"id":"2e7f6f67-6406-48ec-ba7b-10f324ffb6e1","arxiv_id":"2506.08273","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper gives simple dyadic-annulus proofs of discrete Hardy inequalities with power weights on Z^d for all p and d, and identifies the optimal exponents for the local case.","lead":"This paper proves a family of discrete Hardy inequalities on the integer lattice in any dimension, covering both nearest-neighbor and fractional long-range differences, with explicit constants. It also pins down exactly which power weights are admissible for the non-fractional inequalities, for every exponent p and dimension d.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimality test function is admissible for d≤p; the real gap is an incorrect summation step in Lemma 2.1 (sp<d) that needs an iteration argument.","rationale":"The paper's positive results are well supported: the dyadic-annulus mechanism in Lemma 2.1 and Proposition 2.3 is coherent, and the fractional and local inequalities are derived consistently. The optimality section also survives the reader's objection: Theorem 1.1(3)-(4) impose u(0)=0 rather than finite left side, so using 1-v_n (which has u(0)=0 and finite gradient energy) is legitimate; the divergence of its left side for t≤d is exactly what proves optimality, and the missing t<d case in Proposition 3.2(3.6) is an obvious extension. The genuine defect is in the proof of Lemma 2.1, case sp<d: the displayed 'sum over Z_+^d\\{0}' after summing (2.5) is not derived from the preceding line, since the second term only runs over shifted annuli. This is load-bearing because Lemma 2.1 is the common source for Theorems 1.1 and 1.3. The gap is repairable by an iteration/limiting argument, and no counterexample to the lemma itself is apparent. Therefore the verdict should remain conditional pending a corrected proof of Lemma 2.1, and the specific concern raised by the reader should be set aside.","tokens_in":14358,"tokens_out":37838,"duration_ms":420338,"concrete_test":"Independently re-derive Lemma 2.1, case sp<d, by summing (2.5) over n=1,...,N and letting N→∞. Check whether the low annuli A_1,...,A_K are controlled by the first double sum after the tail term vanishes. If the printed line 'sum over Z_+^d\\{0}' is reproduced verbatim, it is false; the corrected iteration should produce the same constant C(d,p,s,K) and should be included in the revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern about u=1-v_n does not land: for d≤p, Theorem 1.1(3)-(4) are asserted for all functions with u(0)=0, not only for functions with finite left side. The test function satisfies u(0)=0 and has finite right side in (4.6), so an infinite left side is a valid contradiction. The actually load-bearing weak point is in Lemma 2.1, case sp<d. Summing (2.5) over n≥1 does not give the printed term 1/2∑_{j∈Z_+^d\\{0}} |u(j)|^p/||j||^sp: the second term of (2.5) runs over annuli A_{n+K}, so the summed contribution covers only ||j||∞≥2^K, missing A_1,...,A_K. The displayed inequality in the proof is therefore not a valid algebraic consequence of the preceding line. The lemma is recoverable by unrolling the recursion L_n ≤ B_n + 1/2 L_{n+K} and using finiteness of the total left side to let the remainder vanish, which yields the same constant, but the written proof needs this correction. Since Lemma 2.1 is the common source of all positive inequalities in Theorems 1.1 and 1.3, the completeness claim rests on a proof step that is not currently justified as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14592,"tokens_out":20583,"duration_ms":249928,"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":[{"comment":"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.","section":"§2, proof of Lemma 2.1, case sp<d"}],"minor_comments":[{"comment":"The first sentence of the abstract contains a punctuation error: 'inequality, Our constants' should read 'inequality; our constants'.","section":"Abstract and Introduction"},{"comment":"There is a typo 'rearraging' in the paragraph after (2.6); it should be 'rearranging'.","section":"§2, proof of Lemma 2.1"},{"comment":"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.","section":"§3, Proposition 3.2(3.6)"},{"comment":"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.","section":"§4, proof of Theorem 1.1(7)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the novelty claims appear plausible. The central issue is the invalid summation step in Lemma 2.1; if the author supplies a correct iteration argument and adjusts the statement or proof accordingly, I would be supportive of publication. I do not see a circularity or novelty problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper gives a clean, self-contained proof of the full power-weight picture for discrete Hardy inequalities on Z_+^d and Z^d, for every 0<p<∞ and every d. The genuinely new results are the multidimensional non-fractional cases for all p and the multidimensional fractional inequalities; the 1D and p=2 cases were already known. The proof is organized around a single dyadic-annulus lemma (Lemma 2.1) with a path-counting argument (Proposition 2.3), which is a nice simplification and yields explicit constants. The optimality claims for the exponents are supported by explicit test functions, and they look correct.\n\nOne caveat in the reader's report does not hold up. The concern that u=1-v_n is not admissible for t≤d when d≤p misses that Theorem 1.1(3)-(4) are stated for all functions with u(0)=0, not only those with finite left side. So an infinite left side is a valid contradiction. The optimality argument is fine as written.\n\nThere is a real, though minor, flaw in the proof of Lemma 2.1. After summing (2.5) over n in the sp<d case, the second term on the right is only the tail Σ_{||x||_∞ ≥ 2^k} |u(x)|^p/||x||^sp, not the full sum, so the displayed inequality is not a legitimate algebraic consequence. The lemma is still true: since the tail is bounded by the entire sum, one gets L ≤ (C/2)D + L/2 and hence L ≤ C D. The sp>d case has a similar tail issue. These are one-line fixes, not structural gaps.\n\nA minor presentation issue: Theorem 1.1(3)-(4) omit the usual 'for which the left side is finite' qualification, which is needed to avoid trivial counterexamples. The optimality discussion does not depend on this, but the statement should be cleaned up.\n\nThe citation pattern is honest, and the author is careful about what is new and what is known. The constants are explicit and the proof is reproducible as written (modulo the small summation correction).\n\nThis is a useful reference result, not a breakthrough. It would benefit from a minor revision, but it definitely deserves a serious referee. I would send it to review and would cite it in work on discrete Hardy inequalities.\n\nCheers.","headline":"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.","tokens_in":15200,"tokens_out":9937,"would_cite":true,"duration_ms":101274,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D15","26A33","35J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Hardy inequality","discrete Hardy inequality","fractional Hardy inequality","integer lattice","power weights","optimal exponents","nearest-neighbour differences","dyadic annuli"],"falsifier":"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.","tokens_in":14095,"feed_emoji":"🧮","tokens_out":17344,"duration_ms":188423,"temperature":0.7,"pith_summary":"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 $0<p<\\infty$ and every dimension $d$. The allowed weight exponents are exactly $1$ when $0<p\\le 1<d$, $p$ when $1\\le p<d$ and when $p>d$, and an arbitrarily small correction $\\varepsilon>0$ in the critical case $d=p$ (and in dimension $1$ for $0<p<1$). The constants are explicit but not optimal. The same dyadic-annulus lemma also proves the fractional analogue, with differences over all pairs weighted by $\\|j-m\\|_\\infty^{-(sp+d)}$, for every $s>0$ 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.","feed_headline":"For every p, Hardy's inequality on the integer lattice is settled","feed_subtitle":"All power-weight exponents are pinned down, including critical cases that need an extra epsilon.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the original one-dimensional Hardy inequality that the discrete results generalize.","marker":"[10]"},{"why":"It is the classical statement of the discrete inequality used as the baseline for the $p>d$ case.","marker":"[11]"},{"why":"It proves the multidimensional $p=2$, $d\\ge3$ case that the paper extends to all $p$.","marker":"[13]"},{"why":"It gives the optimal one-dimensional discrete inequality for $p=2$, against which the local constants are measured.","marker":"[16]"},{"why":"It establishes the one-dimensional discrete Hardy inequality for general $p>1$, the range covered by the paper's $p>d$ case.","marker":"[6]"},{"why":"It provides the optimal one-dimensional weight and constant, the baseline for the explicit power weights.","marker":"[18]"},{"why":"It develops non-local Hardy weights via Green functions, the construction that the paper's explicit weights avoid.","marker":"[4]"},{"why":"It studies optimality and decay of Hardy weights on graphs, the context that makes an explicit power-weight statement non-obvious.","marker":"[5]"},{"why":"It proves the fractional Hardy inequality on the line for $p=2$ and $0<s<1/2$, the case Theorem 1.3 generalizes.","marker":"[2]"},{"why":"It obtains the optimal fractional Hardy constant on the line, the sharp counterpart to the explicit constants here.","marker":"[15]"}],"fun_headline_variants":["Integer lattice Hardy inequality: complete power-weight picture","All p and weights: Hardy inequality on Z^d settled","Hardy on lattice: optimal exponents and critical epsilon cases","Discrete Hardy inequality: complete classification of power weights","Fractional and non-fractional Hardy on Z^d fully classified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Integer lattice Hardy inequality: complete power-weight picture","All p and weights: Hardy inequality on Z^d settled","Hardy on lattice: optimal exponents and critical epsilon cases","Discrete Hardy inequality: complete classification of power weights","Fractional and non-fractional Hardy on Z^d fully classified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3165,"prompt_tokens":845,"completion_tokens":2320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":2238}},"tokens_in":461,"tokens_out":2320,"duration_ms":20600,"temperature":1.0,"reasoning_tokens":2238,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:18:12.813286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the original one-dimensional Hardy inequality that the discrete results generalize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the classical statement of the discrete inequality used as the baseline for the $p>d$ case."},{"cited_title":"Kapitanski and A","cited_arxiv_id":null,"evidence_quote":"It proves the multidimensional $p=2$, $d\\ge3$ case that the paper extends to all $p$."},{"cited_title":"Keller, Y","cited_arxiv_id":null,"evidence_quote":"It gives the optimal one-dimensional discrete inequality for $p=2$, against which the local constants are measured."},{"cited_title":"Fischer, M","cited_arxiv_id":null,"evidence_quote":"It establishes the one-dimensional discrete Hardy inequality for general $p>1$, the range covered by the paper's $p>d$ case."},{"cited_title":"Roychowdhury and D","cited_arxiv_id":null,"evidence_quote":"It provides the optimal one-dimensional weight and constant, the baseline for the explicit power weights."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It develops non-local Hardy weights via Green functions, the construction that the paper's explicit weights avoid."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It studies optimality and decay of Hardy weights on graphs, the context that makes an explicit power-weight statement non-obvious."},{"cited_title":"Ciaurri and L","cited_arxiv_id":null,"evidence_quote":"It proves the fractional Hardy inequality on the line for $p=2$ and $0<s<1/2$, the case Theorem 1.3 generalizes."},{"cited_title":"Keller and M","cited_arxiv_id":null,"evidence_quote":"It obtains the optimal fractional Hardy constant on the line, the sharp counterpart to the explicit constants here."}],"review_version":1}