The optimal range of the Calder\`{o}n operator and its applications

D. Zanin; F. Sukochev; K. Tulenov

REVIEW 2 major objections 3 minor 43 references

The optimal range of the Calder\`{o}n operator and its applications

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

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Pith's one-line read The paper identifies the optimal (smallest) symmetric quasi-Banach range space for the Calderón operator, the Hilbert transform, and the triangular truncation operator, for every symmetric quasi-Banach domain inside the Lorentz space…

desk verdict Genuinely new framework for optimal range of the Hilbert transform and triangular truncation in symmetric quasi-Banach spaces, but the discrete lower-bound proof has a concrete parity error in Lemma 23 that needs repair before Theorem 34 is proved as written. read the letter →

arxiv 1908.09548 v1 pith:UEEJTJTW submitted 2019-08-26 math.FA

F. Sukochev , K. Tulenov , D. Zanin This is my paper · ORCID

classification math.FA MSC 46E3047B1046L5146L5244A1547L2047C15

keywords symmetricquasi-BanachspacesCalderónoperatorHilberttransformtriangulartruncationoptimalrangeLipschitzfunctionscommutatorestimatesLorentz

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 settles a long-standing question: given a symmetric quasi-Banach space $E$ of functions or sequences inside the Lorentz space $\Lambda_{\log}$, what is the smallest symmetric quasi-Banach space that receives the Calderón operator $S$, the Hilbert transform, and the triangular truncation operator? The answer is the same space $F$ in all three cases, defined by the condition that the rearrangement of the output is dominated by $S$ applied to the rearrangement of some element of $E$. This extends the classical reduction in [7] from the Banach-space setting to the full quasi-Banach setting, and it sharpens the classical weak-$L^1$ result: for $E = L^1$ the optimal range of the Hilbert transform is the closure of bounded functions in $L^{1,\infty}$.

What carries the argument

The Calderón operator $S$, defined by $(Sx)(t) = \frac{1}{t}\int_0^t x(s)\,ds + \int_t^\infty \frac{x(s)}{s}\,ds$, is the carrying mechanism. Theorem 14 converts quantitative $L^p$ bounds of an operator $T$ into a universal domination $\mu(T(A)) \preceq c_{abs} S\mu(A)$; this is what makes $S$ the universal upper bound. Lemma 23 supplies the matching lower bound via the discrete operator $H_d$ and a rearrangement-preserving construction of a sequence $c$ from $a$, proving that triangular truncation is strong enough to force $F$ to be the smallest possible range.

What would settle it

Take $a$ with $\mu(k,a)=1/(k+1)$ and let $c$ be as in (5.12). For every even $n$, $(H_d c)(n)=0$ because the parity condition only allows odd $k$, where $c$ vanishes; the claimed formula replacing the sum by $\sum_{k\ge0}\mu(k,a)/(n+2k)$ is inconsistent with (5.12)–(5.13). Computing $S_d\mu(a)(n)\sim \log n/n$ and comparing it with the decreasing rearrangement of the odd-index values of $H_d c$ would settle whether any absolute constant can make the domination hold.

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Extended reading notes

Core claim

The central discovery is that for every symmetric quasi-Banach space $E \subset \Lambda_{\log}$, the space $F = \{x : \mu(x) \le S\mu(y) \text{ for some } y \in E\}$, equipped with the natural infimum norm, is itself a symmetric quasi-Banach space and is the optimal symmetric quasi-Banach range for the Calderón operator $S$. The same $F$ serves as the optimal range for the non-commutative Hilbert transform $1\otimes H$ acting on $E(M \bar{\otimes} L^\infty(\mathbb{R}))$ when $M$ is atomless and semifinite, and for the triangular truncation operator $T$ acting on the ideal $E(H)$. The upper bound is proved through an abstract domination theorem: any self-adjoint contraction on $L^2(M)$ whose norm on $L^p(M)$ grows at most like $1/(p-1)$ satisfies $\mu(T(A)) \preceq c_{abs} S\mu(A)$ in the submajorization sense. The lower bound, proving that no smaller symmetric quasi-Banach space can work, is built from a discrete Fourier construction showing that triangular truncation can already reproduce the Calderón operator on rearrangements.

Load-bearing premise

The proof that $F$ is the smallest possible range depends on Lemma 23, whose construction of the sequence $c$ assigns nonzero values only to even indices, while the parity condition $k \equiv n+1 \pmod{2}$ in the definition of $H_d$ excludes those indices when $n$ is even; as written, the displayed lower estimate for $(H_d c)(n)$ does not follow from the stated definitions.

Editorial extensions

If this is right

For $E=L^1$, the optimal range of the Hilbert transform is $(L^{1,\infty})_0$, the closure of bounded functions in $L^{1,\infty}$, refining the classical weak-$L^1$ theorem.
For $E=L^1(H)$, the optimal range of the triangular truncation operator is the weak trace ideal $L^{1,\infty}(H)$, confirming the classical prediction for triangular truncation on the trace class.
For every Lipschitz function $f$ and self-adjoint operators $A,B$, the double operator integral maps $E(H)$ into $F(H)$ and satisfies $\|[f(A),B]\|_{F(H)} \le c\,\|f'\|_\infty \|[A,B]\|_{E(H)}$; the same holds for $f(X)-f(Y)$ with $\|X-Y\|_{E(H)}$.
A noncommutative logarithmic integrability theorem follows: if $\|\mu(A)\log^+(\mu(A))\|_{L^1}<\infty$, then $T(A)\in L^1(M)$ with the corresponding norm bound, covering both the Hilbert transform and triangular truncation.

Reading between the lines

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

Editorial inference: the same formula for $F$ may describe optimal range spaces for any operator dominated by $S$ in rearrangement, including Riesz projections and the absolute value map, giving a uniform template for rearrangement-invariant optimal range problems.
Editorial inference: replacing the triangular truncation lower bound with a direct construction for the Hilbert transform would isolate whether the parity issue in Lemma 23 is essential or an artifact; such a construction would make the proof robust and testable.
Editorial inference: should the Lemma 23 lower bound fail for some $E$, the minimality claim would still hold for symmetric spaces with Fatou norm via the classical argument; the quasi-Banach extension would then require a different lower-bound mechanism.

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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

Short version: this is a substantial paper and the main construction is real. The idea of defining the optimal range space F through domination by the Calderón operator S is clean and genuinely new, and it correctly removes the Fatou/Banach assumptions that Boyd, Gohberg–Krein, and Arazy all needed. The upper-bound machinery in Theorem 14 is careful and mostly convincing, and the applications to operator Lipschitz functions and commutator estimates are valuable. If the proof gaps I name below are fixed, this will be a significant paper.

The soft spots are localized but real. Lemma 23 is the load-bearing step for the lower bound in Theorem 34. As printed, (5.12) defines c(k) to be nonzero only for even nonpositive k, with c(k)=0 for all odd k and all positive k. But the operator Hd in (5.4) sums only over k ≡ n+1 (mod 2). For even n, that means k is odd, so every term vanishes and (Hdc)(n)=0. This directly contradicts the displayed estimate |(Hdc)(n)| ≥ (1/2π) Sdμ(n,a) for even n. The repair is almost certainly to change the parity condition to k ≡ n (mod 2), or equivalently to place the values on the opposite parity; with that change the displayed sum matches the intended argument. But as it stands, Theorem 21 and the optimality half of Theorem 34 are not established by the written proof. The sign inconsistency between (5.4) and Lemma 22 is immaterial for absolute values but should be cleaned up.

There is also a smaller overclaim in Theorem 14(i): the statement covers all of Λ log, but the proof through Lemma 19 only reaches the smaller space Λψ. The later use in Theorem 14(ii) is saved because the element A2 is bounded and does lie in Λψ, so the applications still go through, but the theorem as stated is stronger than what is proved.

The optimality proof for the Hilbert transform in Theorem 33 rests on the classical Bennett–Sharpley lower bound, not on Lemma 23, so that part is on solid ground. The citation pattern looks honest and the new results are clearly separated from old ones.

Who should read this: specialists in symmetric spaces, noncommutative integration, and operator ideals. It deserves a serious referee — the defects are concrete but localized and almost certainly repairable, and the framework is worth engaging with carefully. I would send it to review with a request to fix Lemma 23 and the statement of Theorem 14(i).

Referee Report

2 major / 3 minor

Summary. The paper studies the optimal symmetric quasi-Banach range spaces for the Calderón operator S on (0,∞), for the noncommutative Hilbert transform 1⊗H on spaces over M⊗L∞(R), and for the triangular truncation T on ideals of compact operators. The central construction is F = {x : μ(x) ≤ Sμ(y) for some y ∈ E}, equipped with a quasi-norm defined by an infimum of ‖y‖E. The authors prove an abstract domination theorem (Theorem 14) showing that a self-adjoint contraction with suitable L^p and weak-L^1 bounds is dominated by S on Λlog(M), and they use this to establish the upper-bound halves of the optimal-range theorems. The lower-bound halves are proved via the classical Bennett–Sharpley result for the Hilbert transform and via a new noncommutative analogue (Theorem 21) for T. Section 8 applies these results to double operator integrals and commutator/Lipschitz estimates. The stated goal is to resolve Problems 1 and 2 in full generality, removing the Fatou-norm assumption used in earlier Boyd-type results.

Significance. If correct, the paper would solve two longstanding problems in the optimal-range theory of symmetric quasi-Banach spaces and would provide a unified noncommutative framework for the Hilbert transform and the triangular truncation. The construction of F is explicit, parameter-free, and defined directly from the operator S, with no fitted constants; the upper-bound arguments in Lemmas 18 and 19 and Theorem 14 are careful and mostly complete. The optimality claim for the triangular truncation, however, rests on Lemma 23, which contains a genuine parity mismatch that invalidates the proof as written. This defect is localized and appears repairable, but it blocks the minimality half of Theorem 34 and the applications that depend on it. The lower bound for the Hilbert transform in Theorem 33 is not affected, since it uses the external classical result [3, Proposition III.4.10].

major comments (2)

[§5, Lemma 23 (Eqs. (5.12)–(5.13))] The sequence c defined in (5.12) is nonzero only at nonpositive even integers and is zero at all positive integers and at all odd integers. For n ≥ 0 even, the parity condition in (5.13) (k ≡ n+1 mod 2) selects only odd k, for which c(k)=0 by (5.12). Hence (Hdc)(n)=0 for every even n ≥ 0, so the displayed lower bound |(Hdc)(n)| ≥ (1/2π) Sdμ(n,a) cannot hold for these n. Since the subsequent rearrangement step relies on this estimate, the proof of Lemma 23 and therefore of Theorem 21 is not valid as written.
[§7, Theorem 34 and §8, Theorem 37] The optimality (minimality) direction of Theorem 34 is proved by invoking Theorem 21, whose proof depends on Lemma 23. With Lemma 23 uncorrected, the conclusion F(H) ⊂ G(H) for an arbitrary symmetric quasi-Banach ideal G with T(E(H)) ⊂ G(H) is not established. The applications in Theorem 37 that cite Theorem 34 are consequently unsupported as written. This defect does not affect the lower bound in Theorem 33, which cites [3, Proposition III.4.10] rather than Lemma 23; however, Theorem 34 is one of the headline results.

minor comments (3)

[§5, Eq. (5.4) vs. Eq. (5.11)] There is a sign inconsistency: (5.4) defines (Hda)(n) with denominator (k−n), whereas (5.11) and Lemma 22 use denominator (n−k) with the same parity condition. Because all later estimates are taken in absolute value, this does not alter the inequalities, but the two definitions should be reconciled.
[§3, after Theorem 14] The sentence 'A ≈ B means A ≲ B and A /greaterorsimilarB' contains an unrendered symbol; it should read 'A ≳ B'.
[§6, Proof of Theorem 26] In the optimality part of the proof, after showing Sμ(x) ∈ G(0,∞) for x ∈ E(0,∞), the text says 'Hence, F(0,∞) ⊂ G(0,∞)' without explicitly invoking the definition of F. The inference is correct but should be spelled out for clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the range space F is constructed from the Calder\'on operator and optimality rests on external lower bounds; self-citations are used only as published lemmas.

full rationale

The paper's central construction is not fitted: F is defined in Definition 25 (and its discrete analogue Definition 29) as the class of x with µ(x) ≤ Sµ(y) for some y ∈ E, and the optimal-range theorems are then proven by (a) showing S maps E into F by definition, (b) proving F is a quasi-Banach symmetric space, and (c) using lower-bound results to show any symmetric range G must contain Sµ(E). For the Hilbert transform (Theorem 33), the lower bound is the classical Bennett–Sharpley result [3, Proposition III.4.10]; for triangular truncation (Theorem 34), the lower bound is Theorem 21, proved in the paper via Lemma 22 and Lemma 23. No parameter is fitted to a subset of data and then renamed a prediction. Citations to the authors' own prior work ([16], [25], [41], etc.) are used as published, parameter-free ingredients (e.g., symmetric-space completeness and dilation bounds) rather than as the target result; none of these citations is invoked to forbid alternatives or to supply the optimality conclusion itself. The possible parity defect in Lemma 23 identified by the reader is a correctness gap in the proof of the lower estimate, not a circular step: it concerns whether the displayed inequality follows from the stated definitions, not whether the conclusion is assumed as an input. Accordingly the paper merits a low circularity score, with the proof defect noted separately as a correctness risk.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are introduced. The proof relies on standard published theorems in noncommutative integration, interpolation, and rearrangement-invariant spaces. The main external dependencies are the unpublished manuscript [18] and the classical lower bound [3, Proposition III.4.10].

assumptions (6)

standard math Kalton-Sukochev theorem: for a symmetric quasi-Banach function space E, the set E(M) = {A : µ(A) ∈ E} is a complete symmetric operator space.
Invoked in Section 2.2 and used throughout to transfer commutative results to noncommutative operator spaces.
standard math Kothe duality identifications, including (M1,∞ + (L1 ∩ L2))× = Λ log ∩ (L2 + L∞) and noncommutative analogues.
Used in Lemma 17 and in Step 2 of Lemma 18 for duality of (L2 + L∞) with (L1 ∩ L2).
standard math Interpolation result [13, Theorem 4.8] giving Lp bounds with constant 1/(p-1) from weak L1 and L2 boundedness.
Used in the proof of Theorem 14(ii) to invoke part (i) for bounded inputs.
domain assumption Equivalence sup_{1<p≤2} (p-1)||a||_{ℓp} ≈ ||a||_{M1,∞} from [9, Theorem 4.5].
Used in Lemma 16; published but partially authored by members of the same group.
standard math Bennett-Sharpley [3, Proposition III.4.10]: for x in the domain of H there exists y with µ(y) = µ(x) and Sµ(x) ≤ c µ(Hy).
Load-bearing for the lower bound in Theorems 33 and 34.
domain assumption Double operator integral estimates for Lipschitz symbols, [11, Theorem 1.2], supply the weak-type and L2 assumptions required by Theorem 14.
Used in Theorem 37 to transfer the optimal range result to double operator integrals and commutator estimates.

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Pith. "Pith review of The optimal range of the Calder\`{o}n operator and its applications." pith.science (2026). https://pith.science/paper/UEEJTJTW

@misc{pith2026190809548,
  author       = {Pith},
  title        = {Pith review of: The optimal range of the Calder\`on operator and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEEJTJTW}},
  note         = {Machine review of arXiv:1908.09548}
}

read the original abstract

We identify the optimal range of the Calder\`{o}n operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.

Discussion (0). Continue with ORCID to comment.

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Pith tools

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

[1] [1]

Arazy, Some remarks on interpolation theorems and the boundedness of the triangular projection in unitary matrix spaces , Integr

J. Arazy, Some remarks on interpolation theorems and the boundedness of the triangular projection in unitary matrix spaces , Integr. Equ. Oper. theory, 1:4 (1978), 453–495

work page 1978

[2] [2]

Astashkin and F.A

S.V. Astashkin and F.A. Sukochev, Series of independent random variables in rearrangement invariant spaces: An operator approach , Israel J. Math. 145 (2005), 125–156

work page 2005

[3] [3]

Bennett and R

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

work page

[4] [4]

Berkson, T.A

E. Berkson, T.A. Gillespie, and P.S. Muhly, Abstract spectral decomposition guaranteed by the Hilbert transform , Proc. London Math. Soc. 53 (1986), 489–517

work page 1986

[5] [5]

Birman, M

M. Birman, M. Solomyak, Spectral theory of selfadjoint operators in Hilbert space , Mathe- matics and its Applications (Soviet Series). D. Reidel Publ ishing Co., Dordrecht, 1987

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[6] [6]

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 (1983), 163–168

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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

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[8] [8]

Burholder, A geometrical characterization of Banach spaces in which ma rtingale dif- ference sequences are unconditional , Ann

D.L. Burholder, A geometrical characterization of Banach spaces in which ma rtingale dif- ference sequences are unconditional , Ann. Prob., 9 (1981), 997–1011

work page 1981