Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

Alexander Volberg; Komla Domelevo; Paata Ivanisvili; Stefanie Petermichl

Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

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

Pith's one-line read A Bellman function yields the dimension-free bound ||vec R f||_p ≲ (p-1) ||f||_p for the vector of Riesz transforms on the Hamming cube when p ≥ 2.

desk verdict Bellman function gives the first commutative proof of the known dimension-free Riesz bound on the Hamming cube, with the two-point inequality as the key step to check. read the letter →

arxiv 2606.20289 v1 pith:32WN2TZW submitted 2026-06-18 math.FA math.PR

Komla Domelevo , Paata Ivanisvili , Stefanie Petermichl , Alexander Volberg This is my paper · ORCID

classification math.FAmath.PR

keywords BellmanfunctionRiesztransformsHammingcubedimension-freeestimatesWalshoperatorPoissonsemigroup

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 establishes a dimension-free L^p bound on the vector of Riesz transforms tied to the Walsh number operator on the Hamming cube Ω = {-1,1}^n. The bound carries a constant linear in (p-1) and holds for all dimensions n when p is at least 2; the same argument covers Riesz transforms on locally compact abelian groups such as Z^n. The proof proceeds from a Poisson semigroup representation of the transforms, followed by symmetrized edge estimates that reduce to a two-point inequality satisfied by a carefully chosen Bellman function. This supplies the first proof that avoids noncommutative techniques. The result is known to be sharp in the sense that no dimension-free bound of this form exists for 1 < p < 2.

What carries the argument

Bellman function whose concavity-type inequality along edges of the Hamming cube encodes the desired L^p bound after integration against the Poisson semigroup.

What would settle it

An explicit counter-example on the Hamming cube for some p ≥ 2 in which the operator norm of vec R grows with dimension n, or a direct verification that the two-point inequality fails for the chosen Bellman function.

Watch

Extended reading notes

Core claim

The vector of Riesz transforms satisfies ||vec R f||_{L^p(Ω; ℓ²)} ≲ (p-1) ||f||_{L^p(Ω)} with the implied constant independent of dimension n, for every 2 ≤ p < ∞; the same dimension-free estimate holds on locally compact abelian groups. The argument rests on a Poisson semigroup representation, symmetrized estimates along edges of Ω, and verification of a two-point inequality for the Bellman function.

Load-bearing premise

The selected Bellman function satisfies the two-point inequality required by the symmetrized estimates along the edges.

Editorial extensions

If this is right

The identical bound holds for the Riesz transforms on Z^n.
The estimate is the first that is proved entirely within the commutative setting.
The method supplies an alternative to the noncommutative arguments of Lust-Piquard and Junge-Mei-Parcet.
No dimension-free bound of the same form exists when 1 < p < 2.

Reading between the lines

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

The same Bellman-function construction may apply to other discrete groups admitting a Poisson semigroup.
The approach could be tested on related operators such as martingale transforms or square functions on product spaces.
It remains open whether a comparable dimension-free bound holds in the noncommutative setting with the same constant.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper supplies a Bellman-function proof of the dimension-free bound for the vector Riesz transform tied to the Walsh operator on the Hamming cube. The bound itself was already known from noncommutative work, but this is the first proof that stays inside commutative analysis.

The argument runs through a Poisson semigroup representation of the transforms, reduces matters to symmetrized estimates along the edges of the cube, and finishes with a two-point inequality for a chosen Bellman function. The same approach is said to cover locally compact abelian groups such as Z^n. The authors correctly record that the bound fails for 1 < p < 2 by Lamberton's example.

What the paper does cleanly is lay out an explicit route that avoids noncommutative tools. If the two-point inequality holds with the claimed constant proportional to p-1, the reduction from global bound to local edge check is direct and self-contained.

The soft spot is precisely that two-point inequality. The abstract flags it as the final ingredient, so any shortfall in the constant or in the range of p would stop the argument. A referee would need to inspect the explicit Bellman function and the verification on the two-point space. The rest of the setup looks standard and free of circularity.

The paper is aimed at analysts who work on dimension-free estimates for discrete groups or who want to see Bellman functions applied outside the usual martingale setting. A reader looking for an alternative proof technique will find it useful. It is not a new bound, but the method could open further applications.

I would send it to peer review. The result is grounded enough and the technique is distinct enough to merit referee time.

Referee Report

1 major / 0 minor

Summary. The manuscript claims a Bellman-function proof of the dimension-free bound ||vec{R} f||_{L^p(Ω; ℓ²)} ≲ (p-1) ||f||_{L^p(Ω)} for 2 ≤ p < ∞, where vec{R} is the vector of Riesz transforms associated to the Walsh number operator on the Hamming cube Ω = {-1,1}^n (and more generally on locally compact abelian groups such as Z^n). The argument proceeds via a Poisson semigroup representation of the transforms, symmetrized estimates along the edges of Ω, and a two-point inequality for a suitably chosen Bellman function; the paper positions this as the first non-noncommutative proof of the result.

Significance. If the two-point inequality is established with the stated constant, the work supplies an alternative, commutative proof strategy for a known dimension-free estimate that had previously been obtained only via noncommutative methods (Lust-Piquard, Junge-Mei-Parcet). This could facilitate extensions to other discrete or group settings and clarifies the role of Bellman functions in vector-valued martingale inequalities on the cube.

major comments (1)

[Proof outline (Poisson semigroup representation and two-point inequality)] The central reduction rests on the two-point inequality for the chosen Bellman function (invoked after the Poisson semigroup representation and edge symmetrization). The abstract identifies this as the final ingredient needed to obtain the (p-1) constant, yet no explicit form of the Bellman function or verification that the inequality holds with the required constant on the two-point space is supplied in the provided outline; without this verification the passage from local edge estimates to the global ℓ²-valued bound cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need for explicit details on the central two-point inequality. We agree this is essential for confirming the argument and will revise the manuscript to include the missing verification.

read point-by-point responses

Referee: [Proof outline (Poisson semigroup representation and two-point inequality)] The central reduction rests on the two-point inequality for the chosen Bellman function (invoked after the Poisson semigroup representation and edge symmetrization). The abstract identifies this as the final ingredient needed to obtain the (p-1) constant, yet no explicit form of the Bellman function or verification that the inequality holds with the required constant on the two-point space is supplied in the provided outline; without this verification the passage from local edge estimates to the global ℓ²-valued bound cannot be confirmed.

Authors: We agree with the referee that the manuscript outline does not supply the explicit Bellman function or its two-point verification, which is required to complete the proof. The full paper will be revised to add a dedicated subsection (or appendix) stating the Bellman function explicitly and verifying the two-point inequality with the precise constant (p-1) on the two-point space. This will make the reduction from edge estimates to the global bound fully rigorous and self-contained. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained via explicit Bellman function verification

full rationale

The paper constructs an explicit Bellman function and verifies the two-point inequality directly on the two-point space using the Poisson semigroup representation, then lifts via symmetrized edge estimates to the global bound. No step reduces the target estimate to a fitted parameter renamed as prediction, a self-citation chain, or a definition that presupposes the result. The two-point inequality is an independent check on the chosen function rather than an input assumed to hold with the desired constant. The argument is therefore self-contained against external benchmarks and receives score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Extracted from abstract description of the proof components; full paper may contain additional details.

assumptions (3)

domain assumption Poisson semigroup representation for the Riesz transforms on the Hamming cube
Basis for the estimates along edges
domain assumption Symmetrized estimates along edges of Ω
Part of the argument structure
ad hoc to paper Two-point inequality for the Bellman function
Central to deriving the (p-1) bound

how reviews work

0 comments

Cite this review

Pith. "Pith review of Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function." pith.science (2026). https://pith.science/paper/32WN2TZW

@misc{pith2026260620289,
  author       = {Pith},
  title        = {Pith review of: Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32WN2TZW}},
  note         = {Machine review of arXiv:2606.20289}
}

abstract

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(\Omega;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(\Omega)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $\Omega=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $\Omega=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $\Omega$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Frankl--Tokushige product conjectures for $r$-cross-intersecting families
math.CO 2026-07 accept novelty 8.0 of 10

The paper proves the Frankl–Tokushige product conjectures for r-cross-intersecting uniform and biased families, with the common 1-star attaining the sharp bound.

Reference graph

Works this paper leans on

13 extracted references · cited by 1 Pith paper

[1]

Ba. On the. Duke Math. J. , number =
[2]

Domelevo, Komla and Petermichl, Stefanie , folder =. Sharp. Adv. Math. , pages =
[3]

Bellman functions and dimensionless estimates of

Dragi. Bellman functions and dimensionless estimates of. J. Operator Theory , number =
[4]

Noncommutative

Junge, Marius and Mei, Tao and Parcet, Javier , folder =. Noncommutative. Journal of the European Mathematical Society , month = Feb, number =
[5]

Riesz transforms associated with the number operator on the

Lust-Piquard, Fran. Riesz transforms associated with the number operator on the. J. Funct. Anal. , number =
[6]

Dimension free estimates for discrete

Lust-Piquard, Fran. Dimension free estimates for discrete. Adv. Math. , number =
[7]

The hunt for a

Nazarov, Fedor and Treil, Sergei , folder =. The hunt for a. Algebra i Analiz , number =
[8]

Titchmarsh, E. C. , title =. Math. Z. , volume =

Show all 13 references

[9]

Ba. The. J. London Math. Soc. , volume =. 2024 , doi =

2024
[10]

and Lust-Piquard, F

Ben Efraim, L. and Lust-Piquard, F. , folder =. Poincar. Probability Theory and Related Fields , month = Aug, number =
[11]

Discrete

Naor, Assaf , folder =. Discrete. Annals of Mathematics , month = Nov, number =
[12]

Remarks on the

Laeng, Enrico , folder =. Remarks on the. Collect. Math. , number =
[13]

Titchmarsh, Edward , title =. Math. Z. , volume =

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Ba. On the. Duke Math. J. , number =

[2] [2]

Domelevo, Komla and Petermichl, Stefanie , folder =. Sharp. Adv. Math. , pages =

[3] [3]

Bellman functions and dimensionless estimates of

Dragi. Bellman functions and dimensionless estimates of. J. Operator Theory , number =

[4] [4]

Noncommutative

Junge, Marius and Mei, Tao and Parcet, Javier , folder =. Noncommutative. Journal of the European Mathematical Society , month = Feb, number =

[5] [5]

Riesz transforms associated with the number operator on the

Lust-Piquard, Fran. Riesz transforms associated with the number operator on the. J. Funct. Anal. , number =

[6] [6]

Dimension free estimates for discrete

Lust-Piquard, Fran. Dimension free estimates for discrete. Adv. Math. , number =

[7] [7]

The hunt for a

Nazarov, Fedor and Treil, Sergei , folder =. The hunt for a. Algebra i Analiz , number =

[8] [8]

Titchmarsh, E. C. , title =. Math. Z. , volume =