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REVIEW 3 major objections 4 minor 20 references

On Herz-Bochkarev limiting problem

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the constant in the Lorentz-space Hausdorff–Young inequality blows up at a precise rate as the integrability index $p$ tends to $2$, and that grand Lorentz norms absorb this singularity to yield endpoint inequalities.

desk verdict The main theorems fail because the orthonormal system is not assumed uniformly bounded; the grand Lorentz framework is salvageable but this version should be rejected. read the letter →

arxiv 2506.07478 v1 pith:URSQQ4JQ submitted 2025-06-09 math.FA math.APmath.CA

Erlan Nursultanov , Arash Ghorbanalizadeh , Durvudkhan Suragan This is my paper
classification math.FAmath.APmath.CA MSC 46E3042A1626D15
keywords Hausdorff-YounginequalityLorentzspacesgrandFouriercoefficientsorthonormalsystemssharpconstantsinterpolationlimitingendpointestimates
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 the precise rate at which the constant in the Lorentz-space Hausdorff–Young inequality must blow up when the integrability parameter $p$ tends to $2$ from below: like $(1/p-1/2)^{-1/2}$ for $2\le q\le\infty$ and like $(1/p-1/2)^{-1/q}$ for $0

What carries the argument

The engine is the grand Lorentz space $G^\theta L_{p,q}$, defined by taking a supremum over $0<\varepsilon<1$ of $\varepsilon^\theta$ times the Lorentz norm computed at index $p+\varepsilon$ (equivalently, shifting $t^{1/p}$ to $t^{1/p+\varepsilon}$), together with its sequence analogue $G^\theta\ell^*_{p,q}$ built from dyadic block averages $(k^{-1}\sum_{m\le k}(a^*_m)^2)^{1/2}$. The $\varepsilon^\theta$ weight is chosen to cancel the singular factor $\varepsilon^{-1/2}$ or $\varepsilon^{-1/q}$ produced by dyadic Hardy-type inequalities, so that the limiting inequality survives at $p=2$. A companion interpolation theorem for these grand sequence spaces (Theorem 1.5) provides the route to the endpoint spaces $\Lambda_{p,q,\tau}$ used in the final result.

What would settle it

Construct an orthonormal system $\{\varphi_k\}$ with sup-norms tending to infinity and a function $f\in L^{p,q}$ such that $a_m(f)=1$ for every $m$; then $\|a\|_{\ell^{p',q}}=\infty$ while $\|f\|_{L^{p,q}}<\infty$, directly contradicting the theorem as stated for unrestricted orthonormal systems.

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Extended reading notes

Core claim

For an orthonormal system satisfying the uniform bound $|\varphi_k|\le 1$ used in the proofs, the paper proves that the Fourier-coefficient map $f\mapsto (a_m)$ from $L_{p,q}$ to $\ell_{p',q}$ has operator norm growing at the exact rate $(1/p-1/2)^{-1/2}$ when $2\le q\le\infty$ and $(1/p-1/2)^{-1/q}$ when $0<q<2$, as $p\uparrow 2$. These rates are packaged into grand Lorentz norms: for $q\ge2$ the endpoint bound is $\|a\|_{G^{\theta+1/2}\ell^*_{2,q}}\lesssim \|f\|_{G^\theta L_{2,q}}$, with the analogous $\theta+1/q$ shift for $q<2$. An interpolation step then yields $\|a\|_{\Lambda_{2,q,\tau}}\lesssim \|f\|_{L_{2,q,\tau}}$ for $q>2$, and the special case $\tau=q$ produces a sup-type expression on the coefficient side that dominates the logarithmic lower bound previously known. The upshot is a sharp quantitative form of the coefficient inequality at the $p=2$ limiting case.

Load-bearing premise

The load-bearing premise is that the orthonormal system is uniformly bounded by $1$, so that $|a_m(f)|\le\|f\|_{L^1}$ as the proofs use; without this, there are orthonormal systems and functions in $L^p$ whose Fourier coefficients are all equal to $1$, which would make the sequence-side norm infinite while the function norm is finite.

Editorial extensions

If this is right

  • Near $p=2$, the coefficient norm carries the exact factor $(1/p-1/2)^{-1/2}$ for $q\ge2$ and $(1/p-1/2)^{-1/q}$ for $q<2$, so any constant-free Hausdorff-Young statement in Lorentz spaces must include that factor.
  • The endpoint inequality $\|a\|_{G^{1/2}\ell^*_{2,q}}\lesssim\|f\|_{L_{2,q}}$ gives a finite grand-norm bound on Fourier coefficients of every function in $L_{2,q}$, extending the $L^2$ coefficient identity to the limiting case.
  • For $2<q\le\infty$, the interpolated bound $\|a\|_{\Lambda_{2,q,\tau}}\lesssim\|f\|_{L_{2,q,\tau}}$ holds for every $\tau$, and at $\tau=q$ it dominates the logarithmic lower bound $(\log(n+1))^{-(1/2-1/q)}$ for dyadic block sums of coefficients.
  • The grand-Lorentz interpolation theorem (1.6) supplies a reusable interpolation step for sequence spaces of grand type, applicable beyond the specific coefficient map studied here.

Reading between the lines

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

  • The same $\varepsilon$-shift-and-$\varepsilon^\theta$-weight construction is likely a template: any operator bound whose constant blows up at a critical exponent could be converted into an endpoint inequality in a corresponding grand space, with the theorem proved here as the model case.
  • The phase transition in the exponent at $q=2$ (from $1/2$ to $1/q$) suggests that the tail behavior of Lorentz sequence norms, not the function-space side, dictates the optimal $p$-dependence; one could test this by computing the same constants for wavelet or Walsh-type systems.
  • The sup-type expression in (1.7) resembles a maximal function over dyadic blocks, so it may admit an equivalent formulation as a weighted Hardy-Littlewood maximal norm; if so, numerical checks of the endpoint inequality become straightforward.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Hausdorff-Young-type inequalities for Fourier coefficients with respect to orthonormal systems on [0,1], with emphasis on the dependence of the constants on p as p→2. The authors introduce grand Lorentz spaces G^θ L_{p,q} and G^θ ℓ^*_{p,q}, and state: a scalar estimate in Lorentz spaces with constant (1/p−1/2)^{-1/2} (Theorem 1.2), endpoint estimates in grand Lorentz spaces (Theorems 1.3 and 1.4), an interpolation theorem (Theorem 1.5), and a final endpoint inequality in Λ_{2,q,τ} (Theorem 1.6). The paper claims these results refine Bochkarev's lower bound and are comparable to the recent work of Saucedo and Tikhonov. The main proofs consist of decomposing f into low- and high-level parts, applying Parseval's identity to the low part, and using dyadic Hardy inequalities.

Significance. The grand-Lorentz machinery and the quantitative blow-up rates near p=2 are potentially interesting, and the stated comparison with Bochkarev's logarithmic lower bound is a worthwhile goal. If the theorems were correct for the systems they claim, the paper would make a substantial contribution to the Herz-Bochkarev problem. However, the central statements as written are false for arbitrary orthonormal systems, and Theorem 1.2 is internally inconsistent with its own proof. The proofs appear plausible for uniformly bounded orthonormal systems and with a corrected exponent, so the core ideas may be salvageable, but the present manuscript requires substantial revision.

major comments (3)
  1. [§3.1, Lemma 3.1] The proof of Lemma 3.1 uses the estimate |a_m(f)| ≤ ||f||_{L^1} in the treatment of I2. This inequality holds only if the orthonormal system is uniformly bounded by 1 in L^∞, a hypothesis that is absent from the statements of Theorems 1.2, 1.3, 1.4, and 1.6, which all assert 'arbitrary given orthonormal system'. The missing hypothesis is load-bearing: take disjoint intervals I_k with |I_k|=2^{-k}, define φ_k=2^{k/2}1_{I_k}, and let f=Σ_{k≥1} φ_k. Then {φ_k} is orthonormal, yet for 1<p<2 the function f belongs to L_{p,q} (since ∫|f|^p = Σ 2^{-k(1-p/2)} < ∞ and the Lorentz norm is finite), while a_m(f)=∫ f φ_m = 1 for every m, so ||a||_{ℓ^{p',q}} = ∞. This contradicts Theorem 1.2 for every 0<q≤∞. The same flaw propagates to Theorems 1.3, 1.4, and 1.6, whose proofs invoke Lemma 3.1 or Lemma 3.2. The theorems must all be restated with the hypothesis that the orthonormal system is uniformly bounded, |φ_k| ≤ 1 a.e., consistent with the classical Riesz theorem mentioned in the introduction.
  2. [§3.2, Theorem 1.2] Theorem 1.2 states the constant in (1.4) as c (1/p − 1/2)^{-1/2} for all 0<q≤∞. However, Lemma 3.2, which is the proof of the q<2 case, yields the constant c (1/p − 1/2)^{-1/q}. The proof of Theorem 1.2 simply says 'Combining Lemmas 3.1 and 3.2', but those lemmas have different exponents, so the stated theorem does not follow. This is not a cosmetic mismatch: for q<2 the exponent −1/q blows up faster than −1/2 as p→2, so the claimed uniform exponent is stronger than what the proof establishes. The statement of Theorem 1.2 (and the abstract's description of optimal bounds) must be reconciled with the exponent actually proved.
  3. [§3.3, proof of Theorem 1.4] In the proof of Theorem 1.4, the text reads 'From Lemma 3.2 with 1/p = 1/q + ε', but the displayed inequality that follows uses the factor t^{1/2+ε} on the right-hand side, which corresponds to 1/p = 1/2 + ε. The written substitution is inconsistent: for 0<q<1, 1/q + ε can exceed 1, which is impossible for a real p>1. This appears to be a typo, but as written it makes the proof of Theorem 1.4 invalid. The substitution should be corrected to 1/p = 1/2 + ε throughout.
minor comments (4)
  1. [§2, before (2.1)] The claim that replacing a^*_k by (1/k Σ_{m=1}^k (a^*_m)^α)^{1/α} 'yields an equivalent norm' for the Lorentz sequence space is not true in general. For example, the sequence (1,0,0,...) has finite ℓ_{p,q} norm for all p>0, but for p<2 and α=2 the norm defined with the averaged square diverges. Since the proofs only use the one-sided inequality ||a||_{ℓ} ≤ ||a||_{ℓ^*}, the arguments are not affected, but the equivalence statement as written is false and should be qualified (e.g., for p>α).
  2. [§3.2, proof of Lemma 3.2] In the displayed estimate for I2, the text 'and the fact 2^{-qm} ≥ 2m' is nonsensical. It should presumably read '2^{-qm} ≥ 2^{-m}' for 0<q<1. Please correct this typo.
  3. [§3.3, proof of Theorem 1.6] The proof of Theorem 1.6 relies on [11, Theorem 2] and refers to [12] ('to appear') for details of the interpolation identity (L_{p,q0}, L_{p,q1})_{ητ} = L_{p,q,τ}. Since [12] is an unpublished monograph, the needed interpolation statement should be stated explicitly in the paper or a published reference should be provided.
  4. [Throughout] There are several typographical issues in formulas: e.g., 'f ∗(2−k)' should be 'f^*(2^{-k})' with proper parentheses; in Definition 2.2 the displayed norm for G^θℓ_{p,q} has an inconsistent appearance; and in Theorem 1.4 the parameter θ_1 is defined as θ+1/q even though the proof uses θ+1/q from Lemma 3.2, which should be highlighted after the exponent correction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fourier-coefficient bounds are derived from Parseval and Hardy inequalities, and the grand-space endpoint statements are corollaries rather than assumptions.

full rationale

The derivation chain is self-contained for the central claim. Theorem 1.2 is proved from Lemmas 3.1 and 3.2, which use only the decomposition f=f0+f1, Parseval's equality, Hölder/Jensen, and Hardy inequalities; no target inequality is assumed. Theorems 1.3 and 1.4 are obtained by substituting 1/p=1/2+ε into those lemmas and then taking the supremum that defines the grand norms, so the 'grandization' is a direct corollary of the scalar estimate, not a circular renaming. Theorem 1.6 rests on the interpolation identity (L_{p,q0},L_{p,q1})_{ητ}=L_{p,q,τ} cited to [11,12]; although these are prior works of the authors, the identity is a general interpolation fact about L_{p,q,τ} spaces and does not contain the Fourier-coefficient conclusion, so it is independent support rather than a self-citation chain. The definition of grand Lorentz spaces is explicit in Definition 2.2; the cited monotonicity property from [13] is a standard consequence. Two non-circular correctness gaps should be noted: the proof of Lemma 3.1 uses |a_m(f)|≤∥f∥_{L1} under the wording 'arbitrary given orthonormal system' (uniform boundedness is missing), and the q<2 constant proved in Lemma 3.2 is (1/p-1/2)^{-1/q} while Theorem 1.2 states (1/p-1/2)^{-1/2}. These are mathematical gaps, not equivalence-by-construction, so they leave the circularity score at 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central estimates are derived from standard Fourier-analytic tools plus two unstated or understated assumptions: uniform boundedness of the orthonormal system and a correction to the q < 2 exponent. No numerical parameters are fitted; the new grand spaces are definitions and are not independently evidenced beyond their use in the proofs.

assumptions (4)
  • ad hoc to paper Uniform boundedness of the orthonormal system, |φ_k| ≤ 1.
    Used implicitly in Lemma 3.1 and Lemma 3.2 via |a_m(f)| ≤ ‖f‖_{L1}; absent from all theorem statements.
  • standard math Parseval's equality and the L2 isometric property of orthonormal systems.
    Used to estimate the high-frequency part I1 in Lemma 3.1.
  • standard math Interpolation identity (L_{2,q0}, L_{2,q1})_{ητ} = L_{2,q,τ}.
    Cited from [11,12] in the proof of Theorem 1.6; not proved in the paper.
  • standard math Grand Lorentz space monotonicity and equivalence of norms from [13].
    Used to define and compare grand Lorentz norms; not central to the new estimates but imported from the authors' prior work.
invented entities (1)
  • Grand Lorentz function spaces G^θ L_{p,q} and grand sequence spaces G^θ ℓ*_{p,q}.
    purpose: Absorb the ε-singularity in p near 2 and express endpoint Hausdorff-Young estimates.
    They are new mathematical definitions (Definition 2.2 and (2.1)); they are not independently evidenced empirical entities, and their value rests entirely on the inequalities proved using them.

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Pith. "Pith review of On Herz-Bochkarev limiting problem." pith.science (2026). https://pith.science/paper/URSQQ4JQ

@misc{pith2026250607478,
  author       = {Pith},
  title        = {Pith review of: On Herz-Bochkarev limiting problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URSQQ4JQ}},
  note         = {Machine review of arXiv:2506.07478}
}
abstract

This paper studies Hausdorff-Young-type inequalities within the framework of Lorentz spaces $L_{p,q}$. Focusing on the dependence of the associated constants on the integrability parameter $p$, we derive optimal bounds in the limiting case $p\rightarrow 2$, addressing the Herz-Bochkarev problem. The results obtained refine the pioneering estimates in [3] and are comparable to recent advances in [16]. The main ingredients of our approach are new grand Lorentz space techniques.

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Works this paper leans on

20 extracted references · 19 canonical work pages

  1. [3]

    S. V . Bochkarev, The Hausdorff-Young-Riesz theorem in Lorentz spaces, and multiplicative inequalities, Tr. Mat. Inst. Steklova219 (1997), no. Teor. Priblizh. Garmon. Anal., 103—114

    work page 1997

  2. [16]

    Saucedo and S

    M. Saucedo and S. Tikhonov, The Fourier Transform is an Extremizer of a Class of Bounded Operators, arXiv:2501.17505 (2025)

    arXiv 2025

  3. [13]

    E. D. Nursultanov, H. Rafeiro, and D. Suragan, Convolution-type operators in grand Lorentz spaces, Anal. Math. Phys. 15 (2025), no. 3, Paper No. 65

    work page 2025

  4. [1]

    Arias-de Reyna, Pointwise convergence of Fourier series, J

    J. Arias-de Reyna, Pointwise convergence of Fourier series, J. London Math. Soc. (2) 65 (2002), no. 1, 139-–153

    work page 2002

  5. [2]

    Bennett and K

    C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.), vol. 175, Pa´nstwowe Wydawnictwo Naukowe, 1980

    work page 1980

  6. [4]

    S. V . Bochkarev, Estimates for the Fourier coefficients of functions in Lorentz spaces,Dokl. Akad. Nauk 360 (1998), no. 6, 730—733

    work page 1998

  7. [5]

    C. P. Calder´on and A. Torchinsky, The Hausdorff-Young inequality forn-dimensional Hermite expansions, Sel. Mat. 9 (2022), no. 2, 227-–233

    work page 2022

  8. [6]

    C. P. Calder´on and A. Torchinsky, Maximal integral inequalities and Hausdorff-Young,J. Fourier Anal. Appl. 30 (2024), no. 5, Paper No. 52, 35 pp

    work page 2024

Show all 20 references
  1. [7]

    C. P. Calder´on and A. Torchinsky, Maximal integral inequalities and maximal Fourier coefficients,Banach J. Math. Anal. 19 (2025), no. 2, Paper No. 22, 20 pp

    2025

  2. [8]

    Dominguez and M

    O. Dominguez and M. Veraar, Extensions of the vector-valued Hausdorff-Young inequalities,Math. Z. 299 (2021), no. 1–2, 373-–425. 20 E. NURSULTANOV, A. GHORBANALIZADEH, AND D. SURAGAN

    2021

  3. [9]

    C. S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms,J. Math. Mech. 18 (1968), 283-–323

    1968

  4. [10]

    R. A. Hunt, On L(p, q) spaces, Enseign. Math. (2) 12 (1966), 249—276

    1966

  5. [11]

    E. D. Nursultanov, A multiparametric interpolation functor and the Lorentz space Lp→⃗ q, ⃗ q= (q1, . . . , qn), Funktsional. Anal. i Prilozhen. 31 (1997), no. 2, 79-–82

    1997

  6. [12]

    E. D. Nursultanov, Inequalities and integral operators in function spaces, CRC Press, 2025 (to appear)

    2025

  7. [14]

    E. D. Nursultanov and D. Suragan, Hardy–Littlewood–Stein inequalities for double trigonometric series, Math. Inequal. Appl. 26 (2023), no. 1, 1-–15

    2023

  8. [15]

    E. D. Nursultanov and D. Suragan, Improved Stein inequalities for the Fourier transform,J. Approx. Theory 306 (2025), Paper No. 106126, 15 pp

    2025

  9. [17]

    Sinnamon, Fourier inequalities and a new Lorentz space, in: Banach and function spaces II, Yokohama Publ., Yokohama, 2008, pp

    G. Sinnamon, Fourier inequalities and a new Lorentz space, in: Banach and function spaces II, Yokohama Publ., Yokohama, 2008, pp. 145—155

    2008

  10. [18]

    E. M. Stein, Interpolation of Linear Operators, Trans. Amer. Math. Soc.83 (1956), 482–492

    1956

  11. [19]

    E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 1971

    1971

  12. [20]

    Zygmund, Trigonometric series

    A. Zygmund, Trigonometric series. Vol. I, II , 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. ERLAN NURSULTANOV , D EPARTMENT OF MATHEMATICS AND INFORMATICS , L OMONOSOV MOSCOW STATE UNIVERSITY , KAZAKHSTAN BRANCH , AND INSTITUTE OF MATHE...

    2002

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