pith. sign in

arxiv: 2509.20296 · v2 · submitted 2025-09-24 · 🧮 math.FA

Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators

Oleksiy Karlovych , Eugene Shargorodsky This is my paper

Pith reviewed 2026-05-18 13:43 UTC · model grok-4.3

classification 🧮 math.FA
keywords Wiener-Hopf operatorsKuratowski measure of noncompactnessBanach function spacesvariable Lebesgue spacesFourier multipliersdoubling propertiescones in R^n
0
0 comments X

The pith

If a Banach function space on a domain satisfies the weak doubling property, then the operator norm of the Wiener-Hopf type operator is at least the L^∞ norm of its Fourier multiplier symbol.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves lower bounds relating the size of a Fourier multiplier to the size of the associated Wiener-Hopf type operator on a measurable set Ω. When the underlying Banach function space satisfies a weak doubling condition, the operator norm from the closure of L²(Ω) inside X(Ω) to X(Ω) is bounded below by the essential supremum of the multiplier. Under the stronger separated doubling condition the same lower bound holds, up to a factor of one half, for the Kuratowski measure of noncompactness of the operator. The results are then applied to variable-exponent Lebesgue spaces equipped with Muckenhoupt-type weights on open cones.

Core claim

For a Fourier multiplier a, the Wiener-Hopf type operator W_Ω(a) defined by restriction, multiplication by a in frequency, and extension by zero satisfies ||a||_{L^∞(R^n)} ≤ ||W_Ω(a)||_{B(X_2(Ω),X(Ω))} whenever X(Ω) has the weak doubling property, and (1/2)||a||_{L^∞} ≤ ||W_Ω(a)||_{B,κ} whenever X(Ω) has the separated doubling property.

What carries the argument

The Wiener-Hopf type operator W_Ω(a) = r_Ω F^{-1} a F e_Ω together with the weak and separated doubling properties imposed on the Banach function space X(Ω).

If this is right

  • The operator norm of W_Ω(a) cannot be smaller than the essential supremum of a when the space obeys weak doubling.
  • The Kuratowski measure of noncompactness of W_Ω(a) is bounded below by half the essential supremum of a under separated doubling.
  • The same lower estimates hold for the concrete case of weighted variable Lebesgue spaces on cones with Muckenhoupt weights.

Where Pith is reading between the lines

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

  • The bounds suggest that any attempt to construct bounded Wiener-Hopf operators with unbounded symbols must fail in spaces that satisfy the doubling hypotheses.
  • The factor-one-half gap between the norm and noncompactness lower bounds may be improvable or sharp depending on the geometry of Ω.

Load-bearing premise

The Banach function space X(Ω) satisfies the weak doubling property or the separated doubling property.

What would settle it

A concrete Banach function space lacking the doubling property together with a bounded multiplier a for which the operator norm of W_Ω(a) is strictly smaller than ||a||_{L^∞}.

read the original abstract

Let $X(\mathbb{R}^n)$ be a Banach function space and $\Omega\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_\Omega(a):=r_\Omega F^{-1}aF e_\Omega$, where $F^{\pm 1}$ are the Fourier transforms, $r_\Omega$ is the operator of restriction from $\mathbb{R}^n$ to $\Omega$ and $e_\Omega$ is the operator of extension by zero from $\Omega$ to $\mathbb{R}^n$. Let $X_2(\Omega)$ be the closure of $L^2(\Omega)\cap X(\Omega)$ in $X(\Omega)$. We show that if $X(\Omega)$ satisfies the so-called weak doubling property, then \[ \|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_\Omega(a)\|_{\mathcal{B}(X_2(\Omega),X(\Omega))}. \] Further, we prove that if $X(\Omega)$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_\Omega(a)$ admits the following lower estimate: \[ \frac{1}{2}\|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_\Omega(a)\|_{\mathcal{B}(X_2(\Omega),X(\Omega)),\kappa}. \] These results are specified to the case of variable Lebesgue spaces $L^{p(\cdot)}(C,w)$ with Muckenhoupt type weights $w$ over open cones $C\subseteq\mathbb{R}^n$ with the vertex at the origin.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper establishes lower bounds for the operator norm and Kuratowski measure of noncompactness of Wiener-Hopf type operators W_Ω(a) = r_Ω F^{-1} a F e_Ω acting between suitable subspaces of a Banach function space X(Ω). Under the weak doubling property on X(Ω), it proves ||a||_∞ ≤ ||W_Ω(a)||_{B(X_2(Ω),X(Ω))}. Under the separated doubling property, it obtains (1/2)||a||_∞ ≤ ||W_Ω(a)||_{B,κ}. The results are specialized to weighted variable-exponent Lebesgue spaces L^{p(·)}(C,w) on open cones C with Muckenhoupt-type weights.

Significance. If the derivations hold, the explicit lower estimates via doubling properties and localized test functions provide a useful tool for obtaining sharp norm and non-compactness information for Fourier-multiplier-based operators on general Banach function spaces. The specialization to variable Lebesgue spaces on cones verifies the hypotheses under standard Muckenhoupt conditions and connects to existing literature on weighted variable-exponent spaces.

minor comments (2)
  1. §3: The precise statement of the weak doubling property (Definition 3.1) is used to control the X-norm of extensions, but an explicit example of a space satisfying it but not the separated version would help readers see the distinction between the two main theorems.
  2. §4, after Theorem 4.2: The factor 1/2 in the Kuratowski-measure lower bound arises from the separation constant; a brief remark on whether this constant is sharp for some concrete a and Ω would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary accurately captures the main contributions concerning lower bounds for the norm and Kuratowski measure of noncompactness of Wiener-Hopf type operators under weak and separated doubling properties on Banach function spaces, with specialization to weighted variable-exponent Lebesgue spaces on cones.

read point-by-point responses

  1. Referee: No major comments were raised in the report.

    Authors: We appreciate the referee's recognition of the utility of the explicit lower estimates via doubling properties and localized test functions, as well as the verification of the hypotheses under standard Muckenhoupt conditions for the variable-exponent setting. No changes to the manuscript are required. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central lower bounds are obtained directly from the stated weak and separated doubling properties of X(Ω) via explicit constructions of localized test functions or sequences that control the X-norm of extensions/restrictions and ensure positive separation under W_Ω(a). These steps rely on standard properties of Fourier multipliers and restriction/extension operators without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations. The specialization to weighted variable Lebesgue spaces over cones verifies that the doubling conditions hold under the given Muckenhoupt-type assumptions, rendering the application consistent with the general theorems rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the weak and separated doubling properties of the space X(Ω), standard boundedness of the Fourier transform on the underlying Banach function space, and the definition of the restriction and extension operators. No free parameters are fitted; the doubling conditions function as domain assumptions.

axioms (2)
  • domain assumption X(Ω) satisfies the weak doubling property (resp. separated doubling property)
    Invoked explicitly to obtain the norm lower bound and the noncompactness lower bound.
  • standard math Fourier multiplier a is bounded on X(R^n)
    Used to define the Wiener-Hopf operator W_Ω(a).

pith-pipeline@v0.9.0 · 5867 in / 1460 out tokens · 38880 ms · 2026-05-18T13:43:20.304301+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    R. R. Akhmerov, M. I. Kamenski˘ ı, A. S. Potapov, A. E. Rodkina, and B. N. Sadovski˘ ı.Measures of noncompactness and condensing operators, volume 55 ofOp- erator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1992

    work page 1992

  2. [2]

    Bana´ s and M

    J. Bana´ s and M. Mursaleen.Sequence spaces and measures of noncompactness with applications to differential and integral equations. Springer, New Delhi, 2014

    work page 2014

  3. [3]

    Bennett and R

    C. Bennett and R. Sharpley.Interpolation of operators, volume 129 ofPure and Applied Mathematics. Academic Press, Boston, MA, 1988

    work page 1988

  4. [4]

    E. I. Berezhnoi. Two-weighted estimations for the Hardy-Littlewood maximal func- tion in ideal Banach spaces.Proc. Amer. Math. Soc., 127(1):79–87, 1999

    work page 1999

  5. [5]

    B¨ ottcher and B

    A. B¨ ottcher and B. Silbermann.Analysis of Toeplitz operators. Springer Monographs in Mathematics. Springer-Verlag, Berlin, second edition, 2006

    work page 2006

  6. [6]

    Cruz-Uribe, L

    D. Cruz-Uribe, L. Diening, and P. H¨ ast¨ o. The maximal operator on weighted variable Lebesgue spaces.Fract. Calc. Appl. Anal., 14(3):361–374, 2011

    work page 2011

  7. [7]

    Cruz-Uribe and A

    D. Cruz-Uribe and A. Fiorenza.Variable Lebesgue spaces. Foundations and har- monic analysis. Applied and Numerical Harmonic Analysis. Birkh¨ auser/Springer, Heidelberg, 2013

    work page 2013

  8. [8]

    Cruz-Uribe, A

    D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces.J. Math. Anal. Appl., 394(2):744– 760, 2012

    work page 2012

  9. [9]

    Diening, P

    L. Diening, P. Harjulehto, P. H¨ ast¨ o, and M. R˚ uˇ ziˇ cka.Lebesgue and Sobolev spaces with variable exponents, volume 2017 ofLecture Notes in Mathematics. Springer, Heidelberg, 2011. LOWER ESTIMATES FOR WIENER-HOPF OPERATORS 17

    work page 2017

  10. [10]

    C. A. Fernandes, A. Y. Karlovich, and Y. I. Karlovich. Noncompactness of Fourier convolution operators on Banach function spaces.Ann. Funct. Anal., 10(4):553–561, 2019

    work page 2019

  11. [11]

    Grafakos.Fundamentals of Fourier analysis, volume 302 ofGraduate Texts in Mathematics

    L. Grafakos.Fundamentals of Fourier analysis, volume 302 ofGraduate Texts in Mathematics. Springer, Cham, 2024

    work page 2024

  12. [12]

    K.-P. Ho. Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces.Anal. Math., 38(3):173–185, 2012

    work page 2012

  13. [13]

    Karlovich and E

    A. Karlovich and E. Shargorodsky. When does the norm of a Fourier multiplier dominate itsL ∞ norm?Proc. Lond. Math. Soc. (3), 118(4):901–941, 2019

    work page 2019

  14. [14]

    A. Y. Karlovich and I. M. Spitkovsky. The Cauchy singular integral operator on weighted variable Lebesgue spaces. InConcrete operators, spectral theory, operators in harmonic analysis and approximation, volume 236 ofOper. Theory Adv. Appl., pages 275–291. Birkh¨ auser/Springer, Basel, 2014

    work page 2014

  15. [15]

    Karlovych and E

    O. Karlovych and E. Shargorodsky. Discrete Riesz transforms on rearrangement- invariant Banach sequence spaces and maximally noncompact operators.Pure Appl. Funct. Anal., 9(1):195–210, 2024

    work page 2024

  16. [16]

    Karlovych and E

    O. Karlovych and E. Shargorodsky. Maximal noncompactness of Wiener-Hopf op- erators. https://arxiv.org/abs/2509.17451 (2025)

    work page arXiv 2025

  17. [17]

    Karlovych and M

    O. Karlovych and M. Valente. On the algebras of Wiener-Hopf operators with con- tinuous symbols acting on some Banach function spaces. InAnalysis without borders, volume 297 ofOper. Theory Adv. Appl., pages 123–144. Birkh¨ auser/Springer, Cham, 2024

    work page 2024

  18. [18]

    Karlovych and M

    O. Karlovych and M. Valente. On the operator and essential norms of Fourier con- volution operators and Wiener-Hopf operators with the same symbol. InOperator Theory, Related Fields, and Applications, volume 307 ofOperator Theory: Advances and Applications. Birkh¨ auser/Springer, Cham, 2025, in press

    work page 2025

  19. [19]

    Lebow and M

    A. Lebow and M. Schechter. Semigroups of operators and measures of noncompact- ness.J. Functional Analysis, 7:1–26, 1971

    work page 1971

  20. [20]

    Lorist and Z

    E. Lorist and Z. Nieraeth. Banach function spaces done right.Indag. Math. (N.S.), 35(2):247–268, 2024

    work page 2024

  21. [21]

    W. A. J. Luxemburg.Banach function spaces. Technische Hogeschool te Delft, Delft, 1955

    work page 1955

  22. [22]

    L. Pick, A. Kufner, O. John, and S. Fuˇ c´ ık.Function spaces. Vol. 1, volume 14 ofDe Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 2013

    work page 2013

  23. [23]

    Rudin.Functional analysis

    W. Rudin.Functional analysis. International Series in Pure and Applied Mathemat- ics. McGraw-Hill, Inc., New York, second edition, 1991

    work page 1991

  24. [24]

    Sawano, G

    Y. Sawano, G. Di Fazio, and D. I. Hakim.Morrey spaces—introduction and appli- cations to integral operators and PDE’s. Vol. I. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2020

    work page 2020

  25. [25]

    M. Valente. Fredholm criteria for Wiener-Hopf operators with continuous symbols acting on some Banach function spaces.J. Math. Sci., to appear, https://arxiv.org/abs/2509.13996 (2025). Centro de Matem´atica e Aplicac ¸˜oes, Departamento de Matem´atica, Fac- uldade de Ci ˆencias e Tecnologia, Universidade Nova de Lisboa, Quita da Torre, 2829–516, Caparica,...

    work page arXiv 2025