{"id":"fee12b56-e632-4178-87cc-332d45bfe3d8","arxiv_id":"2608.15453","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each cube Q, two-weight Poincaré-Sobolev estimates hold with constants given by a new dyadic local A^{r,~r} seminorm, including p>q cases without Muckenhoupt weights.","lead":"This math paper proves weighted Poincaré-Sobolev inequalities on every cube using new local weight classes that do not require the Muckenhoupt condition. It also derives weighted Sobolev embeddings and Gagliardo-Nirenberg interpolation inequalities as applications.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5's Hedberg argument conflates χ_R with χ_{E_R}: for x∈R\\E_R the small-cube oscillation term is not dominated by OSC_{S,α}f, so the critical-case applications are not established by the displayed proof.","rationale":"The reader's weakest_assumption targeted Lemma 4.3 itself, asking whether the sparse-domination decomposition can be applied with constants compatible with the dyadic A-seminorm. My reading accepts Lemma 4.3 as a standard tool for Theorems 3.1 and 3.2, where the proof only uses χ_R and the disjointness of E_R enters through the maximal-function estimate. The more concrete problem appears in Theorem 3.5, a main theorem and the route to the advertised critical-case and Fefferman–Phong-type results. There the proof passes from a χ_R-sum to the χ_E-based OSC_{S,α}f without justification; the sparse condition |R|≤2|E_R| permits large gaps R\\E_R where the two quantities differ by a full oscillation term. Lemma 4.4 is built around the disjointness of E_R, so this is not a harmless notational change. The fix would require either proving a pointwise domination by a maximal-function-modified OSC or developing an L^q estimate for overlapping χ_R sums in Morrey spaces; neither is present. Since Theorems 3.1–3.2 may still be correct and the deficiency is localized to Theorem 3.5 and its applications, the appropriate verdict remains conditional: the central pair of estimates is plausible but the paper's advertised critical-case consequences are not yet supported. This is why I partially agree with the reader: the same sparse-domination machinery is fragile, but the actual failure mode I found is the χ_R/χ_{E_R} mismatch in the Hedberg step rather than the applicability of Lemma 4.3 to Lipschitz functions.","tokens_in":13602,"tokens_out":44787,"duration_ms":394263,"concrete_test":"Check the exact statement of Lemma 5.1 in Lerner–Ombrosi–Rivera-Ríos [12]: if its right-hand side uses χ_{E_R}, then Lemma 4.3 as quoted is stronger than its source and needs an independent proof; if it uses χ_R, then the step |R0|^α OSC_{S,α}f in §5.3 is invalid. Settle the pointwise issue directly: take a one-dimensional cube R=[0,1], sparse set E_R=[0,1/2], f(x)=x, and x=3/4. Then χ_R(x)=1 but χ_{E_R}(x)=0, so the asserted domination by |R0|^α OSC_{S,α}f(x) fails; recompute the Hedberg inequality in §5.3 with this example to verify whether the missing term is absorbed elsewhere.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.3 gives |f(x)-f_Q| ≲ Σ_{R∈S} χ_R(x) ffl_R |f-f_R|. Lemma 4.4 then defines OSC_{S,α}f(x) = Σ_{R∈S} χ_{E_R}(x) |R|^{-α} ffl_R |f-f_R|, where E_R is only required to satisfy |R| ≤ 2|E_R|. In the proof of Theorem 3.5 (§5.3), the small-cube part Σ_{R⊂R0, x∈R} χ_R(x) ffl_R |f-f_R| is asserted to be ≲ |R0|^α OSC_{S,α}f(x). This step requires the pointwise inequality χ_R(x) ≲ χ_{E_R}(x), which is false: E_R may be a proper subset of R, so for x∈R\\E_R the right-hand side vanishes while the left-hand term can be positive. Lemma 4.4's proof relies on the disjointness of E_R and computes ∫_{E_R}; replacing E_R by R introduces overlapping cubes and invalidates the sequence-Hölder step. Thus Theorem 3.5, and the critical-case/Fefferman–Phong consequences that depend on the Hedberg reduction, are not proven by the argument as written. The gap is not a cosmetic typo: the L^q norm of the χ_R-sum is not controlled by the χ_E-sum through Lemma 4.4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a local two-weight class A^{r,~r}_{p,q,ρ}(Q) defined by a dyadic sum over subcubes of Q, and proves local weighted Poincaré–Sobolev inequalities for Lipschitz functions on every cube Q, with the constant given by the new seminorm. Theorem 3.1 covers p ≤ q; Theorem 3.2 covers p > q, with q > 1 using r̃ = r and q ≤ 1 using r̃ = 1. The paper then derives Fefferman–Phong-type weighted inequalities in local Morrey spaces (Theorems 3.3–3.5), states a negative result (Theorem 3.6), and applies the inequalities to two-weighted Sobolev embeddings and Gagliardo–Nirenberg inequalities (Section 7). Section 6 explains how known results of Fabes–Kenig–Serapioni, Chanillo–Wheeden, and Pérez–Rela are recovered by bounding the new seminorm.","tokens_in":13929,"tokens_out":18709,"duration_ms":169935,"significance":"If all claims were proven, the paper would provide a genuinely local and Muckenhoupt-free framework for weighted Poincaré–Sobolev inequalities. The explicit dyadic seminorm gives quantitative constants, the p > q range is not covered by previous two-weight results, and the reduction of known theorems to a single seminorm estimate is conceptually clean. The paper also contains explicit power-weight examples and several applications. However, the validation of the critical cases and of the Sobolev-embedding corollaries is incomplete, as detailed below, so the contribution is not yet established in its full stated form.","major_comments":[{"comment":"The proof of Theorem 3.5 hinges on the pointwise bound osc_S f(x) ≲ |R_0|^α OSC_{S,α}f(x) + Σ_{R⊋R_0} ... . This requires χ_R(x) ≲ χ_{E_R}(x) for the cubes in the sparse family, but the sparse condition |R| ≤ 2|E_R| does not imply that containment. For x ∈ R \\ E_R the term χ_R(x) ffl_R |f-f_R| can be positive while OSC_{S,α}f(x) vanishes. Lemma 4.4 is proved by integrating over the disjoint sets E_R, so the displayed argument cannot be repaired by merely replacing χ_{E_R} with χ_R, since the cubes R overlap and the sequence Hölder step would break. Consequently, the proof of Theorem 3.5, and the critical-case applications that depend on the Hedberg reduction, are not established by the manuscript as written.","section":"§5.3, Lemma 4.4"},{"comment":"The embeddings stated in Corollary 7.6 have the wrong indices. The inequality derived in §7.2 is ||g f||_{L^1} ≲ ||g||_{M^{(p*)'}_{1,p'}} ||∇f||_{L^p}, which gives the embeddings M^{(p*)'}_{1,p'} ⊂ \\dot{W}^{-1,p'} and m^{(p*)'}_{1,p'} ⊂ W^{-1,p'}. Corollary 7.6 instead states M^p_{1,p*} ⊂ \\dot{W}^{-1,p*} and m^p_{1,p*} ⊂ W^{-1,p*}, swapping the upper and lower indices and using the wrong target space. The claimed improvement of the Lorentz–Sobolev embedding is therefore not justified as stated.","section":"Corollary 7.6 and §7.2"},{"comment":"Theorems 3.3 and 3.4 are stated as corollaries of Theorems 3.1 and 3.2, but no proof is given. To apply Theorems 3.1 and 3.2 with w = 1 and v = g, one must bound the A-seminorm [1,g]_{A^{r,~r}_{p,q,ρ}(Q)} by the corresponding Morrey norm ||g||_{M^s_{r,ρ}(Q)}; this is not immediate because the seminorm contains L^{q~r} averages while the Morrey norms use L^r (or L^q) integrability, and the parameter matching changes between the cases p ≤ q and p > q. The reduction should be written out rather than left to the reader.","section":"§3, Theorems 3.3–3.4"},{"comment":"The proof of Theorem 3.6 appeals to 'the argument in Section 7 below' to pass from the local estimate to the global inequality ||g f||_{L^p} ≲ ||g||_{M^n_p} ||∇f||_{L^p}. The limiting argument is not given: one needs to justify the behaviour of f_Q and of the seminorm as |Q| → ∞ for an arbitrary increasing family of cubes, and to show that the constant in the assumed local inequality is uniform. As written, the negative result is not fully demonstrated.","section":"§5.4, Theorem 3.6"}],"minor_comments":[{"comment":"There are numerous typos that should be corrected, including 'Lipchitz', 'Sobolv', 'fmaily', 'amlgam norm', 'disscusion', and 'inequailties'.","section":"Abstract and throughout"},{"comment":"In the displayed Poincaré–Sobolev inequality in the introduction, the right-hand side is written as (1/|Q| ∫_Q |∇f(x)| dx)^{1/p}; the exponent p on the integrand is missing and should be |∇f(x)|^p dx.","section":"§1, displayed inequality"},{"comment":"The displayed definition of ||g||_{M^p_{q,r}} in Section 7.2 contains both a supremum over Q ∈ D and a sum over Q ∈ D, which are not equal; this should be aligned with Definition 2.5.","section":"§7.2, displayed norm"},{"comment":"In Corollary 7.4(1), the assumption uses A^{r,~r,loc}_{p,q,ρ}(R^n_+) but the conclusion uses the non-local seminorm [w,v]_{A^{r,~r}_{p,q,ρ}(R^n_+)}; presumably the assumption should be A^{r,~r}_{p,q,ρ}(R^n_+).","section":"Corollary 7.4(1)"},{"comment":"The sentence 'the double critical case p = q = r does not hold' refers to an undefined variable r; either r should be introduced in the statement or the sentence should be rephrased.","section":"Theorem 3.5"}],"recommendation":"major_revision","confidential_remarks":"The main gap in the proof of Theorem 3.5 is substantive and not a typo; the sparse-family argument in Lemma 4.4 cannot be extended to the χ_R-sum by a cosmetic change. The applications in Section 7.2 contain a systematic index swap that should be corrected. Theorems 3.3 and 3.4 need explicit proofs or a complete reduction to Theorems 3.1 and 3.2. The central Theorems 3.1 and 3.2 look plausible and are worth preserving, so a careful revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [colleague],\n\nThe headline: the paper gives a real extension of two-weight local Poincaré–Sobolev inequalities, including the p>q case, using cube-dependent weight classes that do not require Muckenhoupt or A_∞. But as posted, Theorem 3.5 has a genuine gap and Corollary 7.6 has an index error. The core Theorems 3.1–3.2 look plausible and worth referee time.\n\nWhat is actually new: the seminorm [w,v]_{A^{r,\\tilde r}_{p,q,\\rho}(Q)} is a sensible local object, and the p>q range is not covered by Lorist–Wagenaar or Pérez–Rela. Section 6 is the right kind of comparison: it recovers Fabes–Kenig–Serapioni, Chanillo–Wheeden, and Pérez–Rela by bounding the new seminorm, which confirms the generalization is meaningful. The proofs of 3.1–3.2 follow the standard sparse-domination route; Sections 5.1–5.2 are compressed but the sequence-Hölder steps look correct on a careful read.\n\nSoft spots, in proportion.\n\nFirst, Theorem 3.5. The Hedberg step as written is not valid. In Section 5.3 the small-cube sum in osc_S f is bounded pointwise by |R_0|^α OSC_{S,α}f(x), but OSC_{S,α} uses χ_{E_R}, not χ_R, and E_R can be a proper subset of R. For x∈R\\E_R the right-hand term vanishes while the left-hand term can be positive. The pointwise inequality fails, so the L^q bound does not follow from Lemma 4.4. This is not a cosmetic typo: Lemma 4.4 relies on disjointness of the E_R, and swapping in χ_R introduces overlaps that break the sequence-Hölder step. The Fefferman–Phong-type critical consequences in Theorem 3.5 are therefore not established as written. A repair may be possible with a sparse maximal-function argument, but it needs to be written.\n\nSecond, Corollary 7.6. The indices look swapped. The L^1 inequality in Section 7.2 gives an embedding of M^{(p^*)'}_{1,p'} into \\dot{W}^{-1,p'}, not M^p_{1,p^*} into \\dot{W}^{-1,p^*}. As stated, both the Morrey parameter and the dual Sobolev exponent are off.\n\nThird, Theorem 3.6 is terse: the forward reference to Section 7 is understandable, but the local-to-global passage should be spelled out.\n\nBottom line: Theorems 3.1–3.2 are the core and appear sound. The applications section is rougher and contains at least one unproven corollary and one index error. A good referee can verify the core and help fix the applications. I would send this to peer review and ask for revision rather than reject.","headline":"Genuine advance on local two-weight Poincaré–Sobolev inequalities, with a real gap in Theorem 3.5 and an index error in Corollary 7.6; the core theorems deserve a serious referee, not a desk reject.","tokens_in":14442,"tokens_out":6857,"would_cite":false,"duration_ms":58153,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["42B35","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a two-weight Poincaré–Sobolev inequality on every cube, with cube-dependent local weights and no Muckenhoupt condition, covering both p ≤ q and p > q.","keywords":["Poincaré–Sobolev inequality","two-weight inequalities","local weights","Muckenhoupt weights","sparse domination","Morrey spaces","Sobolev embedding theorem","dyadic weight classes"],"falsifier":"Take $n=1$, $p=2$, $q=1$, $r=1$, $\\rho=1$, and the power weights $(|x|^\\alpha, |x|^\\beta)$ with parameters strictly inside the region allowed by Example 2.6, so the seminorm $[w,v]_{A^{1,1}_{2,1,1}(Q)}$ is finite and fixed. If a sequence of Lipschitz functions $f_j$ makes the ratio $\\|v(f_j-(f_j)_Q)\\|_{L^1(Q)} / \\| |x|^\\alpha \\nabla f_j\\|_{L^2(Q)}$ grow without bound, the main theorem fails. Alternatively, test Lemma 4.3 directly on a fixed nonsmooth Lipschitz function $f$; the sparse domination must hold with an absolute constant independent of $f$ and $Q$, and any example where that constant must depend on $f$ destroys the proof.","tokens_in":13403,"feed_emoji":"📐","tokens_out":16015,"duration_ms":136985,"temperature":0.7,"pith_summary":"This paper proves a two-weight local Poincaré–Sobolev inequality: for any cube $Q\\subset \\mathbb{R}^n$, any Lipschitz function $f$, and any pair of weights $(w,v)$ that may depend on $Q$, the $v$-weighted oscillation of $f$ in $L^q(Q)$ is controlled by the $w$-weighted $L^p$ norm of $\\nabla f$, with the constant given by a new dyadic seminorm $[w,v]_{A^{r,\\tilde r}_{p,q,\\rho}(Q)}$. The result covers both $p\\le q$ and $p>q$ (including $q\\le 1$), and it requires no Muckenhoupt ($A_p$) or local $A_\\infty$ condition on the weights. Because the weight condition is local — a sum over dyadic subcubes of $Q$ — singular or degenerate weights are admissible. A limiting double critical case ($p=q$) is shown to fail, marking the boundary of the method. From the main estimate the author derives Fefferman–Phong-type inequalities with Morrey norms, two-weight Sobolev embeddings on $\\mathbb{R}^n$ and on the half-space, and weighted Gagliardo–Nirenberg interpolations that improve the classical Lorentz–Sobolev embedding.","feed_headline":"Cube-local weights prove two-weight Poincaré–Sobolev inequality","feed_subtitle":"A dyadic seminorm on each cube controls the v-weighted oscillation by the w-weighted gradient in L^p, for p<q and p>q.","key_machinery":"The decisive object is the dyadic local weight class $A^{r,\\tilde r}_{p,q,\\rho}(Q)$, a seminorm that measures, on each dyadic subcube $R$ of $Q$, the mixed averages of $w^{-p'r}$ and $v^{q\\tilde r}$ scaled by $\\ell_R |R|^{1/p-1/q}$, and then takes the $\\ell^\\rho$ sum over the dyadic tree. The proof of the main theorems runs as follows: a pointwise sparse domination lemma (Lemma 4.3) rewrites the oscillation $|f-f_Q|$ as a sum over a sparse family of dyadic cubes; an integral representation (Lemma 4.2) extracts $|\\nabla f|$ from the oscillation; Hölder steps place the averages of $w^{-p'r}$ and $v^{q\\tilde r}$ into the seminorm; and the $L^p$ boundedness of the local Hardy–Littlewood maximal operator (Lemma 4.1) converts the sparse sum into $\\|w|\\nabla f|\\|_{L^p(Q)}$. The critical cases (Theorem 3.5) additionally use a Hedberg-type truncation (Lemma 4.4) on the modified sparse oscillation $\\mathrm{OSC}_{S,\\alpha}f$.","core_discovery":"The central assertion is Theorems 3.1 and 3.2: for $1<p<\\infty$ and $0<q<\\infty$, whenever $(w,v)$ belongs to the dyadic local class $A^{r,\\tilde r}_{p,q,\\rho}(Q)$ — the class whose seminorm is the $\\ell^\\rho$ sum over dyadic subcubes $R$ of $Q$ of $\\ell_R |R|^{1/p-1/q} (\\fint_R w^{-p'r})^{1/(p'r)} (\\fint_R v^{q\\tilde r})^{1/(q\\tilde r)}$ — every Lipschitz $f$ on $Q$ satisfies $\\|v(f-f_Q)\\|_{L^q(Q)} \\le C\\, [w,v]_{A^{r,\\tilde r}_{p,q,\\rho}(Q)} \\|w|\\nabla f|\\|_{L^p(Q)}$. The case $p\\le q$ uses $\\rho=\\infty$ and $\\tilde r=r$; the case $p>q$ uses $1/q=1/p+1/\\rho$, with $\\tilde r=1$ when $q\\le 1$. No Muckenhoupt, local $A_\\infty$, or Fujii–Wilson-type condition is assumed, and the weight pair is allowed to depend on the cube $Q$ itself. Corollaries convert the left-hand weight $v$ into a Morrey-space multiplier $g$, giving $\\|g(f-f_Q)\\|_{L^q(Q)} \\le \\|g\\|_{M^s_r(Q)}\\|\\nabla f\\|_{L^p(Q)}$, and from these the author obtains homogeneous and inhomogeneous two-weight Sobolev embedding theorems, including on the upper half-space.","pith_inferences":["The conditions in Example 2.6 look necessary as well as sufficient for power weights: at the stated boundary the dyadic sum diverges while the inequality should fail, so the $A^{r,\\tilde r}$ seminorm likely characterizes the estimate quantitatively, although the paper does not prove necessity.","The same sparse-domination route should transfer to Riesz potentials of fractional order, giving fractional (two-weight) Poincaré–Sobolev inequalities with an analogous dyadic class; the parameter $\\alpha$ in Lemma 4.4 already anticipates this extension.","Since the weight condition is local to each cube, the inequality is a natural tool for Moser–Harnack arguments in degenerate elliptic equations whose coefficients satisfy the dyadic local condition rather than uniform ellipticity; the paper does not make this connection.","The new inhomogeneous class $A^{r,\\tilde r,\\mathrm{loc}}_{p,q,\\rho}(\\mathbb{R}^n)$ should provide two-weight local Sobolev embeddings on bounded domains where the typical $A_\\infty$ hypothesis is replaced by a finite dyadic sum over cubes of size at most one."],"forward_implications":["Weighted oscillation estimates now hold for singular or degenerate pairs $(w,v)$ as long as the dyadic $A^{r,\\tilde r}$ sum is finite, going beyond the Muckenhoupt framework.","The range $p>q$ is included, with the parameter $\\rho$ linking $p$ and $q$; previous local two-weight results covered only $1\\le p\\le q$ and assumed a Fujii–Wilson-type local $A_\\infty$ condition.","Letting the cube expand to $\\mathbb{R}^n$ yields two-weight Sobolev embeddings: homogeneous ones under a global $A^{r,\\tilde r}$ condition and inhomogeneous ones under the new local condition (cubes of size at most one), both also on the half-space.","The Fefferman–Phong-type corollaries give $\\|g(f-f_Q)\\|_{L^q(Q)} \\lesssim \\|g\\|_{M^s_r(Q)}\\|\\nabla f\\|_{L^p(Q)}$; the case $q=1$ embeds Bourgain–Morrey spaces into negative Sobolev spaces, improving the Lorentz–Sobolev embedding $L^{p,p^*}\\hookrightarrow \\dot{W}^{-1,p^*}$.","The double critical case $p=q$ in Theorem 3.5 is shown to fail (Theorem 3.6), marking the boundary of what the dyadic method can prove."],"supporting_citations":[{"why":"Supplies the pointwise sparse-domination lemma (Lemma 4.3) that is the foundation of the proof of Theorems 3.1 and 3.2.","marker":"[12]"},{"why":"Supplies the integral representation (Lemma 4.2) that extracts the gradient from the oscillation and is used throughout the proof.","marker":"[11]"},{"why":"Origin of the $A^{r,\\tilde r}_{p,q}$ weight family that the paper adapts to a cube-dependent local seminorm.","marker":"[14]"},{"why":"The prior local two-weight result (with local $A_\\infty$) that the paper's theorems extend by removing the $A_\\infty$ condition.","marker":"[13]"},{"why":"The degenerate one-weight framework whose corollaries the paper refines into the local two-weight results.","marker":"[20]"},{"why":"Supplies the counterexample quoted in Theorem 3.6 that shows the double critical case fails.","marker":"[23]"}],"fun_headline_variants":["Local cube weights yield Poincaré–Sobolev without Muckenhoupt","Two-weight Poincaré–Sobolev holds with cube-dependent weights","Local weight class replaces Muckenhoupt in Poincaré–Sobolev","Dyadic local seminorm alone proves two-weight Poincaré–Sobolev","Without Muckenhoupt: cube-local weights give Poincaré–Sobolev"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a quoted sparse-domination lemma: the oscillation $|f-f_Q|$ of any Lipschitz function is dominated pointwise on $Q$ by a sparse sum of dyadic average oscillations, with a constant independent of $f$ and $Q$; if that lemma cannot be applied to arbitrary Lipschitz functions with the constant compatible with the dyadic $A$-seminorm, the main theorems do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Local cube weights yield Poincaré–Sobolev without Muckenhoupt","Two-weight Poincaré–Sobolev holds with cube-dependent weights","Local weight class replaces Muckenhoupt in Poincaré–Sobolev","Dyadic local seminorm alone proves two-weight Poincaré–Sobolev","Without Muckenhoupt: cube-local weights give Poincaré–Sobolev"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000913,"raw_usage":{"total_tokens":3955,"prompt_tokens":1015,"completion_tokens":2940,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":2827}},"tokens_in":631,"tokens_out":2940,"duration_ms":21129,"temperature":1.0,"reasoning_tokens":2827,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T20:52:50.003720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $n=1$, $p=2$, $q=1$, $r=1$, $\\rho=1$, and the power weights $(|x|^\\alpha, |x|^\\beta)$ with parameters strictly inside the region allowed by Example 2.6, so the seminorm $[w,v]_{A^{1,1}_{2,1,1}(Q)}$ is finite and fixed. If a sequence of Lipschitz functions $f_j$ makes the ratio $\\|v(f_j-(f_j)_Q)\\|_{L^1(Q)} / \\| |x|^\\alpha \\nabla f_j\\|_{L^2(Q)}$ grow without bound, the main theorem fails. Alternatively, test Lemma 4.3 directly on a fixed nonsmooth Lipschitz function $f$; the sparse domination must hold with an absolute constant independent of $f$ and $Q$, and any example where that constant must depend on $f$ destroys the proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise sparse-domination lemma (Lemma 4.3) that is the foundation of the proof of Theorems 3.1 and 3.2."},{"cited_title":"Jost, Partial Differential Equations, Springer New York, GTM,214(2012)","cited_arxiv_id":null,"evidence_quote":"Supplies the integral representation (Lemma 4.2) that extracts the gradient from the oscillation and is used throughout the proof."},{"cited_title":"Moen, Weighted inequalities for multilinear fractional integral operators, Collect","cited_arxiv_id":null,"evidence_quote":"Origin of the $A^{r,\\tilde r}_{p,q}$ weight family that the paper adapts to a cube-dependent local seminorm."},{"cited_title":"P´ erez and E","cited_arxiv_id":null,"evidence_quote":"The degenerate one-weight framework whose corollaries the paper refines into the local two-weight results."},{"cited_title":"Sawano, S","cited_arxiv_id":null,"evidence_quote":"Supplies the counterexample quoted in Theorem 3.6 that shows the double critical case fails."}],"review_version":1}