Dimension-free estimates for semigroup BMO and $A_p$

Leonid Slavin; Pavel Zatitskii

REVIEW 1 major objections 3 minor 12 references

Dimension-free estimates for semigroup BMO and $A_p$

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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).

desk verdict 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. read the letter →

arxiv 1908.02602 v2 pith:4V7TFEYR submitted 2019-08-07 math.CA

Leonid Slavin , Pavel Zatitskii This is my paper · ORCID

classification math.CA MSC 42A0542B3542A6149K20

keywords BMOApweightsdimension-freeestimatesBellmanfunctionsheatkernelPoissonJohn-Nirenbergconstantsemigrouptransference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\varepsilon^{\mathrm{JN}}_*(n)\gtrsim n^{-1/2}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

This 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.

Referee Report

1 major / 3 minor

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).

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 (1)

[Section 5, Theorem 5.1 and Theorem 3.1; also Section 4, Theorem 4.4] 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.

minor comments (3)

[Corollary 4.3] 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 2, (2.2) and (2.5)] 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.
[Lemma 5.2, general case] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the transference theorem is a genuine consequence of the imported local-concavity theorem plus a semigroup Jensen lemma; no estimate is fitted or assumed as its own conclusion.

full rationale

The derivation chain is not circular. Theorem 3.1 proves B^K(x;μ̃,f,n) ≤ B_*(x;μ,f) by combining Theorem 5.1 (B_* is the minimal locally concave function with boundary datum f, quoted from [10]) with Lemma 5.2, which establishes the Jensen-type inequality U(φ(z0),φ²(z0)) ≥ U(φ,φ²)(z0) from the martingale representation of the heat/Poisson kernels. No estimate of the form (3.3) is assumed in proving (3.1); the Bellman functions are defined as extremal problems, and the inequality is derived for arbitrary non-negative measurable f. Corollary 3.3 is a direct consequence of (3.1), not a restatement of its hypothesis: the equivalence between (3.3) and B_*(0,x2;μ,f)≤C_f(μ) is a simple reformulation, and the conclusion (3.4) is a different object (K-averages on R^n). The John–Nirenberg application uses the external interval result [11] and transfers it through Corollary 3.3; Propositions 4.1 and 4.2 supply the explicit √n factors. The only soft spot is whether Theorem 5.1, imported from [10] (whose second author overlaps with the present paper), covers discontinuous f=χ_{|s|>λ}; if not, that is a hypothesis-checking correctness gap, not circularity, because no step reduces to its own input and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results are pure mathematics; no free parameters are fitted to data. The proof relies on published Bellman function theory (including the local concavity theorem of [10]) and classical probabilistic representations of heat and Poisson kernels. The authors' self-citations are to established, peer-reviewed results that do not depend on the present paper, so the transference is not circular. All constants are explicit and dimension independent.

assumptions (4)

domain assumption The Bellman function B_*(x;µ,f) (and its A_p analog D_p) is the minimal locally concave function on its parabolic domain Ω_µ with boundary value f (Theorem 5.1, quoted from [10]).
This characterization is the bridge that lets the authors apply Lemma 5.2 with U = B_*. The theorem is not proved in the paper; it is taken from prior work of Stolyarov and Zatitskiy, one of the present authors.
standard math The heat and Poisson kernels admit a probabilistic representation as terminal distributions of stopped Brownian motions, and the semigroup extensions of φ and φ² form martingales for the corresponding processes (equation (5.2)).
This classical fact (see [3]) underlies the key submartingale inequality in Lemma 5.2. It is a standard tool in harmonic analysis and diffusion theory.
standard math Sharp one-dimensional interval estimates are available: ⟨e^{|η-⟨η⟩_I|}⟩_I ≤ 1/(1-||η||_{*,I}) from [9], and the weak-type John-Nirenberg estimate from [11] with C_f(s) = e^{1-λ/s}.
These prior sharp estimates are fed into the transfer principle (Corollary 3.3) to produce the heat-vs-ball comparison in Proposition 4.2 and the John-Nirenberg bound in Theorem 4.4.
standard math The ball mean oscillation difference satisfies |⟨φ⟩_{B_{r1}} - ⟨φ⟩_{B_{r2}}| ≲ n |log(r1/r2)| ||φ||_* for concentric balls (used in Proposition 4.1).
This elementary-looking estimate is used to evaluate the Gaussian integral giving the √n bound for ||φ||_H vs ||φ||_*. It follows from John-Nirenberg theory but is stated without proof.

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Cite this review

Pith. "Pith review of Dimension-free estimates for semigroup BMO and $A_p$." pith.science (2026). https://pith.science/paper/4V7TFEYR

@misc{pith2026190802602,
  author       = {Pith},
  title        = {Pith review of: Dimension-free estimates for semigroup BMO and $A_p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4V7TFEYR}},
  note         = {Machine review of arXiv:1908.02602}
}

abstract

Let $K_t$ be either the heat or the Poisson kernel on $\mathbb{R}^n.$ Let $\mathcal{A}$ stand either for BMO equipped with the quadratic seminorm or for $A_p,$ $1< p\le\infty.$ We establish the following transference between the class $\mathcal{A}$ on an interval $I\subset\mathbb{R}$ and its $K$-version, $\mathcal{A}^K,$ on $\mathbb{R}^n$: If a given integral functional admits an estimate on $\mathcal{A}(I),$ then the same estimate holds for $\mathcal{A}^K(\mathbb{R}^n),$ with all Lebesgue averages replaced by $K$-averages. In particular, all such estimates are dimension-free. As an application, via the heat kernel, we obtain a weakly-dimensional theory for ${\rm BMO}(\mathbb{R}^n)$ on balls. In particular, we show that the John--Nirenberg constant of this space decays with dimension no faster than $n^{-1/2}.$

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Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages

[10]

D. M. Stolyarov, P. B. Zatitskiy. Theory of locally conc ave functions and its applications to sharp estimates of integral functionals. Advances in Mathematics , Vol. 291 (2016), pp. 228-273

work page 2016
[1]

Unions of cubes in $\mathbb{R}^{n}$, combinatorics in $\mathbb{Z}^{n}$ and the John-Nirenberg and John-Str\"omberg inequalities

M. Cwikel, Unions of cubes in Rn, combinatorics in Zn and the John–Nirenberg and John-Str¨ omberg inequalities. arXiv:1702.00602v1

work page Pith review arXiv
[2]

Cwikel, Y

M. Cwikel, Y. Sagher, P. Shvartsman, A new look at the John -Nirenberg and John-Str¨ omberg theorems for BMO. J. Funct. Anal. Vol. 263 (2012), No. 1, pp. 129-166

work page 2012
[3]

Ikeda, S

N. Ikeda, S. Watanabe. Stochastic diﬀerential equation s and diﬀusion processes. North-Holland Mathe- matical Library, 24. North-Holland Publishing Co., Amster dam-New York; Kodansha, Ltd., Tokyo, 1981. 464 pp

work page 1981
[4]

Ivanishvili, N

P. Ivanishvili, N. Osipov, D. Stolyarov, V.Vasyunin, P. Zatitskiy. Bellman function for extremal problems in BMO. Trans. Amer. Math. Soc. Vol. 368 (2016), No. 5, pp. 3415-3468

work page 2016
[5]

Ivanisvili, D

P. Ivanisvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zati tskiy. Bellman function for extremal problems in BMO II : evolution. Mem. Amer. Math. Soc. Vol. 255 (2018), No. 1220, v+133 pp

work page 2018
[6]

Ivanishvili, N

P. Ivanishvili, N. Osipov, D. Stolyarov, V.Vasyunin, P. Zatitskiy. Sharp estimates of integral functionals on classes of functions with small mean oscillation. C. R. Math. Acad. Sci. Paris Vol. 353 (2015), No. 12, pp. 1081-1085

work page 2015
[7]

F. John, L. Nirenberg. On functions of bounded mean oscil lation. Comm. Pure Appl. Math. , Vol. 14 (1961), pp. 415-426

work page 1961

Show all 12 references

[8]

Slavin, V

L. Slavin, V. Vasyunin. Sharp results in the integral-fo rm John–Nirenberg inequality. Trans. Amer. Math. Soc., Vol. 363 (2011), No. 8, pp. 4135-4169

2011
[9]

Slavin, V

L. Slavin, V. Vasyunin. Sharp Lp estimates on BMO. Indiana Univ. Math. J. , Vol. 61 (2012), No. 3, pp. 1051-1110

2012
[11]

Vasyunin, A

V. Vasyunin, A. Volberg, Sharp constants in the classic al weak form of the John–Nirenberg inequality. Proc. Lond. Math. Soc. (3) , Vol. 108 (2014), No. 6, pp. 1417-1434

2014
[12]

I. Wik. On John and Nirenberg’s theorem. Ark. Mat. Vol. 28 (1990), No. 1, pp. 193-200 University of Cincinnati, P.O. Box 210025, OH 45221-0025, U SA E-mail address : leonid.slavin@uc.edu St. Petersburg State University, 29B 14th Line V.O., Saint P etersburg, 199178, Russia and...

1990

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

D. M. Stolyarov, P. B. Zatitskiy. Theory of locally conc ave functions and its applications to sharp estimates of integral functionals. Advances in Mathematics , Vol. 291 (2016), pp. 228-273

work page 2016

[2] [1]

Unions of cubes in $\mathbb{R}^{n}$, combinatorics in $\mathbb{Z}^{n}$ and the John-Nirenberg and John-Str\"omberg inequalities

M. Cwikel, Unions of cubes in Rn, combinatorics in Zn and the John–Nirenberg and John-Str¨ omberg inequalities. arXiv:1702.00602v1

work page Pith review arXiv

[3] [2]

Cwikel, Y

M. Cwikel, Y. Sagher, P. Shvartsman, A new look at the John -Nirenberg and John-Str¨ omberg theorems for BMO. J. Funct. Anal. Vol. 263 (2012), No. 1, pp. 129-166

work page 2012

[4] [3]

Ikeda, S

N. Ikeda, S. Watanabe. Stochastic diﬀerential equation s and diﬀusion processes. North-Holland Mathe- matical Library, 24. North-Holland Publishing Co., Amster dam-New York; Kodansha, Ltd., Tokyo, 1981. 464 pp

work page 1981

[5] [4]

Ivanishvili, N

P. Ivanishvili, N. Osipov, D. Stolyarov, V.Vasyunin, P. Zatitskiy. Bellman function for extremal problems in BMO. Trans. Amer. Math. Soc. Vol. 368 (2016), No. 5, pp. 3415-3468

work page 2016

[6] [5]

Ivanisvili, D

P. Ivanisvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zati tskiy. Bellman function for extremal problems in BMO II : evolution. Mem. Amer. Math. Soc. Vol. 255 (2018), No. 1220, v+133 pp

work page 2018

[7] [6]

Ivanishvili, N

P. Ivanishvili, N. Osipov, D. Stolyarov, V.Vasyunin, P. Zatitskiy. Sharp estimates of integral functionals on classes of functions with small mean oscillation. C. R. Math. Acad. Sci. Paris Vol. 353 (2015), No. 12, pp. 1081-1085

work page 2015

[8] [7]

F. John, L. Nirenberg. On functions of bounded mean oscil lation. Comm. Pure Appl. Math. , Vol. 14 (1961), pp. 415-426

work page 1961