The optimal symmetric quasi-Banach range of the discrete Hilbert transform

K.S. Tulenov

REVIEW 3 major objections 5 minor 17 references

The optimal symmetric quasi-Banach range of the discrete Hilbert transform

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any symmetric quasi-Banach sequence space $E$ contained in $\ell_{\log}$, the least symmetric quasi-Banach range $F$ for the discrete Hilbert transform is characterized by $\mu(x) \le S_d\mu(y)$ for some $y\in E$.

desk verdict The paper has the right shape for solving a real open problem, but two load-bearing proof steps are unsupported and one is simply false as written. read the letter →

arxiv 1908.09542 v1 pith:W5AH7JKK submitted 2019-08-26 math.FA

K.S. Tulenov This is my paper · ORCID

classification math.FA MSC 46E3047B1046L5146L5244A1547L2047C15

keywords symmetricquasi-BanachsequencespacesdiscreteHilberttransformCalderónoperatoroptimalrangeLorentzweakℓ1decreasingrearrangementℓlogspace

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 addresses Problem 1: given a symmetric quasi-Banach sequence space $E(\mathbb{Z})$, find the smallest symmetric quasi-Banach space $F(\mathbb{Z})$ that contains the image of the discrete Hilbert transform $H_d$. The answer, under the mild assumption $E(\mathbb{Z}_+)\subset \ell_{\log}(\mathbb{Z}_+)$, is the space $F(\mathbb{Z})$ of sequences $x$ whose decreasing rearrangement satisfies $\mu(x)\le S_d\mu(y)$ for some $y\in E(\mathbb{Z})$, where $S_d$ is the discrete Calderón operator and the quasi-norm on $F$ is the infimum of $\|y\|_E$ over such $y$. The paper proves that $F$ is symmetric quasi-Banach, that $S_d$ maps $E$ into $F$, and that every symmetric quasi-Banach range for $H_d$ on $E$ contains $F$, so $F$ is optimal. For $E=\ell_{1,\infty}$, the optimal range is the set of sequences with $\mu(n,x)\le c_x\log(n+2)/(n+1)$, which makes Komori's weak-$\ell_1$ estimate optimal rather than merely qualitative. Read sympathetically, the main result solves Problem 1 for every symmetric quasi-Banach domain contained in $\ell_{\log}$.

What carries the argument

The load-bearing object is the discrete Calderón operator $S_d$, defined by $(S_dx)(n)=\frac{1}{n+1}\sum_{k=0}^n x(k)+\sum_{k=n+1}^\infty \frac{x(k)}{k}$. The optimal range $F$ is generated by $S_d$ acting on decreasing rearrangements: $x\in F$ exactly when $\mu(x)$ is dominated by $S_d\mu(y)$ for some $y\in E$. This operator is positive, its kernel is decreasing in $k$, it commutes with the discrete dilations used in the completeness proof, and it dominates the discrete Hilbert transform on rearrangements, which is why the range question for $H_d$ collapses to the range question for $S_d$. Completeness of $F$ is assembled from a convergence-in-measure rearrangement inequality, boundedness of dilation operators in quasi-Banach symmetric spaces, and Aoki–Rolewicz metrization.

What would settle it

Compute the quantity $\sup\{\|S_d\mu(y)\|_{\ell_{1,\infty}+\ell_\infty}/\|y\|_E : y\in E\}$ for a symmetric quasi-Banach space $E$ strictly inside $\ell_{\log}$, such as a Lorentz space with logarithmic weight and trivial Boyd indices. If this supremum is infinite, the separation argument in Lemma 8 collapses and $F$ would not be a quasi-Banach space, contradicting Theorem 6; if it is always finite, the gap is only in the exposition.

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Extended reading notes

Core claim

Given a symmetric quasi-Banach sequence space $E$ on $\mathbb{Z}$ with $E(\mathbb{Z}_+)\subset \ell_{\log}(\mathbb{Z}_+)$, define $F(\mathbb{Z})$ by $\mu(x)\le S_d\mu(y)$ for some $y\in E(\mathbb{Z})$ and $\|x\|_F=\inf\{\|y\|_E: \mu(x)\le S_d\mu(y)\}$. The paper's central claim is that $F(\mathbb{Z})$ is the optimal symmetric quasi-Banach range for the discrete Hilbert transform on $E$: the map $H_d:E(\mathbb{Z})\to F(\mathbb{Z})$ is bounded, and any symmetric quasi-Banach space $G(\mathbb{Z})$ with $H_d:E(\mathbb{Z})\to G(\mathbb{Z})$ must contain $F(\mathbb{Z})$. The proof first establishes the analogous optimal-range statement for the discrete Calderón operator $S_d$ on $\mathbb{Z}_+$, then transfers it to $H_d$ through the rearrangement domination $\mu(H_dx)\le c\,S_d\mu(x)$. For the domain $E=\ell_{1,\infty}$, the optimal range is explicitly the set of sequences $a$ with $\mu(n,a)\le c_a\log(n+2)/(n+1)$.

Load-bearing premise

The load-bearing premise is that the positive operator $S_d$ is automatically bounded from every symmetric quasi-Banach space $E\subset\ell_{\log}$ into $\ell_{1,\infty}+\ell_\infty$, a step the proof borrows from a Banach-lattice theorem without adapting it to quasi-Banach spaces.

Editorial extensions

If this is right

Problem 1 is solved for every symmetric quasi-Banach sequence space $E$ with $E(\mathbb{Z}_+)\subset \ell_{\log}(\mathbb{Z}_+)$: the optimal range is given explicitly by the $S_d$-submajorization construction.
For $E=\ell_{1,\infty}$, the optimal range for both $S_d$ and $H_d$ is the set with $\mu(n,x)\le c\log(n+2)/(n+1)$, so the logarithmic correction to weak-$\ell_1$ is now known to be optimal.
Any symmetric quasi-Banach space $G$ containing $H_d(E)$ must contain $F$, so testing a candidate range reduces to comparing it with $F$; boundedness $H_d:E\to G$ and $F\subset G$ are equivalent.
The same space $F$ serves as the optimal range for the Calderón operator $S_d$ on $\mathbb{Z}_+$, so the construction covers both operators in one framework.

Reading between the lines

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

The same template—define a range by $\mu(x)\le T\mu(y)$ for a positive integral operator $T$, prove minimality, then transfer via pointwise domination by a singular transform—should apply to other discrete singular operators whose kernels are controlled by a monotone model operator.
For Lorentz domains $\ell_{p,\infty}$, evaluating $S_d$ on $\mu(k)=(k+1)^{-1/p}$ gives a concrete conjecture for the optimal range, namely $\mu(n)\le C n^{-1/p}\log(n+2)$, interpolating between the paper's weak-$\ell_1$ example and Hardy's inequality.
The same submajorization construction on $\mathbb{Z}$ suggests a continuous analogue: for symmetric quasi-Banach function spaces on $\mathbb{R}$, the continuous Calderón operator should identify optimal rearrangement-invariant ranges for the Hilbert transform.

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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 discrete optimal-range result is genuinely new and worth knowing about: it extends the Banach-space theorem of Andersen and the weak-ℓ1 result of Komori to symmetric quasi-Banach sequence spaces, and it gives a concrete range formula for the weak-ℓ1 domain. The construction via the Calderón operator is standard and the main idea is plausible. If the result stands, it answers Problem 1 for every symmetric quasi-Banach sequence space contained in ℓlog.

The paper is also honest: it mostly adapts the continuous version from [16] and the discrete obstacles are real. But the proof as written does not close those obstacles.

First, Lemma 8 concludes that the positive operator Sd is bounded from the quasi-Banach lattice E into ℓ1,∞+ℓ∞ by citing Meyer-Nieberg [12, Prop 1.3.5]. That proposition is a Banach-lattice result; E is only quasi-Banach, and ℓ1,∞+ℓ∞ is not a Banach lattice. Positivity does not automatically give boundedness in this setting, and no substitute argument is supplied. This boundedness is then used to show the quasi-norm on F separates points, so the whole space F rests on an unproved step.

Second, (3.7) asserts Sdσ_{2^k} = σ_{2^k}Sd for the block-repetition dilation defined in the paper. That equality is false. With z=(1,0,0,...), σ2Sdz(0)=1 while Sdσ2z(0)=2. The inequality σSdz ≤ Sdσz appears to hold, but the proof uses the equality to interchange Sd with an infinite sum, so the argument for completeness of F is broken.

These are not cosmetic issues. They sit in the proof of Theorem 6, which is the foundation of everything else. The result may well be true and a repair may be possible, but the manuscript does not currently support it.

Who is this for? Specialists in rearrangement-invariant spaces and quasi-Banach lattices. A serious referee could work through the gaps and tell whether the theorem is salvageable, so it deserves referee time rather than a desk rejection. I would not cite it in my own work until the proof is fixed.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a construction of the optimal symmetric quasi-Banach range space F for the discrete Calderón operator Sd and the discrete Hilbert transform Hd. For a symmetric quasi-Banach sequence space E with E(Z+) contained in ℓlog(Z+), it defines F by the condition µ(x) ≤ Sdµ(y) for some y ∈ E, with a natural quasi-norm. Theorem 6 claims that F is the minimal symmetric quasi-Banach range for Sd: Sd maps E boundedly into F, and F embeds into every other symmetric quasi-Banach range space. Theorem 10 transfers this result to the discrete Hilbert transform, and Proposition 11 gives an explicit description of the range when E = ℓ1,∞. The core strategy is a direct domination argument using the Calderón operator and the decreasing rearrangement, similar to constructions in the continuous case.

Significance. If the proof gaps are repaired, the result would solve Problem 1 for all symmetric quasi-Banach sequence spaces contained in the maximal domain ℓlog, substantially extending classical results of Boyd, Andersen, Komori, and the author's prior work. The construction is parameter-free, the optimality argument is the natural minimality-by-domination argument, and the explicit example for ℓ1,∞ is a concrete and useful application. However, the current proof relies on a Banach-lattice positivity theorem in a quasi-Banach setting and on a commutation identity that is false as stated; these are load-bearing issues that must be addressed before the central claims can be accepted.

major comments (3)

[Section 3, Lemma 8 and Theorem 6] The boundedness of Sd from E(Z+) into (ℓ1,∞+ℓ∞)(Z+) is asserted by citing [12, Proposition 1.3.5], a Banach-lattice theorem. Since E is only a quasi-Banach lattice, this proposition does not apply directly, and no alternative argument is supplied. This boundedness is then used in Lemma 8 to show that the quasi-norm on F separates points, and again in the proof of Theorem 6 when the phrase 'Sd is continuous on E by assumption' is invoked. If Sd is not known to be bounded, the space F may not be a quasi-Banach space, which would invalidate the optimal-range claim in Theorem 6. The author should either prove the required positivity-boundedness implication in the quasi-Banach setting or replace this step with a direct argument.
[Section 3, Eq. (3.7) in the proof of Theorem 6] The proof asserts that Sd commutes with the dilation operator σ_{2^k}, i.e., that σ_{2^k} Sd = Sd σ_{2^k} for the block-repetition dilation defined in Section 2.1. This equality is false: for x = δ0 (the sequence with 1 at 0 and 0 elsewhere), σ2 Sd x(0) = 1, whereas Sd σ2 x(0) = 2. The inequality σ_{2^k} Sd z ≤ Sd σ_{2^k} z appears to hold and might suffice for the argument, but the equality as written is not available. The interchange of Sd with the infinite sum in (3.7) therefore needs a corrected justification, as does the conclusion that the series ∑ σ_{2^k} µ(x_{k+1} − x_k) belongs to F(Z+).
[Theorem 10] Theorem 10 states an optimal-range result for the Hilbert transform on E(Z), but the hypotheses are inherited from Theorem 6, which concerns a symmetric quasi-Banach sequence space on Z+ with E(Z+) ⊂ ℓlog(Z+). No assumption is stated on the two-sided space E(Z), such as E(Z) ⊂ ℓlog(Z), although the introduction and Problem 1 explicitly require E(Z) ⊂ ℓlog(Z). As written, the statement of Theorem 10 does not have well-defined hypotheses: E is a space on Z+ in Theorem 6, while Hd acts on sequences indexed by Z. The author should state the assumptions on E(Z) explicitly, or clarify how a symmetric space on Z+ gives a space on Z with the properties used in the proof.

minor comments (5)

[Abstract and keywords] The abstract and keywords contain typographical errors: 'Calder´on' appears as 'Calde r´on', 'range' as 'ran ge', 'Hilbert transform' as 'Hilbe rt transform' in the keywords. These should be fixed.
[Definition 5] The wording of Definition 5 is awkward: the set F(Z+) is defined by a condition involving existence of y, and then a norm is defined separately. The phrase 'such that (3.4)' makes it appear that the norm is part of the set definition. It would be clearer to define F(Z+) as a set and then define the functional ‖·‖F by (3.4).
[Proof of Theorem 6] In the proof of Theorem 6, the text 'the series ∑∞ k=1(xk+1 − kn) converges in measure' appears to contain a typo; it should read ∑∞ k=1(xk+1 − xk).
[Remark 9 and Theorem 10] The passage from the one-sided space F(Z+) to the two-sided space F(Z) in Remark 9 is announced without proof. Since the symmetric sequence space on Z has a different rearrangement structure, a brief explanation or reference is needed to justify that the same construction and completeness argument work on Z.
[References] The proof of Theorem 6 uses [15, Remark 18] for the quasi-triangle inequality and dilation estimates; it would be helpful to state the precise content of that remark in the text, since it is used in a load-bearing way.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the optimal range is constructed explicitly from E and Sd, and optimality is verified against arbitrary competitors.

full rationale

The paper's central construction, Definition 5, defines the candidate optimal range F(Z+) directly as the symmetric hull of Sd(E): a sequence x belongs to F exactly when μ(x) ≤ Sdμ(y) for some y in E, with the quasi-norm given by the infimum of ‖y‖_E over such y. Theorem 6(ii) then proves boundedness of Sd: E → F using the already-established inequality μ(Sdx) ≤ Sdμ(x) from (2.4), and proves optimality by taking an arbitrary competitor G with Sd: E → G bounded: from μ(x) ≤ Sdμ(y) one immediately gets ‖x‖_G ≤ ‖Sd‖ ‖y‖_E, and taking the infimum gives ‖x‖_G ≤ ‖Sd‖ ‖x‖_F. This is a genuine construction-and-verification argument, not a fitted parameter renamed as a prediction and not a conclusion imported by definition. The citations to the author's prior work [16] are for continuous or noncommutative analogues and are not load-bearing for the discrete result; the proof of Lemma 4 is supplied in the paper, and Proposition 11 is an application rather than a premise. The main weaknesses identified by a skeptical reader — the use of a Banach-lattice positivity argument [12, Proposition 1.3.5] in a quasi-Banach setting and the asserted commutation σ2k Sd = Sd σ2k in (3.7) — are correctness gaps in the proof of completeness, not circular reductions. No equation or constructed space reduces by construction to its own target, so the circularity score is 0.

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

The paper introduces no free parameters and no new postulated entities. Its central claim rests on the domain inclusion E subset ell-log, on a positivity-boundedness step that is not fully justified for quasi-Banach lattices, and on two discrete Bennett-Sharpley propositions taken from external references.

assumptions (3)

domain assumption E is a symmetric quasi-Banach sequence space with E(Z+) contained in ell-log(Z+).
This is the standing hypothesis of Theorems 6 and 10, and it ensures that the Calderon operator and Hilbert transform are defined on E.
ad hoc to paper The inclusion of E into ell-log is continuous and Sd maps E boundedly into ell-1,infinity plus ell-infinity.
Invoked in Lemma 8 to prove that the quasi-norm on F separates points, via [12, Proposition 1.3.5]. The paper applies a Banach-lattice theorem to a quasi-Banach lattice without proving the extension.
domain assumption The discrete versions of Bennett-Sharpley Propositions III.4.8 and III.4.10 hold as stated.
Theorem 10 uses discrete versions of these propositions to bound Hd by Sd and to majorize Sd by a Hilbert transform of an equimeasurable function. The paper cites them without stating or proving them.

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Cite this review

Pith. "Pith review of The optimal symmetric quasi-Banach range of the discrete Hilbert transform." pith.science (2026). https://pith.science/paper/W5AH7JKK

@misc{pith2026190809542,
  author       = {Pith},
  title        = {Pith review of: The optimal symmetric quasi-Banach range of the discrete Hilbert transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5AH7JKK}},
  note         = {Machine review of arXiv:1908.09542}
}

abstract

We identify symmetric quasi-Banach range of the discrete Calder\'{o}n operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

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

[1] [16]

Sukochev, K

F. Sukochev, K. Tulenov, and D. Zanin, The optimal range of the Calder´ on operator and its applications. J. Func. Anal. (2019), 1–47. doi.org/10.1016/j.jfa.2019. 05.012 (preprint) THE OPTIMAL SYMMETRIC QUASI-BANACH RANGE OF THE DISCRETE HI LBERT TRANSFORM 11 Al-F arabi Kazakh National University, 050040 Almaty, Kazakhst an; Institute of Mathematics and M...

doi:10.1016/j.jfa.2019 2019

[2] [1]

Andersen, Discrete Hilbert Transforms and Rearrangement invariant S equence Spaces, Applicable analysis, 5 (1976), 193–200

K.F. Andersen, Discrete Hilbert Transforms and Rearrangement invariant S equence Spaces, Applicable analysis, 5 (1976), 193–200

work page 1976

[3] [2]

Bennett and R

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics,

work page

[4] [3]

Boyd, The Hilbert transform on rearrangement-invariant spaces , Can

D. Boyd, The Hilbert transform on rearrangement-invariant spaces , Can. J. Math. 19 (1967), 599–616

work page 1967

[5] [4]

Hudzik, L

H. Hudzik, L. Maligranda, An interpolation theorem in symmetric function F-spaces , Proc. Amer. Math. Soc. 110 (1) (1990), 89–96

work page 1990

[6] [5]

Katznelson, An introduction to Harmonic analysis , Third Corrected Edition, Cambridge University Press, (2012)

Y. Katznelson, An introduction to Harmonic analysis , Third Corrected Edition, Cambridge University Press, (2012)

work page 2012

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Kalton, N

N. Kalton, N. Peck, J. Rogers, An F-space Sampler , London Math. Soc. Lecture Note Ser., 89, Cambridge University Press, Cambridge, 1985

work page 1985

[8] [7]

Kolmogorov, Sur les fonctions harmoniques conjuguees et les series de Fo urier, Fun- damenta Math

A.N. Kolmogorov, Sur les fonctions harmoniques conjuguees et les series de Fo urier, Fun- damenta Math. 7 (1925), 23–28

work page 1925