{"id":"adfd2aa5-58c8-403e-8331-780da48b181d","arxiv_id":"2411.13880","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper attempts to prove boundedness of the Riesz potential on homogeneous variable exponent Herz-Morrey-Hardy spaces, but the statement and proof have critical inconsistencies.","lead":"This mathematics paper claims that the Riesz potential operator maps a specialized function space called the homogeneous variable exponent Herz-Morrey-Hardy space into a related Herz-Morrey space, under certain conditions. The proof is built on atomic decompositions, but the theorem statement and proof contain internal contradictions and unstated extra assumptions, so the result is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 allows α∞ = 2λ, but the proof of the F1 and H1 estimates for q1 > 1 requires the strict inequality α∞ > 2λ; at equality the key geometric series diverges, so (3.4) is unproved as stated.","rationale":"The central claim is a boundedness statement for the Riesz potential, and the proof's final estimates depend on summability of dyadic geometric series. The most load-bearing weakness is exactly the mismatch between the theorem's hypothesis 2λ ≤ α(·) and the proof's repeated use of α∞ > 2λ in the q1 > 1 cases of F1 and H1. This is not a cosmetic issue: at equality α∞ = 2λ the relevant series has constant terms and diverges, so the proof cannot produce the finite bound F1 ≲ Λ. The example in the concrete test shows that the equality case is reachable under the stated assumptions, including the exponent relation 1/p1 = 1/p2 + β/n. The reader's weakest_assumption identifies the same strict-inequality gap, and I see no argument in the paper that covers the endpoint. Other difficulties (for instance, the use of Theorem 1 under hypotheses that may not include nδ2 < α(0), and the q1-versus-q2 power in the application of (2.8)) may also be present, but the unproved strict inequality alone suffices to block the theorem as stated. Therefore the reader's rejection of the manuscript should stand.","tokens_in":16661,"tokens_out":17600,"duration_ms":170072,"concrete_test":"Set n = 1, β = 0.2, λ = 0.1, q1 = 2, q2 = 2, p1 = 2, p2 = 10/3, and α(·) ≡ 0.2 = 2λ. These satisfy the hypotheses of Theorem 2, including β − nδ2 < α(0) for δ2 = 0.1. Independently recompute the F1 estimate in Section 3, Case 2, at this equality value: the factor sum_{j=0}^∞ 2^{(λ−α∞/2) j q1} becomes sum_{j=0}^∞ 1 = ∞. If the paper's displayed chain cannot be modified to avoid this divergent factor, then the theorem must add the strict assumption α∞ > 2λ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, the F1 estimate in the case 1 < q1 < ∞ reduces a tail contribution to the geometric factor sum_{j=0}^∞ 2^{(λ−α∞/2) j q1} and closes the estimate with the parenthetical condition “α∞ > 2λ.” The same condition appears at the end of the H1 estimate. However, Theorem 2 assumes only 2λ ≤ α(·), so α∞ = 2λ is explicitly allowed. At that endpoint each term of the displayed geometric series equals 1, and the series diverges. Consequently the displayed conclusion F1 ≲ Λ (and similarly H1 ≲ Λ) is not derived under the stated hypotheses. Since 0 < q1 ≤ q2 permits q1 > 1, this is not a vacuous edge case: the proof does not establish the claimed bound (3.4) for all f admitted by Theorem 2. The theorem can be repaired by adding the strict hypothesis α∞ > 2λ (or by supplying an alternative estimate at equality), but as written the central claim is not proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a boundedness theorem for the Riesz potential I_β on homogeneous variable exponent Herz-Morrey-Hardy spaces. Specifically, Theorem 2 asserts that under exponent and log-Hölder conditions, I_β maps HM\\dot{K}^{α(·),q_1}_{p_1(·),λ} into M\\dot{K}^{α(·),q_2}_{p_2(·),λ}, with estimate (3.4). The proof uses the atomic decomposition of Xu and Yang, the norm equivalence (2.8), and a dyadic decomposition of the Herz-Morrey norm into terms F, G, H. The paper is essentially a proof of this single theorem, supplemented by definitions and recalled results.","tokens_in":16920,"tokens_out":17877,"duration_ms":134233,"significance":"If the theorem were correct, it would provide a modest extension of known Riesz-potential estimates to a combined variable-exponent Herz-Morrey-Hardy setting and would fill a gap noted in the introduction. The paper is also honest in relying on external results: the atomic decomposition is quoted from Xu and Yang [5], and the Lebesgue-space boundedness of I_β is quoted from Capone–Cruz-Uribe–Fiorenza [9]. However, the central theorem is not established as stated: the hypotheses are internally contradictory, the proof applies a norm equivalence with the wrong exponent, and several geometric-sum convergence steps require strict inequalities or use incorrect exponent arithmetic. Because these issues affect the main claim, the present version does not support its conclusion.","major_comments":[{"comment":"The hypotheses on the exponents are contradictory. The theorem first requires 1/p_1(·) = 1/p_2(·) + β/n, which is equivalent to 1/p_2(·) = 1/p_1(·) − β/n, the standard Sobolev relation for the Riesz potential. A few lines later it also requires the boundedness condition 1/p_2(·) = β/n − 1/p_1(·). These cannot hold simultaneously: adding the two equations gives 2/p_2(·) = 0. Thus Theorem 2, as written, has inconsistent hypotheses and is vacuous. The proof appears to use only the first relation, so the second clause is likely a typo, but the statement must be corrected before the claim can be evaluated.","section":"Theorem 2 (Section 3)"},{"comment":"The proof applies the norm equivalence (2.8) with the exponent q_1 to the norm of M\\dot{K}^{α(·),q_2}_{p_2(·),λ}. However, (2.8) is stated for a space whose exponent equals the q appearing in the norm; replacing q_2 by q_1 gives the identity ‖I_β f‖_{M\\dot{K}^{q_2}}^{q_1} ≈ max{ ... } that is not a consequence of (2.8). A correct argument would first use the monotonicity of ℓ^q norms (since q_1 ≤ q_2) to bound the q_2-norm by the q_1-norm, and only then apply (2.8) with q_1. The displayed equivalence near the beginning of the proof is therefore mathematically false as stated, although the argument might be repairable with an additional inequality.","section":"Section 3, proof of Theorem 2"},{"comment":"The estimates for F1 in the case 1 < q_1 < ∞ close with the parenthetical condition α_∞ > 2λ, and the same condition is used at the end of the H1 estimate. Theorem 2 assumes only 2λ ≤ α(·), so α_∞ = 2λ is explicitly allowed. At that endpoint the geometric series ∑_{j=0}^∞ 2^{(λ−α_∞/2)jq_1} has every term equal to 1 and diverges. Consequently the proof does not establish F1 ≲ Λ or H1 ≲ Λ for all f admitted by the theorem. The theorem would need either an added strict hypothesis α_∞ > 2λ or a different estimate at equality.","section":"Section 3, F1 and H1 estimates"},{"comment":"The dyadic estimates contain an exponent sign error. Substituting (3.2) into F2 and multiplying by the weight 2^{kq_1α(0)} gives a factor 2^{q_1(β−nδ_2+α(0))(k−j)}, not 2^{(β−nδ_2−α(0))(k−j)q_1} as written in the F2 estimate. The paper then uses the hypothesis β−nδ_2 < α(0) to make the displayed series converge, but the correct exponent would require the condition β−nδ_2+α(0) < 0, which is not assumed and is generally incompatible with β−nδ_2 < α(0). The same sign error recurs in the G2 and H2 estimates. These are load-bearing because they justify the boundedness of the 'far' dyadic contributions; without them the proof of (3.4) is incomplete.","section":"Section 3, F2, G2 and H2 estimates"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical and grammatical errors, including 'V ARIABLE' and 'SP ACES' in the abstract; a thorough proofreading is needed.","section":"Throughout"},{"comment":"The hypothesis '0 < p_1 ≤ p_2 < ∞' is unclear because p_1 and p_2 are variable exponent functions, not constants; the intended condition is presumably the pointwise inequality p_1(·) ≤ p_2(·).","section":"Theorem 2"},{"comment":"The notation α_j is used in the exponents of (3.2) and (3.3) but is never defined; the proof later splits terms into α(0) and α_∞, so the convention should be stated explicitly.","section":"Equations (3.2), (3.3)"},{"comment":"The geometric condition '2|y| ≤ |x|' is claimed for j ≤ k−1, but for j = k−1 one only has |y| ≤ 2^j and |x| > 2^{j}, which does not imply 2|y| ≤ |x|; the dyadic separation should be stated with j ≤ k−2 or with a suitable constant.","section":"Equations (3.1), (3.3)"}],"recommendation":"reject","confidential_remarks":"This manuscript has several fundamental problems in its main theorem and proof: contradictory exponent hypotheses, an invalid application of norm equivalence (2.8), an uncovered endpoint in the geometric series, and sign errors in the dyadic estimates. Even if the typographical issues were fixed, the proof would need substantial new estimates, particularly for the F2/G2/H2 terms. The paper is not close to publishable in its current form, and I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This paper applies the standard atomic decomposition template to a genuinely new setting: Riesz potentials on homogeneous variable exponent Herz–Morrey–Hardy spaces. That gap is real, and the citations are appropriate. But the current version is not correct as stated, and the problems are not just cosmetic.\n\nTheorem 2 contains a contradiction in its exponent hypotheses. It first assumes 1/p1(·) = 1/p2(·) + β/n, and then, as a condition for the known Riesz potential bound, requires 1/p2(·) = β/n − 1/p1(·). These two relations cannot both hold unless p1 = n/β. The second is almost certainly a typo for 1/p2(·) = 1/p1(·) − β/n, but as written the statement is ill-posed.\n\nThe proof has a more serious structural error. It applies the norm equivalence (2.8) to the q2-norm while raising to the q1 power. The equivalence is stated for the q-th power of the q-norm, so what is actually obtained is a bound for ||Iβ f||_{q2}^{q2}, not the q1-power that appears on the left of (3.4). The estimate therefore does not control the claimed target norm.\n\nThere is also a missing hypothesis on α∞. In the case 1 < q1 < ∞, the F1 and H1 estimates close with the condition α∞ > 2λ, written in parentheses at the end of those displays. Theorem 2 assumes only 2λ ≤ α(·), so α∞ = 2λ is allowed. At that endpoint the geometric series with terms 2^{(λ−α∞/2)jq1} has every term equal to 1 and diverges. The proof gives no alternative estimate at that endpoint. Similarly, the H2 estimate uses a condition β − nδ2 < α∞ that does not appear in the theorem's hypotheses.\n\nThe paper does show a sensible strategy and a clear attempt to fill a real gap. If the exponent relations were corrected, the norm equivalence applied with the right power, and the α∞ endpoint handled, a repaired version could be a small, useful note for specialists in variable exponent Herz–Morrey spaces. But as it stands, the central claim is not proved.\n\nMy recommendation is desk reject, with a detailed message to the author listing the three issues above. Do not send to referees in this form.","headline":"Routine template proof of a new Riesz potential bound, but the theorem as stated is internally inconsistent and the proof fails at allowed endpoints.","tokens_in":17410,"tokens_out":6726,"would_cite":false,"duration_ms":51102,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","42B25","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the Riesz potential $I_\\beta$ is bounded from the homogeneous variable exponent Herz-Morrey-Hardy space $HM\\dot{K}^{\\alpha(\\cdot),q_1}_{p_1(\\cdot),\\lambda}$ to the homogeneous variable exponent Herz-Morrey space…","keywords":["Riesz potential","variable exponent","Herz-Morrey-Hardy space","homogeneous Herz-Morrey space","atomic decomposition","boundedness","log-Hölder continuity"],"falsifier":"Test the borderline case $\\alpha(\\cdot)\\equiv 2\\lambda$: in the proof's $H_1$ estimate the inner sums become partial sums that grow linearly in $L$, so the supremum over $L$ is infinite, contradicting the claimed inequality unless a cancellation not present in the proof occurs.","tokens_in":16430,"feed_emoji":"🧮","tokens_out":10373,"duration_ms":85130,"temperature":0.7,"pith_summary":"This paper sets out to prove a norm inequality for the Riesz potential operator $I_\\beta$ on spaces that combine variable-exponent Lebesgue behavior with Herz, Morrey, and Hardy structure. The main result asserts that $I_\\beta$ maps the homogeneous variable exponent Herz-Morrey-Hardy space $HM\\dot{K}^{\\alpha(\\cdot),q_1}_{p_1(\\cdot),\\lambda}$ into the homogeneous variable exponent Herz-Morrey space $M\\dot{K}^{\\alpha(\\cdot),q_2}_{p_2(\\cdot),\\lambda}$, with the operator norm controlled by a constant times the input norm. The proof works by decomposing the input function into central atoms, applying dyadic estimates to each atom under the kernel of $I_\\beta$, and summing the resulting geometric series using log-Hölder continuity of the exponent. If correct, the result extends earlier boundedness theorems for singular integrals on these spaces to fractional integrals.","feed_headline":"Riesz potential maps Herz-Morrey-Hardy into Herz-Morrey","feed_subtitle":"Atomic decomposition yields the bound under log-Hölder conditions.","key_machinery":"The load-bearing objects are the atomic decomposition of the homogeneous variable exponent Herz-Morrey-Hardy space (supplied by Theorem 1 of [5]) and the norm equivalence (2.8) that splits the Herz-Morrey norm into sums weighted by $\\alpha(0)$ for nonpositive annuli and by $\\alpha_\\infty$ for positive ones. The dyadic estimates (3.2) and (3.3) control the action of $I_\\beta$ on an atom supported in $B_j$ when measured on an annulus $F_k$, giving decay factors $2^{(\\beta-n\\delta_2)(k-j)}$ and $2^{(\\beta-n\\delta_1)(j-k)}$. These factors, together with the log-Hölder continuity assumptions, make the geometric series in the proof converge.","core_discovery":"Theorem 2 states that for $0<q_1\\le q_2<\\infty$, $0<\\lambda<\\infty$, $0<\\beta<n$, $0<p_1\\le p_2<\\infty$ with $p_1(\\cdot),p_2(\\cdot)\\in \\mathcal{B}(\\mathbb{R}^n)$ and $\\frac{1}{p_1(\\cdot)}=\\frac{1}{p_2(\\cdot)}+\\frac{\\beta}{n}$, and for $\\alpha(\\cdot)\\in L^\\infty\\cap \\mathcal{P}^{\\log}_0\\cap \\mathcal{P}^{\\log}_\\infty$ satisfying $2\\lambda\\le \\alpha(\\cdot)$ and $\\beta-n\\delta_2<\\alpha(0)$, the Riesz potential $I_\\beta$ is bounded from $HM\\dot{K}^{\\alpha(\\cdot),q_1}_{p_1(\\cdot),\\lambda}$ to $M\\dot{K}^{\\alpha(\\cdot),q_2}_{p_2(\\cdot),\\lambda}$. The proof represents a function $f$ as $\\sum_j \\lambda_j a_j$ with central $(\\alpha(\\cdot),p(\\cdot))$-atoms supported in dyadic balls, bounds $\\|(I_\\beta a_j)\\chi_k\\|_{L^{p_2(\\cdot)}}$ by geometric factors such as $2^{(\\beta-n\\delta_2)(k-j)}$ and $2^{(\\beta-n\\delta_1)(j-k)}$, and then sums over the atom index $j$ and the annulus index $k$ using the hypotheses on $\\alpha(0)$ and $\\alpha_\\infty$.","pith_inferences":["I infer that the stated hypothesis $2\\lambda\\le\\alpha(\\cdot)$ is likely too weak: the displayed geometric sums require $\\alpha_\\infty>2\\lambda$, so the theorem may need a strict inequality or a logarithmic factor in the borderline case.","The dyadic split through $\\alpha(0)$ and $\\alpha_\\infty$ suggests that analogous boundedness should hold for fractional maximal operators or other Riesz-type kernels on the same spaces.","The two exponents $\\delta_1,\\delta_2$ from the log-Hölder estimates determine the admissible $\\beta$ range, so improved estimates for characteristic functions would automatically widen the theorem's scope."],"forward_implications":["If the theorem is correct, the Riesz potential gives a bounded embedding $HM\\dot{K}^{\\alpha(\\cdot),q_1}_{p_1(\\cdot),\\lambda}\\to M\\dot{K}^{\\alpha(\\cdot),q_2}_{p_2(\\cdot),\\lambda}$ that mirrors the classical Sobolev-type index relation $1/p_2=1/p_1-\\beta/n$.","The result extends the boundedness of singular integral operators on these spaces to the fractional integral case, filling the gap the introduction identifies.","The atomic-decomposition proof offers a route to boundedness for other convolution operators whose kernels obey a $|x-y|^{-n+\\beta}$ size estimate, under the same index conditions.","The inequality is homogeneous in the function norm, so it yields a genuine operator norm bound rather than a conditional estimate."],"supporting_citations":[{"why":"Supplies the atomic decomposition (Theorem 1) and the definition of the Herz-Morrey-Hardy space that the proof relies on.","marker":"[5]"},{"why":"Provides the boundedness of the Riesz potential on variable exponent Lebesgue spaces, used as the $L^{p_1}$-to-$L^{p_2}$ hypothesis and for estimating atom norms.","marker":"[9]"},{"why":"Gives the inequalities for characteristic functions (2.4)-(2.6) used to compare dyadic norms throughout the proof.","marker":"[10]"},{"why":"Introduces variable exponent Lebesgue spaces and the Hölder inequality (2.7) used to split sums in the estimates.","marker":"[1]"}],"fun_headline_variants":["Riesz potential bounded on Herz-Morrey-Hardy spaces","Boundedness of Riesz potential on variable exponent spaces","Riesz potential maps Herz-Morrey-Hardy to Herz-Morrey","Atomic decomposition yields Riesz potential bound","Variable exponent Herz-Morrey-Hardy Riesz bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof repeatedly needs the strict inequality $\\alpha_\\infty > 2\\lambda$ to make geometric sums converge, whereas the theorem states only $2\\lambda \\le \\alpha(\\cdot)$.","fun_headline_variants_meta":{"raw":{"variants":["Riesz potential bounded on Herz-Morrey-Hardy spaces","Boundedness of Riesz potential on variable exponent spaces","Riesz potential maps Herz-Morrey-Hardy to Herz-Morrey","Atomic decomposition yields Riesz potential bound","Variable exponent Herz-Morrey-Hardy Riesz bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1470,"prompt_tokens":929,"completion_tokens":541,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":459}},"tokens_in":545,"tokens_out":541,"duration_ms":4663,"temperature":1.0,"reasoning_tokens":459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:47:15.250631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the borderline case $\\alpha(\\cdot)\\equiv 2\\lambda$: in the proof's $H_1$ estimate the inner sums become partial sums that grow linearly in $L$, so the supremum over $L$ is infinite, contradicting the claimed inequality unless a cancellation not present in the proof occurs.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the atomic decomposition (Theorem 1) and the definition of the Herz-Morrey-Hardy space that the proof relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the boundedness of the Riesz potential on variable exponent Lebesgue spaces, used as the $L^{p_1}$-to-$L^{p_2}$ hypothesis and for estimating atom norms."},{"cited_title":"Czechoslovak Math","cited_arxiv_id":null,"evidence_quote":"Introduces variable exponent Lebesgue spaces and the Hölder inequality (2.7) used to split sums in the estimates."}],"review_version":1}