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arxiv: 2510.05469 · v3 · submitted 2025-10-07 · 🧮 math.FA

On inclusion relations of weighted L^p-type spaces defined in terms of weight function matrices

Gerhard Schindl This is my paper

Pith reviewed 2026-05-18 09:54 UTC · model grok-4.3

classification 🧮 math.FA
keywords weighted L^p spacesweight function matricesinclusion relationsBeurling-BjörckBraun-Meise-Taylorultradifferentiable functionstranslation invariance
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The pith

Inclusion relations of weighted L^p spaces are determined by relations between their weight function matrices.

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

The paper defines new weighted L^p spaces using matrices whose entries are weight functions and provides a complete characterization of when one space is contained in another solely in terms of the matrices. A sympathetic reader would care because this unifies the description of ultradifferentiable function classes under the Beurling-Björck approach, which the paper shows is more general than the Braun-Meise-Taylor setting by constructing a weight that works for the former but violates its convexity requirement. The results also determine when these spaces coincide with ordinary L^p, when they are nontrivial, and how they behave under translations of the functions. Extensions to weighted subspaces of L^p and to weights on the Fourier transform side are treated within the same framework.

Core claim

Weighted L^p-type spaces are introduced via weight function matrices, and the inclusion between two such spaces holds exactly when the matrices satisfy a certain comparison condition. This condition is made explicit and applied to matrices coming from a single weight function together with a real parameter. In the ultradifferentiable context the same characterization shows that Beurling-Björck weights need not obey the convexity condition that is standard in the Braun-Meise-Taylor theory, as demonstrated by an explicit counterexample.

What carries the argument

The weight function matrix, a collection of individual weight functions arranged in matrix form, which enters the definition of the space norm and thereby controls the size of functions in the space.

If this is right

  • Inclusions between the spaces reduce to verifiable conditions on the entries of the matrices.
  • For the special case of matrices built from one weight function and a parameter, the inclusions simplify to relations involving that weight and the parameter value.
  • The spaces coincide with standard L^p spaces precisely when the matrix entries satisfy appropriate boundedness or decay conditions.
  • Translation invariance of the spaces depends on the growth properties of the weight functions in the matrix.
  • The same matrix comparison applies when the weighting is transferred to the Fourier transform side.

Where Pith is reading between the lines

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

  • The matrix-based description could be used to compare weighted versions in other classes of function spaces.
  • The explicit counterexample separating the two ultradifferentiable weight settings can be used to check which properties require convexity.
  • The treatment of Fourier-side weighting indicates the framework extends naturally to harmonic analysis.

Load-bearing premise

The weight function matrices satisfy positivity, measurability, and growth conditions that ensure the weighted expressions define valid Banach space norms.

What would settle it

A pair of weight function matrices where the L^p spaces include each other even though the matrix comparison condition fails.

read the original abstract

We introduce new weighted $L^p$-type spaces defined in terms of weight function matrices and characterize the inclusion relations in terms of the defining matrices. Moreover, we provide a detailed study concerning the coincidence with the common (non-weighted) $L^p$-spaces, the (non-)triviality of such weighted spaces and investigate their translation invariance. The obtained results are then applied to particular weight function matrices which are expressed in terms of one single weight function and a positive real parameter. Also variations of this new weighted setting are discussed; more precisely weighted Banach (sub-)spaces of $L^p$ and when weighting the Fourier image of appropriate Banach spaces of functions. The general framework allows to describe the known ultradifferentiable weight function setting by Beurling-Bj\"{o}rck which is more original than the approach presented by Braun, Meise and Taylor. When applying the characterization of the inclusion relations to Beurling-Bj\"{o}rck-type spaces we are able to emphasize the difference between both ultradifferentiable weight function settings: We construct a technical (counter-)example which is a weight in the sense of Beurling Bj\"{o}rck but violates the standard and crucial convexity condition needed in the Braun-Meise-Taylor setting.

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

1 major / 2 minor

Summary. The manuscript introduces weighted L^p-type spaces via matrices of weight functions satisfying positivity, measurability and growth conditions. It characterizes inclusion relations between these spaces by direct comparison of the defining matrix entries, studies their coincidence with ordinary L^p spaces, (non-)triviality and translation invariance, and applies the results to matrices generated from a single weight function together with a positive parameter. Variations such as weighted Banach subspaces of L^p and weighted Fourier images are discussed. The framework is then used to embed the Beurling-Björck ultradifferentiable setting and to distinguish it from the Braun-Meise-Taylor setting by exhibiting an explicit weight that satisfies the former axioms but violates the convexity condition required by the latter.

Significance. If the matrix-based characterization and the counter-example construction are correct, the work supplies a uniform language for comparing weighted L^p spaces and clarifies a concrete distinction between two standard ultradifferentiable weight-function frameworks. The explicit counter-example is a concrete, falsifiable contribution that can be checked independently.

major comments (1)
  1. [§4] §4 (counter-example construction): the verification that the constructed weight satisfies all Beurling-Björck axioms (subadditivity, continuity, etc.) while failing the Braun-Meise-Taylor convexity condition is only sketched; an explicit computation of the associated matrix entries and a direct check of the convexity inequality at a concrete point would strengthen the claim that the two settings are genuinely distinct.
minor comments (2)
  1. [§2] Notation for the matrix-valued weight functions is introduced in §2 but the precise measurability and growth hypotheses are restated in several later sections; a single consolidated list of standing assumptions would improve readability.
  2. [Theorem 3.2] The statement of Theorem 3.2 on inclusion relations refers to “the natural ordering of matrices” without recalling the precise partial order used; a brief reminder or cross-reference to Definition 2.4 would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestion regarding the counter-example. We address the comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [§4] §4 (counter-example construction): the verification that the constructed weight satisfies all Beurling-Björck axioms (subadditivity, continuity, etc.) while failing the Braun-Meise-Taylor convexity condition is only sketched; an explicit computation of the associated matrix entries and a direct check of the convexity inequality at a concrete point would strengthen the claim that the two settings are genuinely distinct.

    Authors: We agree that the verification of the counter-example in §4 is presented in a condensed form. In the revised manuscript we will expand this part by supplying explicit formulas for the matrix entries generated by the chosen weight function. We will also perform a direct numerical check of the convexity inequality at a concrete point (e.g., evaluating the relevant expressions at a specific positive real number) to demonstrate explicitly that the Braun-Meise-Taylor condition fails while all Beurling-Björck axioms continue to hold. These additions will make the distinction between the two ultradifferentiable frameworks fully verifiable without changing any of the stated theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces weighted L^p-type spaces via weight function matrices satisfying standard positivity, measurability and growth conditions, then characterizes inclusion relations by direct comparison of those matrix entries. It further constructs an explicit technical counterexample weight that satisfies the Beurling-Björck axioms while violating the convexity condition required in the Braun-Meise-Taylor framework. These steps rest on external prior definitions of the two ultradifferentiable settings and ordinary functional-analysis axioms; no load-bearing claim reduces by construction to a fitted parameter, self-definition, or self-citation chain internal to the paper. The derivation therefore remains self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions for weight functions in functional analysis; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Weight function matrices satisfy positivity, measurability, and suitable growth conditions so that the associated spaces are well-defined Banach spaces.
    Required for the inclusion characterization and non-triviality statements to make sense; invoked implicitly when defining the spaces.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

  1. [1]

    Beurling

    A. Beurling. Quasi-analyticity and general distributions. Lecture 4 and 5, AMS Summer Institute, Stanford, 1961

    work page 1961

  2. [2]

    Bingham, C.M

    N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular variation. Encyclopedia of Mathematics and its Appli- cations, Cambridge University Press, Cambridge, 1989

    work page 1989

  3. [3]

    G. Björck. Linear partial differential operators and generalized distributions.Ark. Mat., 6:351–407 (1966), 1966

    work page 1966

  4. [4]

    Boiti, D

    C. Boiti, D. Jornet, A. Oliaro, and G. Schindl. Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting.Banach J. of Math. Anal., 15(1):art. no. 14, 2021

    work page 2021

  5. [5]

    Bonet, R

    J. Bonet, R. W. Braun, R. Meise, and B. A. Taylor. Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions.Studia Math., 99(2):155–184, 1991

    work page 1991

  6. [6]

    Bonet, R

    J. Bonet, R. Meise, and S. N. Melikhov. A comparison of two different ways to define classes of ultradifferentiable functions.Bull. Belg. Math. Soc. Simon Stevin, 14:424–444, 2007. 44 G. SCHINDL

    work page 2007

  7. [7]

    R. W. Braun, R. Meise, and B. A. Taylor. Ultradifferentiable functions and Fourier analysis.Results Math., 17(3-4):206–237, 1990

    work page 1990

  8. [8]

    Debrouwere, L

    A. Debrouwere, L. Neyt, and J. Vindas. On the inclusion relations between Gelfand-Shilov spaces.Studia Math., 283:237–256, 2025

    work page 2025

  9. [9]

    U. Franken. Kerne von Faltungsoperatoren auf Räumen von Ultradistributionen. Diplomarbeit, Universität Düsseldorf, 1988

    work page 1988

  10. [10]

    U. Franken. Weight functions for classes of ultradifferentiable functions.Res. Math., 25:50–53, 1994

    work page 1994

  11. [11]

    Jiménez-Garrido, D

    J. Jiménez-Garrido, D. N. Nenning, and G. Schindl. On generalized definitions of ultradifferentiable classes.J. Math. Anal. Appl., 526:127260, 2023

    work page 2023

  12. [12]

    Jiménez-Garrido, J

    J. Jiménez-Garrido, J. Sanz, and G. Schindl. Indices of O-regular variation for weight functions and weight sequences.Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. RACSAM, 113(4):3659–3697, 2019

    work page 2019

  13. [13]

    Jiménez-Garrido, J

    J. Jiménez-Garrido, J. Sanz, and G. Schindl. Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions.Results Math., 74(27), 2019

    work page 2019

  14. [14]

    Jiménez-Garrido, J

    J. Jiménez-Garrido, J. Sanz, and G. Schindl. Sectorial extensions for ultraholomorphic classes defined by weight functions.Math. Nachr., 293(11):2140–2174, 2020

    work page 2020

  15. [15]

    H. Komatsu. Ultradistributions. I. Structure theorems and a characterization.J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20:25–105, 1973

    work page 1973

  16. [16]

    Mandelbrojt.Séries adhérentes, Régularisation des suites, Applications

    S. Mandelbrojt.Séries adhérentes, Régularisation des suites, Applications. Gauthier-Villars, Paris, 1952

    work page 1952

  17. [17]

    Meise and B

    R. Meise and B. A. Taylor. Whitney’s extension theorem for ultradifferentiable functions of Beurling type.Ark. Mat., 26(2):265–287, 1988

    work page 1988

  18. [18]

    D. N. Nenning, A. Rainer, and G. Schindl. On optimal solutions of the Borel problem in the Roumieu case. Bull. Belg. Math. Soc. - Simon Stevin, 29(4):509–531, 2022

    work page 2022

  19. [19]

    D. N. Nenning, A. Rainer, and G. Schindl. The Borel map in the mixed Beurling setting.Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 117(art. no. 40), 2023

    work page 2023

  20. [20]

    Neyt and J

    L. Neyt and J. Vindas. Kernel theorems for Beurling-Björck type spaces.Bull. Sci. Math., 187:103309, 2023

    work page 2023

  21. [21]

    J. Peetre. Concave majorants of positive functions.Acta Mathematica Academiae Scientiarum Hungaricae, Tomus 21 (3-4):327–333, 1970

    work page 1970

  22. [22]

    Petzsche

    H.-J. Petzsche. On E. Borel’s theorem.Math. Ann., 282(2):299–313, 1988

    work page 1988

  23. [23]

    Petzsche and D

    H.-J. Petzsche and D. Vogt. Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions.Math. Ann., 267:17–35, 1984

    work page 1984

  24. [24]

    Rainer and G

    A. Rainer and G. Schindl. Composition in ultradifferentiable classes.Studia Math., 224(2):97–131, 2014

    work page 2014

  25. [25]

    Rainer and G

    A. Rainer and G. Schindl. Equivalence of stability properties for ultradifferentiable function classes.Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A. Matemáticas, 110(1):17–32, 2016

    work page 2016

  26. [26]

    G. Schindl. Generalized upper and lower Legendre conjugates for Braun-Meise-Taylor weight functions. 2025, available online athttps://arxiv.org/pdf/2505.17725.pdf

    work page arXiv 2025

  27. [27]

    G. Schindl. Generalized upper and lower Legendre conjugates for weight functions. 2025, available online at https://arxiv.org/pdf/2505.07497.pdf

    work page arXiv 2025

  28. [28]

    G. Schindl. On the equivalence between moderate growth-type conditions in the weight matrix setting II. 2025, available online athttps://arxiv.org/pdf/2505.08839.pdf

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    G. Schindl. Exponential laws for classes of Denjoy-Carleman-differentiable mappings, 2014. PhD Thesis, Uni- versität Wien, available online athttp://othes.univie.ac.at/32755/1/2014-01-26_0304518.pdf

    work page 2014

  30. [30]

    G. Schindl. On subadditivity-like conditions for associated weight functions.Bull. Belg. Math. Soc. Simon Stevin, 28(3):399–427, 2022

    work page 2022

  31. [31]

    G. Schindl. On inclusion relations between weighted spaces of entire functions.Bull. Sci. Math., 190:103375, 2024

    work page 2024

  32. [32]

    G. Schindl. On strong growth conditions for weighted spaces of entire functions.Bull. Sci. Math., 196:103490, 2024

    work page 2024

  33. [33]

    G. Schindl. On the regularization of sequences and associated weight functions.Bull. Belg. Math. Soc. Simon Stevin, 31(2):174–210, 2024. G. Schindl: F akultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Aus- tria. Email address:gerhard.schindl@univie.ac.at

    work page 2024