{"id":"6be15456-ebfe-43c1-b8fe-0a44b6662bf4","arxiv_id":"2411.12849","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A quantitative reverse Hölder inequality is proved for scalar A_{p(·)} weights, and it is applied to show matrix A_{p(·)} weights are both right and left open.","lead":"Researchers proved a reverse Hölder inequality for variable exponent Muckenhoupt weights, a key structural property that was missing for this class. The result yields right and left 'openness' for these weight classes, extending known matrix weight theory and giving a new result even for scalar weights.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof has a homogenization gap: the per-cube normalization is undone incorrectly, so the claimed Q-independent constant is not justified.","rationale":"The reader's weakest_assumption targets log-Hölder continuity. That hypothesis is certainly necessary for the comparison lemmas, but it is not the place where the proof of Theorem 1.1 breaks: the homogenization error occurs even under the full log-Hölder assumption and in the constant-exponent case. The paper's Lemma 3.5 and the modular estimates appear plausible, and the endpoint s=r discrepancy flagged by the reader is easily repaired by applying Theorem 1.1 directly. However, the per-cube normalization in the proof of Theorem 1.1 is not undone correctly: the constants obtained under the normalization depend on the normalized weight's norm on the fixed cube Q0, and the subsequent global normalization by Q0 cannot remove that dependence because the two normalizations interact. A concrete family such as w(x)=|x|^α with constant exponent shows the intermediate constant would blow up for small cubes, so the claimed Q-uniform bound does not follow from the argument as written. The central claim may be true and repairable, but the current proof leaves it unverified.","tokens_in":28878,"tokens_out":23254,"duration_ms":227949,"concrete_test":"Track the homogenization factor explicitly through (4.5)-(4.13). Let λ_Q = |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} and w0 = w/λ_Q. Reproduce the final bound of the second case with w replaced by w0, then multiply the resulting inequality by λ_Q and compare the constant with the displayed J in the paper. If the correct constant is C max{1, (∥wχ_{Q0}∥/λ_Q)^{p_+/p_-}} rather than C max{1, ∥wχ_{Q0}∥^{p_+/p_-}}, the 'undo the homogenization' step is invalid. A concrete sanity check: n=1, p(·)=2, w(x)=|x|^{1/4}, Q=[-ℓ,ℓ], Q0=[-e,e]; then w∈A_2, λ_Q≈ℓ^{1/4}, while ∥wχ_{Q0}∥_2 is a fixed positive constant, so the correct constant tends to infinity as ℓ→0, contradicting the claimed uniformity.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The main theorem's proof is not internally sound at the homogenization step. In Section 4, after fixing Q, the authors normalize so that λ_Q := |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} = 1. Under this per-cube normalization, the displayed estimate before the word 'Therefore' contains max{1, ∥wχ_{Q0}∥_{L^{p(·)}}^{p_+/p_-}}. The correct way to undo this normalization is to replace w by w/λ_Q; the constant for the original w then becomes max{1, (∥wχ_{Q0}∥/λ_Q)^{p_+/p_-}}. The paper instead writes max{1, ∥wχ_{Q0}∥^{p_+/p_-}} and then removes this dependence by a second, global normalization ∥wχ_{Q0}∥ = 1. The two normalizations are incompatible: after the global normalization, the per-cube normalized weight has Q0-norm 1/λ_Q, which is not controlled. This is a proof gap that occurs under the stated hypotheses, not a failure of log-Hölder continuity. The reader's endpoint issue is real but minor by comparison.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a quantitative reverse Hölder inequality for variable-exponent Muckenhoupt weights: for p(·) log-Hölder continuous with p_+<∞ and w∈A_{p(·)}, there is r>1 and C_{p(·)} such that |Q|^{-1/(r p_Q)}∥wχ_Q∥_{L^{r p(·)}} ≤ C_{p(·)} |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} for all cubes Q. The proof strategy is to convert the A_{p(·)} condition into a quantitative A_∞-type condition for the measure w(x)^{p(x)}dx (Lemma 3.5), apply a known classical reverse Hölder inequality (Lemma 4.1), and then pass from modular estimates to norm estimates using log-Hölder continuity. The paper then derives right- and left-openness for scalar and matrix A_{p(·)} classes via reducing operators and averaging operators.","tokens_in":29075,"tokens_out":14595,"duration_ms":136736,"significance":"If the main theorem is correct, this is the first quantitative reverse Hölder inequality for variable-exponent Muckenhoupt weights, and the resulting openness statements for matrix weights would be new even in the scalar case. The manuscript is carefully written and largely self-contained: constants are tracked explicitly in Lemmas 2.11, 3.3, and 3.5, and the reduction to the classical sharp reverse Hölder inequality is conceptually sound. However, the proof of Theorem 1.1 contains a normalization gap that is load-bearing, and since Corollary 4.4 and the matrix results in Section 6 depend on Theorem 1.1, the current version does not establish the stated conclusions.","major_comments":[{"comment":"The normalization argument has a gap. The reduction \"we may assume |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}} = 1\" is per cube, so when the normalization is undone, every constant that was computed under that assumption must be evaluated at w/λ_Q, where λ_Q = |Q|^{-1/p_Q}∥wχ_Q∥_{L^{p(·)}}. In particular, the factor max{1, ∥wχ_{Q_0}∥_{L^{p(·)}}^{p_+/p_-}} appearing in the estimate just before the word \"Therefore\" should be replaced by max{1, (∥wχ_{Q_0}∥_{L^{p(·)}}/λ_Q)^{p_+/p_-}}. This quantity is not controlled by [w]_{A_{p(·)}}. The subsequent global normalization v = w/∥wχ_{Q_0}∥_{L^{p(·)}} does not repair the issue: in proving inequality (4.14) for v, the per-cube normalization would introduce the factor 1/λ_Q^v, where λ_Q^v = |Q|^{-1/p_Q}∥vχ_Q∥_{L^{p(·)}}, and no bound for this factor is supplied. Thus the claimed Q-independent constants C_{p(·)} and r in Theorem 1.1 are not justified by the proof as written.","section":"Section 4, proof of Theorem 1.1"},{"comment":"The applications inherit the normalization gap. Corollary 4.4 uses Theorem 1.1 directly, and Lemmas 5.11 and 5.12 use Corollary 4.4, so Theorems 1.5 and 1.6 are conditional on a repaired proof of Theorem 1.1. The manuscript should either supply the missing control of the per-cube normalization or state explicitly the weaker reverse Hölder inequality that the present argument actually proves, with constants depending on the uncontrolled ratio, and then revisit which applications survive.","section":"Section 6 and Corollary 4.4"}],"minor_comments":[{"comment":"The name \"Neugebauer\" is misspelled as \"Neugeabauer\" in the abstract and in the first paragraph of Section 1.","section":"Abstract and Section 1"},{"comment":"The heading \"The Reverse H\\\"older Inequality in V ariable Lebesgue Spaces\" contains an erroneous space in \"V ariable\"; it should read \"Variable\".","section":"Section 4 heading"},{"comment":"Reference [12] is formatted as \"arXiv2408.12745\"; it should be \"arXiv:2408.12745\".","section":"References"},{"comment":"The statement of Lemma 5.11 says the implicit constant depends only on d, p(·), C_∞, C_* and [W]_{A_{p(·)}}, but the proof invokes Corollary 4.4, whose constant contains [1]_{A_{v(·)}} with v(·) depending on s. Therefore the constant depends on s through v(·). This does not invalidate the openness applications, but the stated uniformity in s is not supported by the proof.","section":"Lemma 5.11"},{"comment":"The explicit constant for C_D is written as exp(C_0(1 + log2√n)), which is ambiguous; it should presumably be exp(C_0(1 + log(2√n))) or the base of the logarithm should be specified.","section":"Lemma 2.7"}],"recommendation":"major_revision","confidential_remarks":"The central idea of the paper is attractive and the exposition is careful, but the normalization gap in the proof of Theorem 1.1 is directly load-bearing. I am not recommending rejection because the gap may be repairable, e.g., by proving a suitable a priori estimate for the ratio ∥wχ_{Q_0}∥/λ_Q or by restructuring the normalization. However, unless that repair is supplied, the main theorem and all its applications should not be considered established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: the main theorem's proof has a homogenization gap, and the paper as written does not justify the Q-independent constant in (1.2). The reader's take missed this; the endpoint issue with s=r is real but minor by comparison.\n\nWhat's genuinely new: a reverse Hölder inequality for scalar A_{p(·)} weights with explicit constants, and the corollaries on right/left openness for scalar and matrix A_{p(·)} classes. The matrix openness results are new even in the scalar case, as the authors say. The proof strategy is sensible: reduce to a quantitative A∞ condition for W=w^{p(·)} (Lemma 3.5), invoke a sharp scalar reverse Hölder result (Lemma 4.1), and then work through modular estimates. The constant tracking is careful and mostly correct. The paper also honestly flags that the auxiliary maximal operator question is deferred to another paper.\n\nThe soft spot is in Section 4, in the \"undo the homogenization\" step. The proof fixes a cube Q and normalizes λ_Q := |Q|^{-1/p_Q}∥wχ_Q∥_{p(·)} = 1. The estimates in the second case then produce a constant that depends on ∥wχ_{Q0}∥_{p(·)}. When you undo the per-cube normalization, the correct constant contains max{1, (∥wχ_{Q0}∥/λ_Q)^{p+/p−}}, not max{1, ∥wχ_{Q0}∥^{p+/p−}}. The later global normalization v = w/∥wχ_{Q0}∥ does not remove this: for v the per-cube λ_Q(v) = λ_Q(w)/∥wχ_{Q0}∥, so (∥vχ_{Q0}∥/λ_Q(v))^{p+/p−} = (∥wχ_{Q0}∥/λ_Q(w))^{p+/p−}, which is not bounded independently of Q. So the claimed uniform constant in (1.2) is not justified by the argument. This is not a failure of log-Hölder continuity; it is a gap in the normalization logic under the stated hypotheses.\n\nI also checked the endpoint issue: Corollary 4.4 states s∈[1,r), while Theorems 1.5 and 1.6 claim s∈[1,r]. The proofs use Lemma 5.11, which states [1,r], but its proof applies Corollary 4.4. This is likely fixable by a limiting argument or by shrinking r slightly, so it's minor.\n\nWho is this for? Specialists in variable exponent weights and matrix weights. The result, if repaired, would be a useful structural tool. I would not cite it in its current form, but it deserves referee time: the question is significant and the gap looks repairable. I'd send it to a serious referee with instructions to focus on the homogenization step.","headline":"Theorem 1.1's proof has a homogenization gap that breaks the claimed Q-independent constant; the result may be true but this version is not there yet.","tokens_in":29665,"tokens_out":6863,"would_cite":false,"duration_ms":58334,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a quantitative reverse Hölder inequality for variable exponent Muckenhoupt weights $A_{p(\\cdot)}$, with explicit constants, and uses it to establish right and left openness for scalar and matrix weights.","keywords":["variable Lebesgue spaces","Muckenhoupt weights","matrix weights","maximal operators","reverse Hölder inequality","log-Hölder continuity","A_{p(·)} weights","right and left openness"],"falsifier":"Take $n=1$, $p(x)=2+\\delta/\\log(e+|x|)$ (log-Hölder at infinity) and $w(x)=|x|^{-\\alpha}$ for small $\\alpha$, and test inequality (1.2) on cubes centered at the origin with the $r$ and $C_{p(\\cdot)}$ given by Theorem 1.1; a single cube where the ratio exceeds $C_{p(\\cdot)}$ refutes the quantitative claim.","tokens_in":28632,"feed_emoji":"📐","tokens_out":10327,"duration_ms":86984,"temperature":0.7,"pith_summary":"This paper proves a quantitative reverse Hölder inequality for variable exponent Muckenhoupt weights $A_{p(\\cdot)}$. It shows that for every log-Hölder continuous exponent function $p(\\cdot)$ with $p_+<\\infty$ and every scalar weight $w\\in A_{p(\\cdot)}$, there is an exponent $r>1$ such that the normalized $L^{rp(\\cdot)}$ norm of $w$ on a cube is bounded by a constant times the normalized $L^{p(\\cdot)}$ norm, with the constant and $r$ expressed explicitly in terms of the $A_{p(\\cdot)}$ characteristic. A sympathetic reader would care because this is the first quantitative structural result of this kind in the variable exponent setting, and it directly implies that the scalar $A_{p(\\cdot)}$ classes are both right and left open. The same reverse Hölder machinery is then applied to matrix $A_{p(\\cdot)}$ weights, proving right and left openness for those classes as well, a result that is new even in the scalar case.","feed_headline":"Reverse Hölder inequality proved for variable exponent weights","feed_subtitle":"Establishes right and left openness for scalar and matrix A_{p(·)} weights, with all constants explicit.","key_machinery":"The proof is carried by a three-step mechanism. First, Lemma 3.5 converts the $A_{p(\\cdot)}$ condition into an $A_\\infty$-type density estimate: for the measure $W(E)=\\int_E w(x)^{p(x)}\\,dx$, one gets $|E|/|Q| \\le L_2 [w]_{A_{p(\\cdot)}}^{1+2C_\\infty p_+/(p_\\infty p_-)} (W(E)/W(Q))^{1/p_+}$. Second, Lemma 4.1, a quantitative reverse Hölder inequality for constant exponents, turns this density estimate into the modular bound $\\int_Q w(x)^{rp(x)}\\,dx \\le 2 (\\int_Q w(x)^{p(x)}\\,dx)^r$. Third, log-Hölder continuity enters through the Diening condition (Lemma 2.7) and comparison lemmas (2.8–2.11), which allow the proof to replace $|Q|^{1/p_Q}$ by $\\|\\chi_Q\\|_{L^{p(\\cdot)}}$ at every scale; a normalization argument removes any dependence on the weight's norm on a fixed large cube. The same machinery, extended with reducing operators and averaging operators, transfers the scalar estimate to matrix weights.","core_discovery":"The paper's central claim is Theorem 1.1: if $p(\\cdot)\\in P(\\mathbb{R}^n)\\cap LH(\\mathbb{R}^n)$ with $p_+<\\infty$ and $w$ is a scalar $A_{p(\\cdot)}$ weight, then there exist $C_{p(\\cdot)}$ and $r>1$ such that for every cube $Q$,\n$$|Q|^{-1/(r p_Q)}\\|w\\chi_Q\\|_{$L^{{rp(\\cdot)}}$} \\le C_{p(\\cdot)} |Q|^{-1/p_Q}\\|w\\chi_Q\\|_{$L^{{p(\\cdot)}}$}.$$\nThe exponent $r$ and the constant $C_{p(\\cdot)}$ are given explicitly in terms of the $A_{p(\\cdot)}$ characteristic and constants that depend only on the dimension, the exponent function, and its log-Hölder constants. When the exponent function is constant, the inequality reduces to the classical reverse Hölder inequality for $A_p$ weights. The paper then derives scalar right and left openness (Corollaries 1.3 and 1.4) and, via reducing and averaging operators, the same openness properties for matrix $A_{p(\\cdot)}$ weights (Theorems 1.5 and 1.6).","pith_inferences":["If the theorem is right, the same three-step mechanism should extend to spaces of homogeneous type, since none of the steps is specific to Euclidean cubes beyond the covering estimates.","The normalization argument that removes the weight's norm on a fixed cube could be reused to sharpen known weighted norm inequalities for the maximal operator on $L^{p(\\cdot)}$.","A natural testable extension, not pursued in the paper, is whether the constant in (1.2) can be made to approach $1$ as $r\\to 1$, which would align the variable-exponent inequality with the sharp constant-exponent reverse Hölder theory.","The matrix results suggest that every scalar $A_{p(\\cdot)}$ consequence, such as weighted degenerate Sobolev inequalities, can be lifted to matrix weights via the reduction to the scalar weights $w_e=|W(\\cdot)e|$ for unit vectors $e$."],"forward_implications":["Scalar $A_{p(\\cdot)}$ classes are right-open: if $w\\in A_{p(\\cdot)}$, then $w\\in A_{sp(\\cdot)}$ for all $s\\in[1,r]$ with $r>1$ from the reverse Hölder inequality (Corollary 1.3).","Scalar $A_{p(\\cdot)}$ classes are left-open: with $p_->1$, if $w\\in A_{p(\\cdot)}$, then $w\\in A_{q(\\cdot)}$ where $q'(\\cdot)=sp'(\\cdot)$ for all $s\\in[1,r]$ (Corollary 1.4).","Matrix $A_{p(\\cdot)}$ classes are right-open and left-open (Theorems 1.5 and 1.6), extending the scalar results and providing the first openness results for matrix variable-exponent weights.","When $p(\\cdot)$ is constant, Theorem 1.1 reduces to the classical reverse Hölder inequality (1.1), so the result is a genuine extension of the constant-exponent theory."],"supporting_citations":[{"why":"Defines $A_{p(\\cdot)}$ weights and contains the qualitative lemmas (3.2, 3.3, 3.5) that this paper reproves quantitatively.","marker":"[8]"},{"why":"Supplies the variable Lebesgue space tools: norm-modular inequalities, generalized Hölder, and the Diening condition Lemma 2.7.","marker":"[7]"},{"why":"Source of Lemma 2.11, used to relate $|Q|^{1/p_\\infty}$ with $\\|\\chi_Q\\|_{L^{p(\\cdot)}}$ for large cubes.","marker":"[14]"},{"why":"Provides the quantitative reverse Hölder inequality for constant exponents used as Lemma 4.1 to produce the modular estimate.","marker":"[4]"},{"why":"Introduced the reverse Hölder inequality for classical $A_p$ weights, the result being generalized here.","marker":"[3]"},{"why":"Introduced the classical $A_p$ condition from which the variable-exponent classes are modeled.","marker":"[23]"},{"why":"Previous paper defining matrix $A_{p(\\cdot)}$ weights and their averaging and reducing operator characterizations.","marker":"[11]"},{"why":"Provides the Roudenko matrix $A_p$ framework and reducing operators used in the matrix part.","marker":"[26]"},{"why":"Source of the reducing operator theorem (Theorem 5.6) used in the matrix argument.","marker":"[2]"}],"fun_headline_variants":["Reverse Hölder inequality proven for variable exponent weights","Quantitative reverse Hölder bound for variable Muckenhoupt weights","Matrix weight openness follows from reverse Hölder inequality","Variable exponent weights get explicit reverse Hölder constants","New openness for matrix weights via reverse Hölder in variable exponents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the exponent function $p(\\cdot)$ being log-Hölder continuous both locally and at infinity, with $p_+<\\infty$; without that continuity, the comparisons between $|Q|^{1/p_Q}$ and the $L^{p(\\cdot)}$ norm of $\\chi_Q$ that carry the proof from modular to norm estimates fail.","fun_headline_variants_meta":{"raw":{"variants":["Reverse Hölder inequality proven for variable exponent weights","Quantitative reverse Hölder bound for variable Muckenhoupt weights","Matrix weight openness follows from reverse Hölder inequality","Variable exponent weights get explicit reverse Hölder constants","New openness for matrix weights via reverse Hölder in variable exponents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001076,"raw_usage":{"total_tokens":4495,"prompt_tokens":930,"completion_tokens":3565,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":3486}},"tokens_in":546,"tokens_out":3565,"duration_ms":25533,"temperature":1.0,"reasoning_tokens":3486,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:10:11.098694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $n=1$, $p(x)=2+\\delta/\\log(e+|x|)$ (log-Hölder at infinity) and $w(x)=|x|^{-\\alpha}$ for small $\\alpha$, and test inequality (1.2) on cubes centered at the origin with the $r$ and $C_{p(\\cdot)}$ given by Theorem 1.1; a single cube where the ratio exceeds $C_{p(\\cdot)}$ refutes the quantitative claim.","supporting_citations":[{"cited_title":"Cruz-Uribe, A","cited_arxiv_id":null,"evidence_quote":"Defines $A_{p(\\cdot)}$ weights and contains the qualitative lemmas (3.2, 3.3, 3.5) that this paper reproves quantitatively."},{"cited_title":"Cruz-Uribe and A","cited_arxiv_id":null,"evidence_quote":"Supplies the variable Lebesgue space tools: norm-modular inequalities, generalized Hölder, and the Diening condition Lemma 2.7."},{"cited_title":"Diening, P","cited_arxiv_id":null,"evidence_quote":"Source of Lemma 2.11, used to relate $|Q|^{1/p_\\infty}$ with $\\|\\chi_Q\\|_{L^{p(\\cdot)}}$ for large cubes."},{"cited_title":"Coifman and C","cited_arxiv_id":null,"evidence_quote":"Introduced the reverse Hölder inequality for classical $A_p$ weights, the result being generalized here."},{"cited_title":"Muckenhoupt","cited_arxiv_id":null,"evidence_quote":"Introduced the classical $A_p$ condition from which the variable-exponent classes are modeled."},{"cited_title":"Cruz-Uribe and M","cited_arxiv_id":null,"evidence_quote":"Previous paper defining matrix $A_{p(\\cdot)}$ weights and their averaging and reducing operator characterizations."},{"cited_title":"Roudenko","cited_arxiv_id":null,"evidence_quote":"Provides the Roudenko matrix $A_p$ framework and reducing operators used in the matrix part."}],"review_version":1}