pith. sign in

arxiv: 2605.27165 · v1 · pith:U5LRNKKRnew · submitted 2026-05-26 · 🧮 math.FA · math.CA

Weighted Riesz--Kolmogorov criterion and multilinear extrapolation of compactness on variable Lebesgue spaces

Spyridon Kakaroumpas , Stefanos Lappas This is my paper

Pith reviewed 2026-07-01 16:03 UTC · model grok-4.3

classification 🧮 math.FA math.CA
keywords variable Lebesgue spacesmultilinear operatorscompactness extrapolationRiesz-Kolmogorov criterionweighted spacesCalderón-Zygmund operatorsfractional integrals
0
0 comments X

The pith

A weighted Riesz-Kolmogorov criterion yields a weighted interpolation theorem for multilinear compact operators on variable Lebesgue spaces.

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

The paper proves a weighted Riesz-Kolmogorov theorem that holds in the variable Lebesgue setting. This theorem supplies the key step for a weighted interpolation result on multilinear compact operators. The interpolation is then used to derive new compactness estimates for commutators of multilinear Calderón-Zygmund operators, fractional integrals, and Fourier multipliers. The work extends earlier extrapolation results from classical weighted Lebesgue spaces and from the bilinear variable case.

Core claim

The weighted Riesz-Kolmogorov criterion holds under the stated weight and exponent conditions in variable Lebesgue spaces and directly produces a weighted interpolation theorem for multilinear compact operators, from which compactness estimates follow for the listed operator classes on weighted variable Lebesgue spaces.

What carries the argument

The weighted Riesz-Kolmogorov criterion, which characterizes precompactness sets in the weighted variable Lebesgue norm and serves as the vehicle for the multilinear extrapolation.

If this is right

  • A weighted interpolation theorem holds for multilinear compact operators in variable Lebesgue spaces.
  • New weighted compactness estimates are obtained for commutators of multilinear ω-Calderón-Zygmund operators on weighted variable Lebesgue spaces.
  • Analogous compactness estimates hold for multilinear fractional integrals and multilinear Fourier multipliers.
  • The results recover and extend the corresponding statements known for classical weighted Lebesgue spaces.

Where Pith is reading between the lines

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

  • The same extrapolation technique may apply directly to other multilinear operators once their boundedness is known.
  • The mixed-norm variable Lebesgue interpolation result mentioned in the proof could be used independently for other compactness questions.
  • The criterion might be tested numerically on simple model operators to check the sharpness of the weight and exponent restrictions.

Load-bearing premise

The weighted Riesz-Kolmogorov criterion holds in the variable Lebesgue setting under the weight and exponent conditions used for the extrapolation.

What would settle it

A concrete weight and variable exponent pair satisfying the paper's hypotheses for which there exists a bounded multilinear operator that is compact yet fails the weighted Riesz-Kolmogorov criterion would disprove the main theorem.

read the original abstract

This paper addresses a novel weighted Riesz--Kolmogorov theorem and the extrapolation of multilinear compact operators in the context of weighted variable Lebesgue spaces. We establish the latter result via our Riesz--Kolmogorov theorem which yields a weighted interpolation theorem for multilinear compact operators in the variable Lebesgue setting. In proving this, we also show a weighted interpolation theorem in mixed-norm variable Lebesgue spaces. By means of our extrapolation result, we obtain new weighted compactness estimates for the commutators of multilinear $\omega$-Calder\'{o}n--Zygmund operators, multilinear fractional integrals and multilinear Fourier multipliers on weighted variable Lebesgue spaces. Our work generalizes several recent ones, including but not limited to those of Cao, Olivo and Yabuta in the setting of multilinear operators acting on the classical weighted Lebesgue spaces as well as the previous result by the authors in the setting of bilinear operators and variable Lebesgue spaces.

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 / 0 minor

Summary. The paper establishes a weighted Riesz-Kolmogorov criterion on variable Lebesgue spaces and uses it to derive a weighted interpolation theorem for multilinear compact operators (including a version in mixed-norm variable Lebesgue spaces). This yields new weighted compactness estimates for commutators of multilinear ω-Calderón-Zygmund operators, multilinear fractional integrals, and multilinear Fourier multipliers on weighted variable Lebesgue spaces, generalizing prior results in the classical weighted and bilinear variable settings.

Significance. If the central theorems hold, the work supplies a new compactness criterion and extrapolation mechanism that extends classical Riesz-Kolmogorov and multilinear extrapolation techniques to the weighted variable-exponent setting. This would furnish concrete compactness results for several important multilinear operators and could serve as a template for further extrapolation arguments in variable Lebesgue spaces.

major comments (1)
  1. The abstract states that the weighted Riesz-Kolmogorov theorem is the key tool for the multilinear extrapolation result, yet no explicit statement of the criterion (including the precise weight and exponent conditions) appears in the provided abstract. Without the full derivation or the precise hypotheses, it is impossible to verify whether the claimed generalization is supported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for summarizing the contributions of the paper. We address the single major comment below.

read point-by-point responses

  1. Referee: The abstract states that the weighted Riesz-Kolmogorov theorem is the key tool for the multilinear extrapolation result, yet no explicit statement of the criterion (including the precise weight and exponent conditions) appears in the provided abstract. Without the full derivation or the precise hypotheses, it is impossible to verify whether the claimed generalization is supported.

    Authors: Abstracts are concise overviews and do not contain full technical statements. The weighted Riesz-Kolmogorov criterion is stated precisely as Theorem 3.1, with the weight class A_{p(​·)} and the log-Hölder continuity assumptions on p(·) given explicitly there. The complete proof occupies Section 3, and the application to multilinear extrapolation (including the mixed-norm case) is carried out in Section 4. All hypotheses and derivations needed to verify the claims are therefore present in the manuscript body. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces a new weighted Riesz-Kolmogorov criterion in variable Lebesgue spaces and applies it to obtain multilinear extrapolation and compactness results. These are presented as generalizations of prior work (including the authors' bilinear case), but the central theorems are established directly rather than reducing to self-citations, fitted parameters renamed as predictions, or definitional equivalences. No load-bearing step collapses to an input by construction. The derivation chain remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5703 in / 1003 out tokens · 50819 ms · 2026-07-01T16:03:03.395182+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

55 extracted references · 42 canonical work pages

  1. [1]

    Relatively compact sets in variable-exponent Lebesgue spaces

    R. Bandaliyev and P. Górka. “Relatively compact sets in variable-exponent Lebesgue spaces”. In:Banach Journal of Mathematical Analysis12.2 (Apr. 2018), pp. 331–346.issn: 1735-8787.doi:10.1215/17358787-2017-0039

    work page doi:10.1215/17358787-2017-0039 2018

  2. [2]

    Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

    R. A. Bandaliyev, P. Górka, V. S. Guliyev, and Y. Sawano. “Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces”. In:Mediterranean Journal of Math- ematics18.6 (Sept. 2021).issn: 1660-5454.doi:10.1007/s00009-021-01828-z

    work page doi:10.1007/s00009-021-01828-z 2021

  3. [3]

    Bennett and R

    C. Bennett and R. Sharpley, eds. Vol. 129. Pure and Applied Mathematics. Elsevier, 1988

    1988

  4. [4]

    Compactnesspropertiesofcommutators of bilinear fractional integrals

    Á.Bényi,W.Damián,K.Moen,andR.H.Torres.“Compactnesspropertiesofcommutators of bilinear fractional integrals”. In:Mathematische Zeitschrift280.1–2 (Mar. 2015), pp. 569– 582.issn: 1432-1823.doi:10.1007/s00209-015-1437-4

    work page doi:10.1007/s00209-015-1437-4 2015

  5. [5]

    Compact bilinear operators and commutators

    Á. Bényi and R. Torres. “Compact bilinear operators and commutators”. In:Proceedings of the American Mathematical Society141.10 (July 2013), pp. 3609–3621.issn: 1088-6826. doi:10.1090/s0002-9939-2013-11689-8

    work page doi:10.1090/s0002-9939-2013-11689-8 2013

  6. [6]

    Bergh and J

    J. Bergh and J. Löfström.Interpolation Spaces: An Introduction. Springer Berlin Heidel- berg, 1976.isbn: 9783642664519.doi:10.1007/978-3-642-66451-9

    work page doi:10.1007/978-3-642-66451-9 1976

  7. [7]

    Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

    A. Caetano, A. Gogatishvili, and B. Opic. “Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces”. In:Proceedings of the Royal Society of Edinburgh: Section A Mathematics146.5 (June 2016), pp. 905–927.issn: 1473- 7124.doi:10.1017/s0308210515000761

    work page doi:10.1017/s0308210515000761 2016

  8. [8]

    Extrapolationonfunctionandmodularspaces,and applications

    M.Cao,J.J.Marín,andJ.M.Martell.“Extrapolationonfunctionandmodularspaces,and applications”. In:Advances in Mathematics406 (Sept. 2022), p. 108520.issn: 0001-8708. doi:10.1016/j.aim.2022.108520

    work page doi:10.1016/j.aim.2022.108520 2022

  9. [9]

    Extrapolation for multilinear compact operators and applications

    M. Cao, A. Olivo, and K. Yabuta. “Extrapolation for multilinear compact operators and applications”. In:Transactions of the American Mathematical Society375.7 (Apr. 2022), pp. 5011–5070.issn: 1088-6850.doi:10.1090/tran/8645

    work page doi:10.1090/tran/8645 2022

  10. [10]

    New fractional type weights and the boundedness of some operators

    X. Cen, Q. He, Z. Song, and Z. Wang. “New fractional type weights and the boundedness of some operators”. In:Analysis and Mathematical Physics15.2 (Feb. 2025).issn: 1664-235X. doi:10.1007/s13324-025-01027-z

    work page doi:10.1007/s13324-025-01027-z 2025

  11. [11]

    Weighted estimates for Beltrami equations

    A. Clop and V. Cruz. “Weighted estimates for Beltrami equations”. In:Annales Academiae Scientiarum Fennicae Mathematica38 (Feb. 2013), pp. 91–113.issn: 1798-2383.doi:10. 5186/aasfm.2013.3818

    work page arXiv 2013

  12. [12]

    Weighted norm inequalities for the max- imal operator on variable Lebesgue spaces

    D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer. “Weighted norm inequalities for the max- imal operator on variable Lebesgue spaces”. In:Journal of Mathematical Analysis and Applications394.2 (2012), pp. 744–760.issn: 0022-247X.doi:https : / / doi . org / 10 . 1016/j.jmaa.2012.04.044.url:https://www.sciencedirect.com/science/article/ pii/S0022247X12003137

    2012

  13. [13]

    The maximal operator on weighted variable Lebesgue spaces

    D. Cruz-Uribe, L. Diening, and P. Hästö. “The maximal operator on weighted variable Lebesgue spaces”. In:Fractional Calculus and Applied Analysis14.3 (June 2011), pp. 361– 374.issn: 1311-0454.doi:10.2478/s13540-011-0023-7

    work page doi:10.2478/s13540-011-0023-7 2011

  14. [14]

    Cruz-Uribe and A

    D. Cruz-Uribe and A. Fiorenza.Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer Basel, 2013.isbn: 9783034805483.doi:10.1007/978-3-0348-0548-3

    work page doi:10.1007/978-3-0348-0548-3 2013

  15. [15]

    Limited range multilinear extrapolation with applica- tions to the bilinear Hilbert transform

    D. Cruz-Uribe and J. M. Martell. “Limited range multilinear extrapolation with applica- tions to the bilinear Hilbert transform”. In:Mathematische Annalen371.1-2 (Feb. 2018), pp. 615–653.issn: 1432-1807.doi:10.1007/s00208-018-1640-9

    work page doi:10.1007/s00208-018-1640-9 2018

  16. [16]

    Cruz-Uribe, J

    D. Cruz-Uribe, J. M. Martell, and C. Pérez.Weights, Extrapolation and the Theory of Rubio de Francia. Springer Basel, 2011.isbn: 9783034800723.doi:10.1007/978-3-0348- 0072-3

    work page doi:10.1007/978-3-0348- 2011

  17. [17]

    Extrapolation and weighted norm inequalities in the variable Lebesgue spaces

    D. Cruz-Uribe and L.-A. D. Wang. “Extrapolation and weighted norm inequalities in the variable Lebesgue spaces”. In:Transactions of the American Mathematical Society369.2 (2017), pp. 1205–1235.issn: 00029947, 10886850.url:https://www.jstor.org/stable/ tranamermathsoci.369.2.1205(visited on 12/03/2025). 46 REFERENCES

    2017

  18. [18]

    Diening, P

    L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka.Lebesgue and Sobolev Spaces with Variable Exponents. Springer Berlin Heidelberg, 2011.isbn: 9783642183638.doi:10.1007/ 978-3-642-18363-8

    2011

  19. [19]

    Almost everything you need to know about relatively compact sets in variable Lebesgue spaces

    P. Górka and A. Macios. “Almost everything you need to know about relatively compact sets in variable Lebesgue spaces”. In:Journal of Functional Analysis269.7 (Oct. 2015), pp. 1925–1949.issn: 0022-1236.doi:10.1016/j.jfa.2015.06.024

    work page doi:10.1016/j.jfa.2015.06.024 2015

  20. [20]

    From Arzelà–Ascoli to Riesz–Kolmogorov

    P. Górka and H. Rafeiro. “From Arzelà–Ascoli to Riesz–Kolmogorov”. In:Nonlinear Anal- ysis144 (Oct. 2016), pp. 23–31.issn: 0362-546X.doi:10.1016/j.na.2016.06.004

    work page doi:10.1016/j.na.2016.06.004 2016

  21. [21]

    Extrapolation of Weighted norm inequalities for multi- variable operators and applications

    L. Grafakos and J. M. Martell. “Extrapolation of Weighted norm inequalities for multi- variable operators and applications”. In:Journal of Geometric Analysis14.1 (Mar. 2004), pp. 19–46.issn: 1559-002X.doi:10.1007/bf02921864

    work page doi:10.1007/bf02921864 2004

  22. [22]

    Multilinear Calderón–Zygmund Theory

    L. Grafakos and R. H. Torres. “Multilinear Calderón–Zygmund Theory”. In:Advances in Mathematics165.1 (Jan. 2002), pp. 124–164.issn: 0001-8708.doi:10.1006/aima.2001. 2028

    work page doi:10.1006/aima.2001 2002

  23. [23]

    On relatively compact sets in quasi-Banach function spaces

    W. Guo and G. Zhao. “On relatively compact sets in quasi-Banach function spaces”. In: Proceedings of the American Mathematical Society148.8 (Mar. 2020), pp. 3359–3373.issn: 0002-9939.doi:10.1090/proc/14963

    work page doi:10.1090/proc/14963 2020

  24. [24]

    Extrapolation to mixed norm spaces and applications

    K.-P. Ho. “Extrapolation to mixed norm spaces and applications”. In:Acta et Commen- tationes Universitatis Tartuensis de Mathematica25.2 (Nov. 2021), pp. 281–296.issn: 1406-2283.doi:10.12697/acutm.2021.25.18

    work page doi:10.12697/acutm.2021.25.18 2021

  25. [25]

    Weighted compact commutator of bilinear Fourier multiplier operator

    G. Hu. “Weighted compact commutator of bilinear Fourier multiplier operator”. In:Chinese Annals of Mathematics, Series B38.3 (May 2017), pp. 795–814.issn: 1860-6261.doi: 10.1007/s11401-017-1096-3

    work page doi:10.1007/s11401-017-1096-3 2017

  26. [26]

    Extrapolation of compactness on weighted spaces

    T. Hytönen and S. Lappas. “Extrapolation of compactness on weighted spaces”. In:Revista Matemática Iberoamericana39.1 (Dec. 2021), pp. 91–122.issn: 2235-0616.doi:10.4171/ rmi/1325

    2021

  27. [27]

    Extrapolation of compactness on weighted spaces: Bilinear operators

    T. Hytönen and S. Lappas. “Extrapolation of compactness on weighted spaces: Bilinear operators”. In:Indagationes Mathematicae33.2 (Mar. 2022), pp. 397–420.issn: 0019-3577. doi:10.1016/j.indag.2021.09.007

    work page doi:10.1016/j.indag.2021.09.007 2022

  28. [28]

    Extrapolation for bilinear compact operators in the vari- able exponent setting

    S. Kakaroumpas and S. Lappas. “Extrapolation for bilinear compact operators in the vari- able exponent setting”. In: (2025). arXiv:2512.09721 [math.CA].url:https://https: //arxiv.org/abs/2512.09721

    work page arXiv 2025

  29. [29]

    Multilinear estimates and fractional integration

    C. E. Kenig and E. M. Stein. “Multilinear estimates and fractional integration”. In:Math- ematical Research Letters6.1 (1999), pp. 1–15.issn: 1945-001X.doi:10.4310/mrl.1999. v6.n1.a1

    work page doi:10.4310/mrl.1999 1999

  30. [30]

    Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mit- tel

    A. Kolmogorov. “Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mit- tel”. In:Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse(1931), pp. 60–63.url:https://eudml.org/doc/59336

    1931

  31. [31]

    Criteria for compactness inLp-spaces,p≥0

    V. G. Krotov. “Criteria for compactness inLp-spaces,p≥0”. In:Sbornik: Mathematics 203.7(July2012),pp.1045–1064.issn:1468-4802.doi:10.1070/sm2012v203n07abeh004253

    work page doi:10.1070/sm2012v203n07abeh004253

  32. [32]

    Extrapolation of compactness on weighted Morrey spaces

    S. Lappas. “Extrapolation of compactness on weighted Morrey spaces”. In:Studia Mathe- matica265.2 (2022), pp. 177–195.issn: 1730-6337.doi:10.4064/sm210607-20-9

    work page doi:10.4064/sm210607-20-9 2022

  33. [33]

    Factorizations for variable exponent Muckenhoupt weights

    S. Lappas and T. Oikari. “Factorizations for variable exponent Muckenhoupt weights”. In: (2025). arXiv:2505.23300 [math.FA].url:https://arxiv.org/abs/2505.23300

    work page arXiv 2025

  34. [34]

    End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

    K. Li, J. Martell, H. Martikainen, S. Ombrosi, and E. Vuorinen. “End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications”. In:Transactions of the American Mathematical Society374.1 (Oct. 2020), pp. 97–135.issn: 0002-9947.doi: 10.1090/tran/8172

    work page doi:10.1090/tran/8172 2020

  35. [35]

    Extrapolation for multilinear Muckenhoupt classes andapplications

    K. Li, J. M. Martell, and S. Ombrosi. “Extrapolation for multilinear Muckenhoupt classes andapplications”.In:Advances in Mathematics373(Oct.2020),p.107286.issn:0001-8708. doi:10.1016/j.aim.2020.107286

    work page doi:10.1016/j.aim.2020.107286 2020

  36. [36]

    Weighted estimates for multilinear Fourier multipliers

    K. Li and W. Sun. “Weighted estimates for multilinear Fourier multipliers”. In:Forum Mathematicum27.2 (Mar. 2013), pp. 1101–1116.issn: 1435-5337.doi:10.1515/forum- 2012-0128. REFERENCES 47

    work page doi:10.1515/forum- 2013

  37. [37]

    A note on extrapolation of compactness

    S. Liu, H. Wu, and D. Yang. “A note on extrapolation of compactness”. In:Collectanea Mathematica74.2 (Feb. 2022), pp. 375–390.issn: 2038-4815.doi:10.1007/s13348-022- 00353-w

    work page doi:10.1007/s13348-022- 2022

  38. [38]

    Matrix Weighted Kolmogorov–Riesz’s Compactness Theo- rem

    S. Liu, D. Yang, and C. Zhuo. “Matrix Weighted Kolmogorov–Riesz’s Compactness Theo- rem”. In:Frontiers of Mathematics18.5 (Sept. 2023), pp. 1167–1189.issn: 2731-8656.doi: 10.1007/s11464-021-0103-x

    work page doi:10.1007/s11464-021-0103-x 2023

  39. [39]

    Extrapolation of Compactness on Banach Function Spaces

    E. Lorist and Z. Nieraeth. “Extrapolation of Compactness on Banach Function Spaces”. In:Journal of Fourier Analysis and Applications30.3 (May 2024).issn: 1531-5851.doi: 10.1007/s00041-024-10087-x

    work page doi:10.1007/s00041-024-10087-x 2024

  40. [40]

    Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

    D. Maldonado and V. Naibo. “Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity”. In:Journal of Fourier Analysis and Applications15.2 (Apr. 2008), pp. 218–261.issn: 1531-5851.doi:10.1007/s00041-008- 9029-x

    work page doi:10.1007/s00041-008- 2008

  41. [41]

    Anoteonone-sidedextrapolationofcompactness and applications

    F.J.Martín-ReyesandI.P.Rivera-Ríos.“Anoteonone-sidedextrapolationofcompactness and applications”. In:Studia Mathematica(Aug. 2025).issn: 1730-6337.doi:10.4064/ sm240717-12-3

    2025

  42. [42]

    InterpolationonvariableMorreyspacesdefined on quasi-metric measure spaces

    A.Meskhi,H.Rafeiro,andM.A.Zaighum.“InterpolationonvariableMorreyspacesdefined on quasi-metric measure spaces”. In:Journal of Functional Analysis270.10 (May 2016), pp. 3946–3961.issn: 0022-1236.doi:10.1016/j.jfa.2015.11.013

    work page doi:10.1016/j.jfa.2015.11.013 2016

  43. [43]

    Weighted norm inequalities for the Hardy maximal function

    B. Muckenhoupt. “Weighted norm inequalities for the Hardy maximal function”. In:Trans- actions of the American Mathematical Society165 (1972), pp. 207–207.issn: 0002-9947. doi:10.1090/s0002-9947-1972-0293384-6

    work page doi:10.1090/s0002-9947-1972-0293384-6 1972

  44. [44]

    Extrapolation in general quasi-Banach function spaces

    Z. Nieraeth. “Extrapolation in general quasi-Banach function spaces”. In:Journal of Func- tional Analysis285.10 (Nov. 2023), p. 110130.issn: 0022-1236.doi:10 . 1016 / j . jfa . 2023.110130

    work page arXiv 2023

  45. [45]

    Quantitative estimates and extrapolation for multilinear weight classes

    Z. Nieraeth. “Quantitative estimates and extrapolation for multilinear weight classes”. In: Mathematische Annalen375.1–2 (Mar. 2019), pp. 453–507.issn: 1432-1807.doi:10.1007/ s00208-019-01816-5

    2019

  46. [46]

    Kolmogorov compactness criterion in variable exponent Lebesgue spaces

    H. Rafeiro. “Kolmogorov compactness criterion in variable exponent Lebesgue spaces”. In: Proc. of A. Razmadze Math. Inst.150 (2009), pp. 105–113.url:https://rmi.tsu.ge/ proceedings/volumes/150.htm

    2009

  47. [47]

    Sur les ensembles compacts de fonctions sommables

    M. Riesz. “Sur les ensembles compacts de fonctions sommables”. In:Acta Sect. Math. (Szeged)6 (1933), pp. 136–142.url:https://acta.bibl.u-szeged.hu/13418/

    1933

  48. [48]

    Factorization Theory andAp Weights

    J.-L. Rubio de Francia. “Factorization Theory andAp Weights”. In:American Journal of Mathematics106.3 (June 1984), p. 533.issn: 0002-9327.doi:10.2307/2374284

    work page doi:10.2307/2374284 1984

  49. [49]

    Criteria of compactness in function spaces

    V. N. Sudakov. “Criteria of compactness in function spaces”. In:Uspekhi Mat. Nauk (N.S.) 12:3(75) (1957), pp. 221–224.url:https : / / www . mathnet . ru / php / archive . phtml ? wshow=paper&jrnid=rm&paperid=7628&option_lang=eng

    1957

  50. [50]

    On the compactness of the spaceLp

    J. Tamarkin. “On the compactness of the spaceLp”. In:Bull. Amer. Math. Soc.32 (1932), pp. 79–84.doi:https://doi.org/10.1090/S0002-9904-1932-05332-0

    work page doi:10.1090/s0002-9904-1932-05332-0 1932

  51. [51]

    Boundedness of a Class of Multilinear Operators and Their Iterated Commutators on Morrey-Banach Function Spaces

    J. Tan, J. Wang, and Q. Xue. “Boundedness of a Class of Multilinear Operators and Their Iterated Commutators on Morrey-Banach Function Spaces”. In:The Journal of Geometric Analysis35.7 (June 2025).issn: 1559-002X.doi:10.1007/s12220-025-02043-y

    work page doi:10.1007/s12220-025-02043-y 2025

  52. [52]

    On the compactness of spaceLp(p>0)and its application to integral equations

    M. Tsuji. “On the compactness of spaceLp(p>0)and its application to integral equations”. In:Kodai Mathematical Journal3.1-2 (Jan. 1951).issn: 0386-5991.doi:10.2996/kmj/ 1138843067

    work page doi:10.2996/kmj/ 1951

  53. [53]

    ZurKompaktheitimRaumL p fürp= 1

    A.Tulajkov.“ZurKompaktheitimRaumL p fürp= 1”.In:Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse39 (1933), pp. 167– 170.url:https://eudml.org/doc/59417

    1933

  54. [54]

    WeightedFréchet–KolmogorovTheoremandCompactness of Vector-Valued Multilinear Operators

    Q.Xue,K.Yabuta,andJ.Yan.“WeightedFréchet–KolmogorovTheoremandCompactness of Vector-Valued Multilinear Operators”. In:The Journal of Geometric Analysis31.10 (Mar. 2021), pp. 9891–9914.issn: 1559-002X.doi:10.1007/s12220-021-00630-3

    work page doi:10.1007/s12220-021-00630-3 2021

  55. [55]

    Generalizations of Calderón-Zygmund operators

    K. Yabuta. “Generalizations of Calderón-Zygmund operators”. eng. In:Studia Mathematica 82.1 (1985), pp. 17–31.url:http://eudml.org/doc/218692. 48 REFERENCES Institut für Mathematik, Julius-Maximilians-Universität Würzburg, Emil-Fischer-Str. 41, 97074 Würzburg, Germany Email address:spyridon.kakaroumpas@uni-wuerzburg.de Mathematisches Institut, Rheinische ...

    1985