{"id":"c393949d-ce93-4ff0-9b75-3f68e9c48d0b","arxiv_id":"1908.02602","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Estimates for BMO and A_p on an interval transfer to identical dimension-free estimates for their heat and Poisson kernel versions, yielding a polynomial lower bound for the John-Nirenberg constant on balls.","lead":"The paper proves that any estimate known for BMO or for Muckenhoupt A_p weights on an interval transfers automatically to their heat-kernel and Poisson-kernel versions on R^n, with the same explicit constant. The authors use this transference to show that the John-Nirenberg constant of BMO on balls decays no faster than n^{-1/2} in dimension n.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central transference (3.1) is a black-box consequence of [10, Th. 5.1]; the paper never verifies that theorem covers the discontinuous f=χ_{|s|>λ} on which Theorem 4.4 depends.","rationale":"The reader's weakest assumption was the imported local-concavity characterization; I agree partially. My concern sharpens it: the theorem is used not only with smooth or exponential f but with an indicator, where boundary discontinuities can interact with the notion 'minimal locally concave function.' No evidence of an internal inconsistency in Lemma 5.2 was found: the stochastic martingale argument, the affine-mollifier construction, and the Fatou/liminf step are coherent. The nonemptiness of F_{x,μ̃,z0} is indeed easy (constant plus a scaled sign-function of a coordinate gives any x∈Ω_μ̃ for heat/Poisson). Thus the entire question is whether [10]'s theorem has the breadth claimed. If it does, the paper's central claim and application stand; if not, the headline result needs an additional approximation argument. I do not regard the concern as disproving the paper, so the reader's ACCEPT remains appropriate, pending the cited theorem's hypothesis check.","tokens_in":11771,"tokens_out":40601,"duration_ms":441322,"concrete_test":"Read the theorem on p.230 of Stolyarov–Zatitskii [10] and list precisely the hypotheses on the boundary data f. If arbitrary measurable f is covered, the objection fails. If not, test the specific f=χ_{|s|>λ} used in Theorem 4.4: approximate it in (3.3) by continuous f_ε with C_{f_ε}(µ)→e^{1−λ/µ}, and check whether the semigroup bound passes to the limit; equivalently, compute B* for f=χ_{|s|>1} on Ω_1 and test whether it equals the minimal locally concave function with that boundary value.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument for (3.1) is sound conditional on Theorem 5.1: given that B* is locally concave, Lemma 5.2 supplies the semigroup/Jensen inequality and the supremum over F_{x,μ̃,z0} yields B^K≤B*. The soft spot is Theorem 5.1 itself, imported from [10] and stated for arbitrary non-negative measurable f. The principal application Theorem 4.4 uses Corollary 3.3 with f(s)=χ_{|s|>λ}, a discontinuous indicator. If the local-concavity characterization in [10] (p.230) is proved only for continuous (or bounded/lower-semicontinuous) boundary data, then the step (3.3)→(3.4) is not justified for this f, and with it the proof of ε_JN^K(n)≥1 and the n^{-1/2} lower bound. An approximation argument might repair the gap, but none is supplied in the paper; the cited theorem's hypotheses are not checked. The internal Lemma 5.2 proof, including the mollifiers (5.3),(5.5), is not the issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a transference principle for integral estimates on BMO and A_p classes. For the heat or Poisson kernel K on R^n, it shows that any one-dimensional estimate of the form ⟨f(η−⟨η⟩_I)⟩_I ≤ C_f(µ) for η∈BMO(I) automatically yields the corresponding K-semigroup estimate f(φ−φ(z))(z) ≤ C_f(µ) for φ∈BMO^K(R^n), and similarly for A_p weights (Theorems 3.1 and 3.2, Corollaries 3.3 and 3.4). The main tool is the local concavity of interval Bellman functions from [10] combined with a semigroup version of Jensen's inequality (Lemma 5.2) proved by Itô calculus and mollification. The applications include a comparison ‖φ‖_H ≲ √n ‖φ‖_∗ (Proposition 4.1), estimates on balls (Corollary 4.3), and a lower bound ε_JN^*(n) ≳ n^{-1/2} for the John–Nirenberg constant of BMO_∗ on balls (Theorem 4.4).","tokens_in":11914,"tokens_out":19886,"duration_ms":220454,"significance":"If the quoted local-concavity theorem covers all nonnegative measurable boundary data, the results are sound and represent a genuine conceptual advance: they replace dyadic dimension-dependent arguments by a Bellman-function transfer that is dimension-free by construction, and they improve the known dimensional behavior of the John–Nirenberg constant for balls from exponential decay to polynomial decay n^{-1/2}. The semigroup/martingale proof of Lemma 5.2 is carefully structured (smooth, continuous, general cases), and the mollification arguments exploit the homogeneity of the BMO and A_p domains in a natural way. The paper also gives explicit, checkable constants and does not rely on any numerical computation. The main caveat is that the central inequality (3.1) is conditional on an imported theorem from [10], and the manuscript does not verify the hypotheses of that theorem for the discontinuous boundary data used in the headline application.","major_comments":[{"comment":"The inequality (3.1) is deduced entirely from Theorem 5.1, which is quoted from [10] and asserts that B_∗(·;µ,f) is the minimal locally concave function on Ω_µ with boundary values f. Theorem 3.1 is stated for arbitrary nonnegative measurable f, and Theorem 4.4 then applies Corollary 3.3 with the discontinuous function f(s)=χ_{|s|>λ}. The manuscript does not check that the cited theorem from [10] indeed covers such boundary data, nor does it supply an approximation argument. If the local-concavity characterization is proved only for continuous or lower-semicontinuous boundary values, then the step from (3.3) to (3.4) is not justified for this f, and with it the proof of (4.9)–(4.10) fails. Please state the exact hypotheses of the theorem from [10], verify that they include the indicator function, or add a separate limiting argument for discontinuous f. This is the only point that I cannot verify from the manuscript itself.","section":"Section 5, Theorem 5.1 and Theorem 3.1; also Section 4, Theorem 4.4"}],"minor_comments":[{"comment":"The displayed hypothesis appears to have a scaling typo: the proof requires ‖φ‖_H < µ, and Proposition 4.1 gives ‖φ‖_H ≲ √n ‖φ‖_∗, so the assumption should read ‖φ‖_∗ ≲ µ/√n, not ‖φ‖_∗ ≲ µ√n.","section":"Corollary 4.3"},{"comment":"The notation (f∘φ)(z) in the definition of B^K and D_{p,K} should be defined explicitly as the K-extension of the map y↦f(φ(y)). It is only implicit from the convention in Section 1 and is central to the interpretation of Corollaries 3.3 and 3.4.","section":"Section 2, (2.2) and (2.5)"},{"comment":"In the construction of V_j via (5.5), the statement that V_j is defined on Ω_{μ̄} holds for all sufficiently large j (for j=1 the shift can reach the upper boundary x_2−x_1^2=µ^2). This is harmless but should be phrased as 'for j large enough'.","section":"Lemma 5.2, general case"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The transference result is the real thing. If an integral functional satisfies an estimate on BMO(I) or A_p(I), the same constant works for the heat- or Poisson-smoothed versions on R^n, with Lebesgue averages replaced by kernel averages. That makes dimension-free estimates routine for a large class of functionals, and the semigroup Bellman functions B^K and D_{p,K} were previously unstudied. The paper is honest about what it does not do—no Bellman functions are computed—and the payoff is a genuinely new lower bound on the John-Nirenberg constant for balls, eps_*^{JN}(n) >~ n^{-1/2}, beating the exponential decay that dyadic methods give.\n\nThe proof of the key lemma is solid. The three-stage approximation—smooth via Ito, continuous via mollifiers (5.3), and general via the shift (5.5)—is carefully handled, and the martingale representation for the heat and Poisson kernels is clean. The A_p side is symmetric and looks correct. I see no circularity: Theorem 5.1 is imported from the authors' own published work, not derived from the present claims, and the new inequalities (3.1)-(3.2) do not reduce to that theorem by construction.\n\nThe one soft spot is exactly the one the stress-test flags. Theorem 5.1 is stated for arbitrary non-negative measurable f, but the paper never verifies that the theorem from [10] covers discontinuous boundary data. Theorem 4.4 uses f(s)=chi_{|s|>lambda}, so if [10]'s local-concavity characterization requires continuity or some other regularity, the step from Corollary 3.3 to (4.8) is unjustified. The internal proof of Lemma 5.2 does not care about f's regularity, so the gap is entirely in the black-box import. An approximation argument (e.g., regularizing f with the same mollifiers) might repair it, but the paper should state it. I would not call this fatal—it is a minor-to-moderate rigor issue that a referee should ask the authors to address.\n\nThis paper is for people who do Bellman functions, BMO, and A_p weights; the transference principle is likely to be used elsewhere. It deserves a serious referee. My own verdict is a qualified accept: clear the hypothesis check on the imported theorem and I'd be happy to see it in print.","headline":"Genuinely new transference principle for semigroup BMO/A_p, clean proofs, real n^{-1/2} John-Nirenberg bound on balls; only real issue is a black-box imported theorem whose hypotheses aren't checked for the discontinuous f used in the application.","tokens_in":12550,"tokens_out":4350,"would_cite":true,"duration_ms":47705,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42A05","42B35","42A61","49K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any integral estimate for BMO or A_p on an interval transfers unchanged to heat- and Poisson-semigroup versions on R^n, giving dimension-free bounds and a John–Nirenberg decay no faster than 1/sqrt(n).","keywords":["BMO","Ap weights","dimension-free estimates","Bellman functions","heat kernel","Poisson kernel","John-Nirenberg constant","semigroup transference"],"falsifier":"Take $f(t)=e^t$, $\\mu=1$, and a point $x=(0,s)$ inside $\\Omega_{1/2}$; from the explicit one-dimensional Bellman function $B_*$ for this $f$, compute the right side of Theorem 3.1, and numerically maximize $(f\\circ\\phi)(z_0)$ over heat-extension $\\phi$ with $\\|\\phi\\|_H<1/2$, $\\phi(z_0)=0$, and $\\phi^2(z_0)=s$. If the maximum exceeds $B_*(0,s;1,f)$, the transference inequality fails.","tokens_in":11480,"feed_emoji":"📏","tokens_out":15097,"duration_ms":144418,"temperature":0.7,"pith_summary":"The paper proves a transference principle for the two classical averaging classes BMO and $A_p$: if an integral estimate holds on an interval with ordinary Lebesgue averages, the same estimate, with the same constant, holds for the heat- or Poisson-semigroup versions on $\\mathbb{R}^n$ with kernel averages. In particular, the constant never depends on the dimension. The proof compares the relevant Bellman functions, showing that the semigroup Bellman function is dominated by the classical one-dimensional one via a local-concavity theorem and a martingale argument. The transfer, applied through the heat kernel and a comparison between heat averages and ball averages, yields a weakly dimensional theory for BMO on balls. Its headline consequence is that the John–Nirenberg constant of BMO on balls decays no faster than $n^{-1/2}$, improving on the exponential decay produced by dyadic arguments.","feed_headline":"BMO estimates transfer from intervals to R^n with no dimension cost","feed_subtitle":"Heat- and Poisson-kernel averaging carries interval inequalities into all dimensions.","key_machinery":"The load-bearing object is the Bellman function $B_*(x;\\mu,f)$ on the parabolic domain $\\Omega_\\mu$, along with its semigroup counterpart $B^K(x;\\tilde\\mu,f,n)$. The proof relies on an imported theorem stating that $B_*$ is the minimal locally concave function on $\\Omega_\\mu$ with boundary value $f$, and on Lemma 5.2, which shows that any non-negative locally concave $U$ satisfies $U(\\phi(z_0),\\phi^2(z_0))\\ge U(\\phi,\\phi^2)(z_0)$. Lemma 5.2 is proved by representing the kernel as the law of an Itô process stopped at the boundary; the process $\\Phi_t=(\\phi(Z_t),\\phi^2(Z_t))$ is a martingale inside $\\Omega_{\\tilde\\mu}$, and Itô's formula together with a non-positive Hessian makes $\\mathbb{E}U(\\Phi_t)$ monotone. Smooth approximations are built with mollifiers that use the additive homogeneity of the BMO domain, and multiplicative homogeneity for $A_p$, so the local concavity of $U$ is preserved.","core_discovery":"On the parabolic domain $\\Omega_\\mu=\\{x_1^2\\le x_2\\le x_1^2+\\mu^2\\}$, the paper defines $B_*(x;\\mu,f)$ as the supremum of $\\langle f\\circ\\eta\\rangle_I$ over one-dimensional BMO functions $\\eta$ with $\\|\\eta\\|_{*,I}\\le\\mu$ and fixed moments, and $B^K(x;\\tilde\\mu,f,n)$ as the analogous supremum over $\\phi\\in\\mathrm{BMO}^K(\\mathbb{R}^n)$ with $\\|\\phi\\|_K\\le\\tilde\\mu$. Theorem 3.1 asserts $B^K(x;\\tilde\\mu,f,n)\\le B_*(x;\\mu,f)$ for every $0<\\tilde\\mu<\\mu$ and $x\\in\\Omega_{\\tilde\\mu}$; Theorem 3.2 is the $A_p$ analogue, $D_{p,K}\\le D_p$, including $p=\\infty$. These inequalities imply that any one-dimensional estimate of the form $\\langle f(\\eta-\\langle\\eta\\rangle_I)\\rangle_I\\le C_f(\\mu)$ holds for semigroup BMO as $f(\\phi-\\phi(z))(z)\\le C_f(\\mu)$, and similarly for $A_p$; all such estimates are dimension-free. Through the heat kernel the paper proves $\\|\\phi\\|_H\\lesssim\\sqrt{n}\\,\\|\\phi\\|_*$, and from that derives Corollary 4.3 and Theorem 4.4: $\\varepsilon^{\\mathrm{JN}}_K(n)\\ge1$ and $\\varepsilon^{\\mathrm{JN}}_*(n)\\gtrsim n^{-1/2}$.","pith_inferences":["Going beyond the paper, the martingale argument only needs a Markovian semigroup with a stochastic representation, so the same transference should hold for other semigroups, such as symmetric stable processes, with constants that depend on the kernel rather than the dimension.","The paper's route suggests that the genuine n-dependence for BMO on balls enters only through the comparison of heat averages with ball averages; sharpening Proposition 4.2 would sharpen the n^{-1/2} exponent, a question the paper leaves open.","For A_p, transference gives a route to dimension-free semigroup versions of reverse-Hölder or Fujii–Wilson-type inequalities by transposing known one-dimensional sharp estimates; the paper does not write these out."],"forward_implications":["Every one-dimensional BMO or A_p estimate with Lebesgue averages becomes the same estimate for heat- and Poisson-semigroup versions on R^n, with identical constants; dimension appears only through the choice of kernel.","A weak John–Nirenberg inequality holds for BMO^K with constant epsilon_JN^K(n) >= 1, independent of dimension, for both heat and Poisson kernels.","For BMO on ordinary balls, the John–Nirenberg constant is at least a constant times n^{-1/2}, a polynomial rate instead of the exponential rate from dyadic proofs.","For any functional whose one-dimensional Bellman function can be computed, the heat-kernel route converts it into a sqrt(n)-dimensional estimate on balls.","The A_p transference makes semigroup A_p^K estimates dimension-free for 1<p<=infinity whenever the corresponding one-dimensional estimate is known."],"supporting_citations":[{"why":"Supplies the theorem that the one-dimensional Bellman functions for BMO and A_p are the minimal locally concave functions with prescribed boundary data; this is the property the transference lemma requires.","marker":"[10]"},{"why":"Provides the Itô stochastic representation of the heat and Poisson kernels, used to turn the extension process into a martingale.","marker":"[3]"},{"why":"Gives the sharp weak John–Nirenberg inequality on intervals, used with Corollary 3.3 to obtain epsilon_JN^K(n) >= 1.","marker":"[11]"},{"why":"Supplies the exponential estimate for BMO on intervals that controls |phi(z_B)-langle phi rangle_B| in Proposition 4.2.","marker":"[9]"},{"why":"Shows the analogous n^{-1/2} lower bound for BMO on cubes, the benchmark the ball result extends beyond product structure.","marker":"[12]"},{"why":"Defines the John–Nirenberg inequality on balls and gives the exponential-in-dimension estimates that Theorem 4.4 improves.","marker":"[7]"}],"fun_headline_variants":["Heat and Poisson kernels give dimension-free BMO bounds","Semigroup BMO: interval inequalities hold in all dimensions","Transference lifts BMO and A_p estimates to R^n free","Kernel averaging removes dimension cost for BMO and A_p","BMO and A_p bounds survive transfer to any dimension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim relies on an imported theorem asserting that the one-dimensional Bellman function for BMO, and likewise for $A_p$, is the minimal locally concave function with the prescribed boundary values; if that characterization fails for some admissible $f$, the transference inequality does not follow from the argument given.","fun_headline_variants_meta":{"raw":{"variants":["Heat and Poisson kernels give dimension-free BMO bounds","Semigroup BMO: interval inequalities hold in all dimensions","Transference lifts BMO and A_p estimates to R^n free","Kernel averaging removes dimension cost for BMO and A_p","BMO and A_p bounds survive transfer to any dimension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1548,"prompt_tokens":1063,"completion_tokens":485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":679,"tokens_out":485,"duration_ms":5529,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:39:26.795809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $f(t)=e^t$, $\\mu=1$, and a point $x=(0,s)$ inside $\\Omega_{1/2}$; from the explicit one-dimensional Bellman function $B_*$ for this $f$, compute the right side of Theorem 3.1, and numerically maximize $(f\\circ\\phi)(z_0)$ over heat-extension $\\phi$ with $\\|\\phi\\|_H<1/2$, $\\phi(z_0)=0$, and $\\phi^2(z_0)=s$. If the maximum exceeds $B_*(0,s;1,f)$, the transference inequality fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that the one-dimensional Bellman functions for BMO and A_p are the minimal locally concave functions with prescribed boundary data; this is the property the transference lemma requires."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"Provides the Itô stochastic representation of the heat and Poisson kernels, used to turn the extension process into a martingale."},{"cited_title":"Vasyunin, A","cited_arxiv_id":null,"evidence_quote":"Gives the sharp weak John–Nirenberg inequality on intervals, used with Corollary 3.3 to obtain epsilon_JN^K(n) >= 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the analogous n^{-1/2} lower bound for BMO on cubes, the benchmark the ball result extends beyond product structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the John–Nirenberg inequality on balls and gives the exponential-in-dimension estimates that Theorem 4.4 improves."}],"review_version":1}