The optimal bound $e$ in a dyadic version of Uchiyama's Lemma

J. Fladung; S. Petermichl

REVIEW 4 major objections 4 minor 13 references

In a dyadic model of the analytic Hardy space, the Uchiyama embedding holds with the sharp constant e, matching the conjectured continuous value.

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In a dyadic martingale model of analytic functions, the optimal constant in Uchiyama's Lemma is e, and the constant is shown to be sharp.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A credible and novel dyadic model of Uchiyama's lemma with sharp constant e, but the optimality proof as written has gaps in the PSD verifications and a compressed lower-bound argument. the 4 major comments →

arxiv 2509.09491 v1 pith:HBNYXE7B submitted 2025-09-11 math.CA

The optimal bound $e$ in a dyadic version of Uchiyama's Lemma

J. Fladung , S. Petermichl This is my paper

classification math.CA MSC 42B2030H1060G42

keywords Uchiyama's lemmadyadic Hardy spaceCarleson embeddingsharp constantBellman functionsliced dyadic martingalereproducing kernel thesis4-adic intervals

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 builds a dyadic analogue of the analytic Hardy space from sliced dyadic martingales whose real and imaginary parts satisfy dyadic Cauchy–Riemann equations. In this model, it proves that the embedding associated to a bounded non-negative sliced dyadic supermartingale has norm at most e times the supermartingale's sup norm, and that e cannot be improved. This mirrors the classical Uchiyama lemma in the unit disk, where e is the best known constant and is conjectured to be sharp. The paper also proves a dyadic reproducing kernel thesis: testing on normalized reproducing kernels yields the full embedding with constant 3e. A careful reader would care because this is a sharp-constant result in a dyadic setting that aligns with the conjectured continuous constant.

Core claim

The central claim, Theorem 2, states that for every bounded non-negative sliced dyadic supermartingale M, with µ_I/|I| = -Δ^dy M_I, the space H^2_dy embeds into ℓ²(µ) with constant e: for every dyadic analytic f = u + iv, Σ_I µ_I (u_I² + v_I²) ≤ e ||M||_∞ ||f||²_{H^2_dy}, and this constant is sharp. The upper bound is proved with the Bellman function B̃(F,r,i,M) = eF − e^{1−M}(r²+i²) and a restricted four-point concavity adapted to dyadic analytic increments. The sharpness is proved through an extremal-problem reduction-of-a-variable argument, which forces any admissible constant to be at least e. The paper further derives a dyadic Uchiyama lemma with measure e^{M_I} Δ^dy M_I and a dyadic re

What carries the argument

The key object is the Bellman function B̃(F,r,i,M) = eF − e^{1−M}(r²+i²) on the domain F ≥ r² + i², 0 ≤ M ≤ 1. It satisfies three properties: it is nonnegative and bounded above by eF; it has an increment inequality in M; and it obeys a four-point 'complex concavity' inequality when the increments obey the dyadic Cauchy–Riemann equations. This restricted concavity, rather than full concavity, is what produces the constant e instead of the classical dyadic constant 4. For sharpness, the extremal problem together with a reduction of a variable yields a logarithmic convexity condition that rules out any constant below e.

Load-bearing premise

The proof and the sharpness result apply only to sliced dyadic supermartingales (equivalently, balanced 4-adic sequences); without this slicing condition the Bellman function fails, so if the true dyadic analogue of Uchiyama's lemma requires non-sliced measures, the constant e may not transfer.

What would settle it

Find a bounded non-negative sliced dyadic supermartingale M and a dyadic analytic f = u + iv with Σ_I µ_I (u_I² + v_I²) > e ||M||_∞ ||f||²_{H^2_dy}; Theorem 2 says none exists, so such an example would refute it. Alternatively, evaluate the restricted-concavity inequality (11) for a non-sliced M with parameters d near 0, d₁ = 0, d₂ = 1/2 − d, where Section 7 shows the required quadratic form can become negative.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

The dyadic Uchiyama lemma (Lemma 2) holds: for a non-positive sliced dyadic submartingale M, the measure ν_I = e^{M_I} Δ^dy M_I is Carleson with embedding constant 1.
The dyadic reproducing kernel thesis holds: if a balanced measure µ satisfies the testing condition on normalized reproducing kernels, then the full H^2_dy-to-ℓ²(µ) embedding holds with constant 3e.
The slicing/balancing condition is essential for the upper-bound proof; the Bellman-function mechanism breaks down without it.
Theorem 4 restates the result as a dyadic embedding theorem for balanced non-negative 4-adic sequences satisfying the packing condition, again with the sharp constant e.
Sharpness of e in this dyadic model gives a rigorous lower bound in the dyadic setting, supporting the conjecture that e is optimal in the continuous Uchiyama lemma.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The optimality proof is non-constructive, so explicit near-extremal dyadic analytic functions and sliced supermartingales whose ratio approaches e would make the sharpness concrete.
Because the Bellman function fails for non-sliced supermartingales, a genuinely new mechanism would be needed to extend the result to the full dyadic setting, if such an extension exists.
The constant 3e in the reproducing kernel thesis is probably not optimal; the paper does not aim to optimize it, so a sharper testing constant may be achievable.
If this dyadic model faithfully captures the continuous mechanism, the sharpness of e here suggests the continuous constant e is genuinely optimal rather than an artifact of the exponential proof.

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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

A credible and novel dyadic model of Uchiyama's lemma with sharp constant e, but the optimality proof as written has gaps in the PSD verifications and a compressed lower-bound argument.

read the letter

Colleague—

This paper is worth a serious look. It sets up a genuinely new dyadic model of Uchiyama's lemma—sliced martingales on 4-adic intervals with dyadic Cauchy–Riemann equations—and shows the sharp constant e in that model. The Bellman function eF − e^{1−M}(r²+i²) is a natural adaptation, and the upper bound follows cleanly from its restricted concavity. The dyadic reproducing kernel thesis with constant 3e is a useful byproduct. This is not a routine transcription of the continuous result; the four-point concavity step is a real new ingredient.

The soft spots are mostly in the detail of the quadratic-form checks. In §4.1 the positive semidefiniteness of the matrix in (12) is asserted after checking only the leading principal minors. That is not a sufficient condition for a general 4×4 symmetric matrix, and no Schur complement or congruence argument is supplied. I suspect the matrix is truly PSD, but the written proof does not establish it. In §4.2, the reduction-of-variable argument (credited to the unpublished [7]) is sketched, and the Taylor/mollification passage is compressed. The use of the four principal minors is defensible there, since PSD implies all principal minors are nonnegative, and the two conditions extracted (Φ≥0 and Φ″Φ−(Φ′)²≥0) are exactly what the later ODE needs. Still, the convergence as the increments go to zero needs more care. These are fixable gaps, not obvious errors. The authors also honestly state that the slicing/balancing condition is essential for their Bellman function, which limits the scope but is a fair limitation.

The citation pattern looks appropriate; the conjecture is cited to [12], and the dyadic proof does not lean on it. If the gaps are patched, this will be a solid contribution to the sharp-constant literature. I would send it to a good referee rather than desk-reject, and I would want a full revision addressing the PSD certificates and the lower-bound details before endorsing it.

Referee Report

4 major / 4 minor

Summary. The paper studies a dyadic analogue of Uchiyama's lemma. It introduces a dyadic Hardy space H^2_dy consisting of pairs (u,v) of sliced dyadic martingales satisfying dyadic Cauchy–Riemann relations, together with sliced dyadic supermartingales M. The main result, Theorem 2, asserts the embedding inequality ∑_I μ_I(u_I^2+v_I^2) ≤ e ||M||_∞ ||f||^2_{H^2_dy} for the measure μ_I/|I| = -Δ^dy M_I, and claims that the constant e is sharp. The upper bound is proved via a Bellman function eF - e^{1-M}(r^2+i^2), while sharpness is approached through an extremal problem and a 'reduction of a variable' argument attributed to Nazarov–Treil–Volberg. The paper also states a balanced-sequence version (Theorem 4), a dyadic Uchiyama lemma with exponential weight (Lemma 2), and a reproducing-kernel thesis with constant 3e (Theorem 6).

Significance. If the proof is completed, the paper gives the first sharp constant e in a nontrivial dyadic model of Uchiyama's lemma, supporting the conjecture that the continuous constant e is optimal. The Bellman function is explicit and the extremal formulation is elegant; the sharpness mechanism is structurally different from the usual two-point Bellman arguments and is an interesting contribution. The main caveat is that the model is restricted to sliced/balanced objects, and the lower-bound proof is currently not fully rigorous. The paper does not provide machine-checked proofs or code, but the arguments are concrete and plausibly repairable.

major comments (4)

[§4.1, Eq. (12)] The verification that the matrix in (12) is positive semidefinite checks only the leading principal minors. For a symmetric 4×4 matrix, nonnegativity of the leading principal minors is not sufficient for positive semidefiniteness; one needs all principal minors or a Schur-complement/congruence argument. The special block structure makes the claim plausible, but as written the proof of (11) is incomplete. Since (11) is the key step for the upper bound, a complete PSD argument should be supplied.
[§4.2, Hessian matrix and M1–M4] The same PSD issue occurs in the sharpness proof: after deriving the 4×4 Hessian matrix, the text computes only four principal minors M1–M4 and concludes that Φ''Φ−(Φ')²≥0 is necessary and sufficient. For a general symmetric 4×4 matrix this is not a valid criterion. In this particular block structure a Schur complement would reduce the condition to the stated one, but that argument is not given. The conclusion that log Φ is convex, and hence C≥e, depends directly on this step, so the lower bound is not yet fully proved.
[§4.2, reduction of a variable] The lower-bound proof relies on several sketched claims: rotation invariance of the extremal function B, preservation of the inequalities (20) and (21) under the mollification, and the passage from the discrete convexity inequality (17) to the PSD property of the Hessian by letting increments go to zero. These steps are not routine and are partly credited to an unpublished communication [7]. Since sharpness is a central claim, the authors should expand this part into a complete, self-contained proof or explicitly state which assertions are taken from [7] and prove them.
[§5, Lemma 2] The proof of the dyadic Uchiyama lemma applies the Bellman-function properties (10) and (11), which were established on the domain 0≤M≤1, to a non-positive submartingale M. Either the algebraic inequalities extend to all real M (the PSD computation in §4.1 suggests they may), or the proof must be reorganized with the sign change M ↦ −M made explicit. As written, the domain mismatch leaves the proof of Lemma 2 incomplete.

minor comments (4)

[§1] There are typos: 'F ormulations using testing' and 'refererred' should be corrected.
[§1, balanced condition (2)] The notation I^x and I^y is used in the definition of a balanced sequence before it is introduced in §2. Please define it at first use.
[§4.2] The interval '1−2ε≤M≤ε' after inequality (20) appears to be a typo; the argument seems to require 1−2ε≤M≤1−ε.
[References] Reference [7] is an unpublished personal communication. Since the lower-bound argument depends on it, the authors should either provide a full proof in the paper or give a public, citable source.

Circularity Check

0 steps flagged

No significant circularity: the dyadic proof is self-contained; self-citations are contextual, not load-bearing.

full rationale

The paper's central result, Theorem 2, is proved by an explicit Bellman function B̃(F,r,i,M)=eF−e^{1−M}(r²+i²), with the necessary inequalities (9)–(11) verified directly by calculation and by a positive-semidefiniteness argument. The sharpness proof in §4.2 is an extremal-problem argument: assuming an embedding constant C, it derives properties of the reduced function b and ultimately shows C≥e. This does not presuppose the sharpness of e; it uses the assumed upper bound as a hypothesis in a contradiction argument, which is standard for Bellman optimality. The only self-citation that could be load-bearing is [12] (Petermichl–Treil–Wick), cited for the conjecture that e is optimal in the continuous case and for the reproducing kernel thesis. However, the dyadic upper and lower bounds do not use [12]; Theorem 6 is explicitly derived 'utilize Theorem 2 directly instead of following the strategy in [12].' The citation to [7] (Nazarov–Treil–Volberg, personal communication) is for the 'reduction of a variable' method, but the argument is re-derived in the text, and the authors are not among the present authors. Section 7's admission that the Bellman function requires the slicing/balancing condition is a stated limitation of the method, not a circular reduction: it restricts the class of admissible measures but does not assume the conclusion. The abbreviated verification of positive semidefiniteness via four principal minors is a possible rigor gap in the lower-bound proof, but it is a correctness concern, not a case of the derivation being equivalent to its inputs. Overall, the proof chain is self-contained and the constant e is derived, not fitted or imported as a conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The paper has no fitted numeric parameters. It introduces a new dyadic model (H^2_dy and the slicing/balancing condition) as domain assumptions. The Bellman function technique and the unpublished 'reduction of a variable' method are relied upon, but the derivations are internal to the paper.

axioms (3)

domain assumption The dyadic model: sliced dyadic martingales with 4-adic intervals and dyadic Cauchy-Riemann equations define the space H^2_dy.
Introduced in Section 2. This is an ad hoc model for analytic functions, not derived from the continuous theory. All theorems apply only to this model.
standard math The Bellman function method and the 'reduction of a variable' argument are valid tools for proving sharp constants.
Used in Sections 4.1 and 4.2. The method is standard in the field, but the 'reduction of a variable' argument is cited as unpublished personal communication [7].
domain assumption Mollification of the function Φ preserves the inequalities (20) and (21), and the limiting argument as the mollification parameter tends to zero is valid.
In Section 4.2, the authors mollify Φ to obtain smooth approximations and then take limits. The technical details are not fully shown, but the argument is standard if the inequalities are stable under convolution.

invented entities (2)

Dyadic Hardy space H^2_dy no independent evidence
purpose: The space of sliced dyadic martingales with dyadic Cauchy-Riemann equations, on which the embedding theorems are stated.
Newly defined in Section 2. No external evidence; its properties are proven in this paper.
Slicing / balancing condition no independent evidence
purpose: A structural condition on the measure sequence µ (or supermartingale M) that restricts the model so the Bellman function works.
Introduced in Section 1 and discussed in Section 7, which shows the condition cannot be removed for the chosen Bellman function. It is an ad hoc condition for this model.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of The optimal bound $e$ in a dyadic version of Uchiyama's Lemma." pith.science (2026). https://pith.science/paper/HBNYXE7B

@misc{pith2026250909491,
  author       = {Pith},
  title        = {Pith review of: The optimal bound $e$ in a dyadic version of Uchiyama's Lemma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBNYXE7B}},
  note         = {Machine review of arXiv:2509.09491}
}

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abstract

The best known constant in Uchiyama's Lemma is $e$. A conjecture states that this cannot be improved. We show that the constant $e$ also stands in a dyadic version of Uchiyama's Lemma. Further, we prove that in the dyadic case, the constant $e$ can indeed not be improved. We deduce a dyadic version of the reproducing kernel thesis for the embedding theorem.

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 1 linked inside Pith

[1]

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Pith/arXiv arXiv
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Nazarov, G

F. Nazarov, G. Pisier, S. Treil, A. Volberg,Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts.J. Reine Angew. Math. 542: 147–171, 2002

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Nazarov, S

F. Nazarov, S. Treil, A. Volberg.The Bellmann functions and optimality in the dyadic Carleson Lemma.Personal communication, 1998

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[8]

Nazarov, S

F. Nazarov, S. Treil, A. Volberg.The Bellmann functions and two-weight inequalities for Haar multipliers.J. Amer. Math. Soc. 12(4): 909–928, 1999

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[9]

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S. Petermichl.Dyadic shift and a logarithmic estimate for Hankel operators with matrix symbol.C. R. Acad. Sci. Paris S´ er. I Math., 330(6): 455–460, 2000

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Petermichl, S

S. Petermichl, S. Pott.A version of Burkholder’s theorem for operator- weighted spaces.Proc. Amer. Math. Soc., 131(11): 3457–3461, 2003

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[12]

Petermichl, S

S. Petermichl, S. Treil, B. Wick.Carleson potentials and the reproducing kernel thesis for embedding theorems.Ill. Math. J., 51(4): 1249–1263, 2007

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[13]

Treil.SharpA 2 estimates of Haar shifts via Bellman function.Theta Ser

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.

[1] [1]

Bourgain.Some remarks on Banach spaces in which martingale difference sequences are unconditional.Ark

J. Bourgain.Some remarks on Banach spaces in which martingale difference sequences are unconditional.Ark. Mat., 21(2): 163–168, 1983

1983

[2] [2]

Carleson.Interpolations by Bounded Analytic Functions and the Corona Problem.Ann

L. Carleson.Interpolations by Bounded Analytic Functions and the Corona Problem.Ann. Math., 76(3): 547–559, 1962

1962

[3] [3]

Culiuc, S

A. Culiuc, S. Treil.The Carleson embedding theorem with matrix weights Int. Math. Res. Not., 2019(11): 3301–3312, 2019

2019

[4] [4]

Domelevo, S

K. Domelevo, S. Petermichl.A dyadic reduction of the UMD conjecture and a two-sided linear estimate

[5] [5]

Domelevo, S

K. Domelevo, S. Petermichl, S. Treil, A. Volberg.The matrixA 2 conjecture fails.arXiv:2402.06961

Pith/arXiv arXiv

[6] [6]

Nazarov, G

F. Nazarov, G. Pisier, S. Treil, A. Volberg,Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts.J. Reine Angew. Math. 542: 147–171, 2002

2002

[7] [7]

Nazarov, S

F. Nazarov, S. Treil, A. Volberg.The Bellmann functions and optimality in the dyadic Carleson Lemma.Personal communication, 1998

1998

[8] [8]

Nazarov, S

F. Nazarov, S. Treil, A. Volberg.The Bellmann functions and two-weight inequalities for Haar multipliers.J. Amer. Math. Soc. 12(4): 909–928, 1999

1999

[9] [9]

Nikolskii.Treatise on the shift operator.Grundlehren der Mathematis- chen Wissenschaften, vol

N. Nikolskii.Treatise on the shift operator.Grundlehren der Mathematis- chen Wissenschaften, vol. 273, Springer-Verlag, Berlin, 1986. 27

1986

[10] [10]

Petermichl.Dyadic shift and a logarithmic estimate for Hankel operators with matrix symbol.C

S. Petermichl.Dyadic shift and a logarithmic estimate for Hankel operators with matrix symbol.C. R. Acad. Sci. Paris S´ er. I Math., 330(6): 455–460, 2000

2000

[11] [11]

Petermichl, S

S. Petermichl, S. Pott.A version of Burkholder’s theorem for operator- weighted spaces.Proc. Amer. Math. Soc., 131(11): 3457–3461, 2003

2003

[12] [12]

Petermichl, S

S. Petermichl, S. Treil, B. Wick.Carleson potentials and the reproducing kernel thesis for embedding theorems.Ill. Math. J., 51(4): 1249–1263, 2007

2007

[13] [13]

Treil.SharpA 2 estimates of Haar shifts via Bellman function.Theta Ser

S. Treil.SharpA 2 estimates of Haar shifts via Bellman function.Theta Ser. Adv. Math., 16: 187–208, 2013. 28

2013

The optimal bound $e$ in a dyadic version of Uchiyama's Lemma

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it