Sharp transference principle for $\mathrm{BMO}$ and $A_p$

Dmitriy Stolyarov; Pavel Zatitskiy

REVIEW 2 major objections 3 minor 22 references

Sharp transference principle for $\mathrm{BMO}$ and $A_p$

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Sharp constants in BMO and A_p are domain-independent

desk verdict A genuinely new transfer principle with a real gap in the BMOp proof: Lemma 4.1's bound on ψλ,n is not justified by the truncation lemma. read the letter →

arxiv 1908.09497 v1 pith:G2E264B3 submitted 2019-08-26 math.CA math.PR

Dmitriy Stolyarov , Pavel Zatitskiy This is my paper · ORCID

classification math.CAmath.PR MSC 42B3542B2546E3060G42

keywords BMOJohn-NirenberginequalityMuckenhouptweightsreverseHoldersharpconstantstransferenceprincipleBellmanfunctionmartingale

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

This paper establishes a transference principle: for a large class of optimization problems defined by local distribution constraints, the best constants are the same whether the underlying functions live on the circle, on an interval, or on the whole real line. In particular, it proves that the sharp John–Nirenberg constants for the natural BMO spaces on these three domains coincide for every p in $[1,\infty)$, that the sharp $L^p$ comparisons between BMO norms on the line equal the interval ones, and that the suprema governing the reverse Hölder inequality for Muckenhoupt $A_p$ weights coincide as well. The point is not that every function on an interval extends to the line with the same norm—that is false—but that the extremal distributions can be reproduced on the circle. A sympathetic reader should care because it turns a collection of sharp interval estimates into sharp line and circle estimates without redoing the hard optimization.

What carries the argument

The central object is the class $A^\circ_s(Y,W)$ of simple functions on the circle whose distribution over every interval lies in a fixed open set $W$ of probability measures on $Y$; this class encodes both BMO$_p$, via the condition that the $p$-th centered moment is small, and $A_p$ weights, via the condition that the product of averages stays below a fixed constant. The argument is carried by two statements: Lemma 2.4, which glues two such functions whose distribution segments lie in $W$ into one function with any prescribed convex combination of their distributions, and Theorem 2.3, which iterates that gluing to realize the terminal distribution of any simple $(W,\Delta)$-martingale. The constructive engine is the $\lambda$-homogenization operator $\Gamma_\lambda$, which rescales a function into nested periodic copies so that, for $\lambda$ close to 1, distributions over both long and short intervals stay close to the original distribution or its convex combinations. This machinery converts an extremal problem whose value is known on an interval into a nearly extremal function on the circle, and hence on the line.

What would settle it

Compute the sharp John–Nirenberg constant for BMO(R) directly by an independent method (e.g., a numerical optimization over periodic functions) and check whether it equals the interval value $C^I_{3,p}$; a strict inequality would refute Theorem 1.3.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.3: every simple martingale taking values in a suitable open set of probability measures on a target space can be realized as the family of local distributions of a simple function on the circle, up to closure. Since BMO$_p$ norms, $A_p$ constants, and their associated extremal problems can all be encoded as local distribution inclusions of this kind, the same realization theorem forces the sharp constants for the circle, the interval, and the line to coincide. The paper states this as coincidence of the sharp John–Nirenberg constants $C^{R}_{3,p}=C^{I}_{3,p}$, the sharp weak-type John–Nirenberg bound for BMO$_2$ on the line, the sharp $L^p$ estimates on BMO over the line, and the reverse Hölder suprema for $A_p$ weights. The proofs combine Bellman-function characterizations with an explicit homogenization construction that packs an extremal function into periodic layers of a circle function.

Load-bearing premise

The paper's equality of constants rests on the previously calculated Bellman-function values and on the lemma that composing with a Lipschitz function never increases a BMO_p norm; if either of those is wrong, the transfer claim fails.

Editorial extensions

If this is right

The sharp John–Nirenberg constants for the naturally defined BMO spaces on the line and circle are exactly the known interval constants, including the classical value $2/e$ for the distributional form with the $L^1$ seminorm.
The sharp $L^p$-comparison inequalities $\|\varphi\|_{2,\mathbb R}\le \|\varphi\|_{p,\mathbb R}\le (p/2\,\Gamma(p))^{1/p}\|\varphi\|_{2,\mathbb R}$ for $p>2$ hold on the line, and the weak-type John–Nirenberg profile for BMO$_2$ transfers to the line with the same three-piece formula.
The suprema governing the reverse Hölder inequality for $A_p$ weights on the line coincide with the interval suprema for every $p$, $q$, and constant $C$.
Any optimization problem that can be encoded as a local distribution constraint of the form $A^\circ(Y,W)$ will have the same sharp value on the circle, the interval, and the line.
Sharpness on the line is achieved not by extending extremal interval functions—which is generally impossible—but by periodic functions whose distribution approximates the extremal martingale terminal distribution.

Reading between the lines

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

The same realization theorem likely transfers sharp constants to higher-dimensional tori and Euclidean spaces, provided the martingale and extremal-function machinery is available there; the paper does not pursue that direction.
Because the transfer works at the level of distributions, it suggests a general recipe: compute an interval sharp constant by extremal-function methods, then realize the extremizer's distribution periodically to obtain explicit near-extremizers on the line.
The appendix's monotonicity and truncation lemma, proved because the authors could not locate it in the literature, may be useful beyond BMO$_p$, for instance in weighted or rearrangement-invariant settings where similar folklore statements are assumed.
The equality of sharp constants for $A_p$ reverse Hölder means that any numerically observed line-versus-interval discrepancy for Muckenhoupt weights must come from non-extremal examples rather than from a genuine difference in the optimal constants.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Let me give you the short version. The transfer principle behind this paper—Theorem 2.3, the martingale-to-circle realization—is genuinely new, and the coincidences for BMO2 and Ap weights follow from it cleanly. The BMOp case, Theorem 1.3, has a gap in the proof that the authors need to fix.

The problem is in Lemma 4.1. The authors need to show the martingale segments lie in W_{p,ε}, and they do it by bounding an auxiliary function ψλ,n. The bound is one line: ‖ψλ,n‖ ≤ ‖log x‖ + |log λ|, attributed to the truncation lemma in Appendix A. But ψλ,n is not a pointwise Lipschitz composition of log. On each cell it replaces log by its interval average, and on the tail it flattens to a constant. Corollary A.2 controls g∘φ for nondecreasing 1-Lipschitz g; it does not control averaging or flattening. This is not cosmetic, because Lemma 4.1 is the only step that transfers the sharp BMOp extremizer to the circle. The proof of Theorem 1.3 currently rests on an unverified estimate.

The write-up of Theorem 1.3 has a second issue. It opens with 'Since C^R_{3,p} ≤ C^I_{3,p}', but restriction (1.4) gives the opposite inequality; the nontrivial direction is the reverse. Also, the step from 'for every m there is a function with integral > m' to 'the constant is not admissible' needs a limiting argument that isn't given.

What the paper does well is real. The homogenization idea, credited to Nazarov, is elegant, Lemma 2.4 is genuinely clever, and Section 3 is honest—it leans on published Bellman function computations and says so. The appendix proves the folklore truncation lemma correctly for the case it actually covers. If the authors supply a direct proof of the ψλ,n bound—which looks plausible, probably via a careful case analysis on intervals—the paper would be strong.

This deserves a serious referee. The transfer principle is a real contribution, and the rest of the paper is solid enough to merit careful review. But as it stands I would not accept it. I'd send it to review with the expectation that the referee asks for a complete proof of Lemma 4.1 and a cleaner limiting argument. I wouldn't cite Theorem 1.3 until that's fixed.

Referee Report

2 major / 3 minor

Summary. The paper proposes a transference principle asserting that certain extremal optimization constants for BMO-type spaces and Muckenhoupt weights coincide on the circle, the interval, and the line. The main results are Theorem 1.3 (equality of the sharp John-Nirenberg constants C^R_{3,p}=C^I_{3,p}), Theorem 1.4 (sharp Lp estimates on BMO over the line), Theorem 1.6 (sharp weak-type John-Nirenberg inequality for BMO2 over the line), and Theorem 1.9 (coincidence of sharp Reverse Holder constants for Ap weights). The engine is Theorem 2.3, which realizes any simple (W,Delta)-martingale as the terminal distribution of a simple function on the circle; Section 3 applies this to Bellman-function extremal problems using known Bellman values from the literature; Section 4 handles the BMOp case by approximating the extremal distribution of log x with specially constructed martingales, relying on a truncation lemma proved in Appendix A.

Significance. If the proof is completed, the paper would establish a useful and nontrivial principle: several sharp constants in classical inequalities do not depend on whether the ambient space is an interval, the line, or the circle. The general martingale-realization theorem (Theorem 2.3) is elegant and likely to have further applications. The derivations in Section 3 are clean and correctly reduce the BMO2 and Ap sharpness results to published Bellman-function computations. The authors are commendably explicit about the folklore status of the truncation lemma and about their reliance on external results. The main unresolved issue is concentrated in Section 4, where the transfer of the BMOp extremizer is not yet fully justified.

major comments (2)

[Section 4, Lemma 4.1] The proof of Lemma 4.1 asserts the bound ||psi_{lambda,n}||_{p,[0,infty)} <= ||log x||_{p,[0,infty)} + |log lambda| and cites the truncation lemma (Appendix A, Corollary A.2). However, psi_{lambda,n} is not obtained from log x by a pointwise nondecreasing 1-Lipschitz composition: on each interval I_k it replaces log x by the interval average (1/|I_k|) int_{I_k} log, and on (lambda^{-n},infty) it is replaced by a constant. Corollary A.2 controls only functions of the form g o phi with g nondecreasing and 1-Lipschitz, e.g., min(phi,N). The step function G(t) that encodes this averaging has jumps of size |log lambda| at the endpoints of the intervals in the log variable, so it is not 1-Lipschitz. Thus the cited lemma does not imply the bound. Since Lemma 4.1 is the only mechanism that transfers the extremal log distribution to the circle, the proof of Theorem 1.3 is incomplete as written. A direct proof of the contraction estimate, or a modified construction to which Corollary A.2 genuinely applies, is needed.
[Section 4, proof of Theorem 1.3] The proof begins with 'Since C^R_{3,p} <= C^I_{3,p}', but with the definitions in (1.7) the restriction inequality (1.4) yields the opposite inequality, C^R_{3,p} >= C^I_{3,p}. The construction that follows is precisely the hard direction C^R_{3,p} <= C^I_{3,p}, so the roles of the two inequalities are reversed and the easy direction is not stated. In addition, the displayed identity int_0^1 exp(C^I_{3,p} log x / ||log x||_{p,[0,1]}) = infty cannot hold as written, because log x < 0 on [0,1] makes the integrand bounded by 1; the intended absolute value around log x - <log x>, or the function -log x, needs to be present for the exponential integral to diverge. These two issues make the write-up of this section unreliable and must be repaired even if Lemma 4.1 is fixed.

minor comments (3)

[Section 4, Lemma 4.1] The equality ||log x||_{p,[0,infty)} = ||log x||_{p,[0,1]} is used without justification; a one-sentence explanation using the scale invariance of the oscillation of log on intervals [a,b] would help the reader.
[Section 3, equation (3.7)] The arrow in '||phi||_{p,T} <= epsilon <= phi in A^circ(R,W_{p,epsilon})' is nonstandard; replacing it by 'phi in A^circ(R,W_{p,epsilon}) implies ||phi||_{p,T} <= epsilon' would avoid ambiguity.
[Section 2, Lemma 2.7] The phrase 'K may be chosen finite dimensional, that is K subset V' is imprecise; K is contained in a finite-dimensional subspace V, not equal to it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper transfers known interval sharp constants via an independent martingale construction; no fitted quantity is relabeled as a prediction.

full rationale

The derivation is not circular. The sharp constants for the interval are taken as external benchmark values from published papers ([13], [15], [16], [20], [21], [22]), and the paper never fits a parameter to the line or circle data it claims to predict. The central transfer engine, Theorem 2.3, is proved here by a self-contained homogenization argument (Lemma 2.4), which constructs circle functions whose interval distributions approximate the terminal distribution of a given martingale. Where the paper relies on the authors' own prior work ([18, Theorem 3.1] and Bellman values from [15], [20], [21], [22]), those are published results with stated assumptions that do not include the target theorems and with derivations independent of the present claims; they are genuine evidence rather than circular support. The only serious issue visible in the write-up is a correctness gap, not a circularity: in Lemma 4.1 the bound ||ψ_{λ,n}||_p ≤ ||log x||_p + |log λ| is justified by the truncation lemma, but ψ_{λ,n} is produced by averaging and flattening, not by a pointwise monotone Lipschitz composition, so the contraction estimate is not directly covered by Corollary A.2. That concern belongs to proof verification and correctness risk, not to circularity analysis. No constant is fitted, renamed, or defined in terms of the claimed conclusion.

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

The central claims have no fitted parameters; the constants are proven equal to known interval values. The load-bearing assumptions are all published external results (Bellman theory and computations) plus the paper's own new truncation lemma, which the authors flag as folklore.

assumptions (4)

domain assumption Bellman function characterization (Theorem 3.1 of [18]): under strict convexity, C2 boundary, and coinciding maximal inscribed cones, the Bellman function is the pointwise minimal locally concave majorant.
The paper restates this as Theorem 3.1 and uses it in Lemmas 3.2 and 3.3 for all Section 3 theorems; no proof is given here.
domain assumption Exact Bellman values: B(0,1)=p/2 Γ(p) in the BMOp norm inequality ([15]), the weak-type values from [22], and R(C,q) from [20],[21].
Sharpness proofs in Section 3 plug in these prior computations; the present paper does not derive them.
domain assumption Monotone rearrangement does not increase the BMOp norm (cited to [7] and [12]).
Used in Appendix A to reduce Lemma A.1 to monotone functions; without it the truncation lemma proof fails.
domain assumption The interval sharp constant C^I_{3,p} is attained at log x on [0,1] ([13], [16]).
Theorem 1.3 builds functions approximating log x; the equality claim inherits the interval extremizer.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp transference principle for $\mathrm{BMO}$ and $A_p$." pith.science (2026). https://pith.science/paper/G2E264B3

@misc{pith2026190809497,
  author       = {Pith},
  title        = {Pith review of: Sharp transference principle for $\mathrmBMO$ and $A_p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2E264B3}},
  note         = {Machine review of arXiv:1908.09497}
}

abstract

We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the John--Nirenberg inequalities for naturally defined $\mathrm{BMO}$-spaces on the circle, the interval, and the line coincide. The same principle holds true for the Reverse H\"older inequality for Muckenhoupt weights.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Brezis, L

H. Brezis, L. Nirenberg, Degree theory and BMO; part 1: compact manifolds without boundaries , Selecta Mathematica 1 (1995), 197–263

work page 1995
[2]

Grafakos, Classical Fourier Analysis , Graduate texts in Mathematics, Springer, 2009

L. Grafakos, Classical Fourier Analysis , Graduate texts in Mathematics, Springer, 2009

work page 2009
[3]

Ivanisvili, D

P. Ivanisvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zati tskiy, Bellman function for extremal problems in BMO II : evolution, Memoirs of the American Mathematical Society 255:1220, 2018

work page 2018
[4]

Ivanishvili, N

P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyu nin, P. B. Zatitskiy, On Bellman function for extremal problems in BMO, Comptes Rendus Mathematique 350:11 (2012), 561–564

work page 2012
[5]

Ivanisvili, N

P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyun in, P. B. Zatitskiy, Sharp estimates of inte- gral functionals on classes of functions with small mean osc illation, Comptes Rendus Mathematique 353:12 (2015), 1081–1085

work page 2015
[6]

Ivanishvili, D

P. Ivanishvili, D. M. Stolyarov, N. N. Osipov, V. I. Vasyu nin, P. B. Zatitskiy, Bellman function for extremal problems in BMO, Transactions of the American Mathematical Society 368 (2016), 3415–3468

work page 2016
[7]

Klemes, A mean oscillation inequality , Proceedings of the American Mathematical Society 93:3 (1985), 497–500

I. Klemes, A mean oscillation inequality , Proceedings of the American Mathematical Society 93:3 (1985), 497–500

work page 1985
[8]

A. A. Korenovskii, On the connection between mean oscillation and exact integr ability classes of functions, Mat. Sb. 181:12 (1990), 1721–1727; Math. USSR-Sb. 71:2 (1992), 561–567

work page 1990

Show all 22 references

[9]

Lerner, The John–Nirenberg inequality with sharp constants , Comptes Rendus Mathemathique 351:11-12 (2013), 463–466

A. Lerner, The John–Nirenberg inequality with sharp constants , Comptes Rendus Mathemathique 351:11-12 (2013), 463–466

2013
[10]

Reznikov, Sharp weak type estimates for weights in the class Ap1,p 2 , Revista Matematica Iberoamericana 29:2 (2013), 433–478

A. Reznikov, Sharp weak type estimates for weights in the class Ap1,p 2 , Revista Matematica Iberoamericana 29:2 (2013), 433–478

2013
[11]

Shanin, Extension of functions with bounded mean oscillation , Ukr

R. Shanin, Extension of functions with bounded mean oscillation , Ukr. Mat. Visn. 10:3 (2013), 397– 411 (in Russian), translated in Journal of Mathematical Sci ences (New York) 196:5 (2014), 693–704. 14

2013
[12]

Shanin, Equimeasurable rearrangements of functions satisfying the reverse H¨ older or the reverse Jensen inequality, Ric

R. Shanin, Equimeasurable rearrangements of functions satisfying the reverse H¨ older or the reverse Jensen inequality, Ric. Mat. 64:1 (2015), 217–228

2015
[13]

Slavin, The John–Nirenberg constant of BMOp, 1 ⩽ p ⩽ 2, https://arxiv.org/abs/1506.04969

L. Slavin, The John–Nirenberg constant of BMOp, 1 ⩽ p ⩽ 2, https://arxiv.org/abs/1506.04969

arXiv
[14]

Slavin, V

L. Slavin, V. Vasyunin, Sharp results in the integral-form John–Nirenberg inequalit y, Transactions of the American Mathematical Society 363:8 (2011), 4135–4169

2011
[15]

Slavin, V

L. Slavin, V. Vasyunin, Sharp Lp estimates on BMO, Indiana University Mathematical Journal 61 (2012), 1051–1110

2012
[16]

Slavin, V

L. Slavin, V. Vasyunin, The John–Nirenberg constant of BMOp,p> 2, Algebra i Analiz 28:2 (2016), 72–96 (in Russian), tranlsated in St. Petersburg Mathemati cal Journal 28 (2017), 181–196

2016
[17]

D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Monotonic rearrangements of functions with small mean oscillations , Studia Mathematica 231:3 (2015), 257–268

2015
[18]

D. M. Stolyarov, P. B. Zatitskiy, Theory of locally concave functions and its applications to s harp estimates of integral functionals , Advances in Mathematics 291 (2016), 228–273

2016
[19]

Vasyunin, The sharp constant in the John–Nirenberg inequality , POMI preprint no

V. Vasyunin, The sharp constant in the John–Nirenberg inequality , POMI preprint no. 20, 2003

2003
[20]

V. I. Vasyunin, The sharp constant in the reverse H¨ older inequality for Muckenhoupt weights, Algebra i analaliz 15:1 (2003), 73–117 (in Russian); translated in St.-Petersbu rg Math. J. 15:1 (2004), 49–79

2003
[21]

V. I. Vasyunin, Mutual estimates of Lp-norms and the Bellman function , Zap. nauchn. sem. POMI 355 (2008), 81–138 (in Russian); translated in Journal of Mathe matical Sciences 156:5 (2009), 766–798

2008
[22]

Vasyunin, A

V. Vasyunin, A. Volberg, Sharp constants in the classical weak form of the John-Nirenbe rg inequality, Proceedings of London Mathematical Society 108:6 (2014), 1417–1434. D. Stolyarov Department of Mathematics and Computer Sciences, St. Peter sburg State University, Russia St....

2014

Pith tools

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

[1] [1]

Brezis, L

H. Brezis, L. Nirenberg, Degree theory and BMO; part 1: compact manifolds without boundaries , Selecta Mathematica 1 (1995), 197–263

work page 1995

[2] [2]

Grafakos, Classical Fourier Analysis , Graduate texts in Mathematics, Springer, 2009

L. Grafakos, Classical Fourier Analysis , Graduate texts in Mathematics, Springer, 2009

work page 2009

[3] [3]

Ivanisvili, D

P. Ivanisvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zati tskiy, Bellman function for extremal problems in BMO II : evolution, Memoirs of the American Mathematical Society 255:1220, 2018

work page 2018

[4] [4]

Ivanishvili, N

P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyu nin, P. B. Zatitskiy, On Bellman function for extremal problems in BMO, Comptes Rendus Mathematique 350:11 (2012), 561–564

work page 2012

[5] [5]

Ivanisvili, N

P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyun in, P. B. Zatitskiy, Sharp estimates of inte- gral functionals on classes of functions with small mean osc illation, Comptes Rendus Mathematique 353:12 (2015), 1081–1085

work page 2015

[6] [6]

Ivanishvili, D

P. Ivanishvili, D. M. Stolyarov, N. N. Osipov, V. I. Vasyu nin, P. B. Zatitskiy, Bellman function for extremal problems in BMO, Transactions of the American Mathematical Society 368 (2016), 3415–3468

work page 2016

[7] [7]

Klemes, A mean oscillation inequality , Proceedings of the American Mathematical Society 93:3 (1985), 497–500

I. Klemes, A mean oscillation inequality , Proceedings of the American Mathematical Society 93:3 (1985), 497–500

work page 1985

[8] [8]

A. A. Korenovskii, On the connection between mean oscillation and exact integr ability classes of functions, Mat. Sb. 181:12 (1990), 1721–1727; Math. USSR-Sb. 71:2 (1992), 561–567

work page 1990