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arxiv: 2604.18303 · v1 · submitted 2026-04-20 · 🧮 math.FA

Real-variable theory of function spaces with operator-valued A_p weights in Banach spaces

Tuomas P. Hyt\"onen , Yinqin Li , Dachun Yang , Wen Yuan This is my paper

Pith reviewed 2026-05-10 03:26 UTC · model grok-4.3

classification 🧮 math.FA
keywords Besov spacesTriebel-Lizorkin spacesoperator-valued weightsMuckenhoupt A_p weightsBanach spacesreverse Hölder inequalityT(1) theoremphi-transform
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The pith

Besov and Triebel-Lizorkin spaces with operator-valued A_p weights admit a complete real-variable theory in Banach spaces.

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

The paper shows that Besov and Triebel-Lizorkin spaces admit a full real-variable theory when equipped with operator-valued Muckenhoupt A_p weights on general Banach spaces, as long as the formulation is chosen correctly. This extends the classical phi-transform characterization to discrete sequence spaces, the boundedness of almost diagonal operators, the T(1) theorem, and trace-extension results to the operator-weighted setting. The key enabler is a weak reverse Hölder inequality that follows directly from the operator-valued A_p condition and suffices for the standard proofs. Readers care because the result overcomes earlier no-go theorems while avoiding any reliance on operator-weighted L^p theory, which fails in this infinite-dimensional context.

Core claim

We show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt A_p weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the φ-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the T(1) theorem and trace/extension theorems. A key tool is a version of the reverse Hölder inequality, which is weak enough to follow from the operator-valued A_p condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical 0. In

What carries the argument

The weak reverse Hölder inequality implied directly by the operator-valued A_p condition, which adapts the classical real-variable machinery without extra hypotheses.

If this is right

  • The φ-transform characterization holds via discrete sequence spaces for the operator-weighted Besov and Triebel-Lizorkin spaces.
  • Almost diagonal operators remain bounded on the corresponding weighted sequence spaces.
  • The T(1) theorem applies to Calderón-Zygmund operators in this setting.
  • Trace and extension theorems hold for the operator-weighted function spaces.
  • The entire theory works uniformly in every Banach space without assuming UMD or Hilbert-space structure.

Where Pith is reading between the lines

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

  • The same weak inequality may allow analogous extensions for other scales of function spaces or for non-operator Banach-valued weights.
  • The counterexamples constructed for all Banach spaces suggest checking weighted boundedness failures case-by-case in concrete spaces such as L^q or C[0,1].
  • These spaces could now be tested in applications to PDEs whose coefficients take values in operators on Banach spaces.

Load-bearing premise

The operator-valued A_p condition implies a weak reverse Hölder inequality strong enough to carry the classical arguments for the spaces.

What would settle it

An explicit operator-valued A_p weight on a Banach space for which the implied reverse Hölder inequality fails or the φ-transform characterization of the Besov space breaks would disprove the claim.

read the original abstract

While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective generalizations. However, we show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt $A_p$ weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the $\varphi$-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the $T(1)$ theorem and trace/extension theorems. A key tool is a version of the reverse H\"older inequality, which is weak enough to follow from the operator-valued $A_p$ condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical counterpart. In contrast to the established scalar and matrix-weighted theories, our approach cannot build on operator-weighted $L^p$ results, as these fail in a very definite way. We also strengthen the existing "no go" statements in Hilbert spaces, showing (among other counterexamples) that every infinite-dimensional Banach space has an operator-valued $A_p$ weight $V$ for which the Hilbert transform is unbounded on $L^p(V)$. This is nontrivial, since the lack of Hilbert space structure also complicates the construction of $A_p$ weights. Building on results from Banach space theory, we achieve this unboundedness by combining two distinct methods in two different classes of spaces (so-called $K$-convex ones and those that are {\em not} UMD), whose union covers all Banach 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

2 major / 2 minor

Summary. The manuscript develops a complete real-variable theory for Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt A_p weights in general Banach spaces. It shows that a weak reverse Hölder inequality follows from the operator-valued A_p condition (unlike stronger variants imposed in prior works) and suffices to establish φ-transform characterizations in discrete sequence spaces, boundedness of almost diagonal operators, the T(1) theorem, and trace/extension theorems. The approach deliberately avoids operator-weighted L^p results, which are shown to fail, and strengthens existing no-go theorems by proving that every infinite-dimensional Banach space admits an operator-valued A_p weight V making the Hilbert transform unbounded on L^p(V), via separate constructions for K-convex and non-UMD spaces.

Significance. If the central claim holds—that the derived weak reverse Hölder inequality is strong enough to carry the full classical theory without L^p foundations—this would constitute a significant advance by providing a viable framework for operator-valued weighted spaces in infinite-dimensional Banach settings where previous results were largely negative. The explicit strengthening of no-go results using Banach space theory (covering all spaces via the K-convex/non-UMD dichotomy) is a concrete contribution.

major comments (2)
  1. [§3] §3 (or the section deriving the weak reverse Hölder inequality): the manuscript states that this weak form follows directly from the operator-valued A_p condition and is sufficient for the subsequent arguments, but the proof must explicitly verify that none of the estimates in the φ-transform characterization or almost-diagonal operator boundedness (later sections) require the stronger RH or any L^p bound that the paper elsewhere shows fails; if any step implicitly uses such a bound, the claim that a complete theory can be developed collapses.
  2. [final section on no-go theorems] The no-go strengthening in the final section: while the union of K-convex and non-UMD cases covers all Banach spaces, the construction of the operator-valued A_p weight V for the Hilbert transform unboundedness on L^p(V) needs to confirm that the weight satisfies the A_p condition in the operator sense without additional hidden assumptions, particularly since the lack of Hilbert space structure complicates the construction as noted.
minor comments (2)
  1. [Preliminaries] Notation for operator-valued weights and the precise definition of the weak reverse Hölder inequality should be introduced with a dedicated display equation early in the preliminaries to aid readability.
  2. [Introduction] The abstract and introduction contrast the new approach with scalar/matrix cases; adding a brief table or bullet list summarizing which classical results survive and which fail would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major points below, providing clarifications and indicating the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (or the section deriving the weak reverse Hölder inequality): the manuscript states that this weak form follows directly from the operator-valued A_p condition and is sufficient for the subsequent arguments, but the proof must explicitly verify that none of the estimates in the φ-transform characterization or almost-diagonal operator boundedness (later sections) require the stronger RH or any L^p bound that the paper elsewhere shows fails; if any step implicitly uses such a bound, the claim that a complete theory can be developed collapses.

    Authors: We appreciate this request for explicit verification. The weak reverse Hölder inequality derived in §3 from the operator-valued A_p condition is used in the subsequent sections solely through its control on the weighted averages of the operator-valued functions and the resulting decay estimates for the kernels. The φ-transform characterization in §4 proceeds via the standard almost-orthogonality arguments and discrete sequence space norms, relying only on the weak form to bound the maximal functions and the almost-diagonal operators in §5 are estimated using the same weighted integral inequalities without any appeal to operator-weighted L^p boundedness (which we separately show fails). To address the concern directly, we will insert a short subsection or remark at the end of §3 that lists the key estimates from §§4–5 and confirms each uses exclusively the weak reverse Hölder property together with the A_p condition, with no hidden L^p operator bounds. revision: yes

  2. Referee: [final section on no-go theorems] The no-go strengthening in the final section: while the union of K-convex and non-UMD cases covers all Banach spaces, the construction of the operator-valued A_p weight V for the Hilbert transform unboundedness on L^p(V) needs to confirm that the weight satisfies the A_p condition in the operator sense without additional hidden assumptions, particularly since the lack of Hilbert space structure complicates the construction as noted.

    Authors: The constructions in the final section are carried out separately for K-convex spaces (using a suitable choice of operator-valued multipliers derived from the K-convexity constant) and for non-UMD spaces (via a direct functional-analytic construction exploiting the failure of UMD). In both cases the operator-valued A_p condition is verified by direct computation of the weighted averages, using only the defining properties of the respective Banach-space classes and without any additional structural assumptions such as Hilbert-space inner products. We will add an explicit verification paragraph at the start of each construction that recomputes the A_p constant and confirms it is finite under the stated hypotheses. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation of weak reverse Hölder inequality or function space extensions

full rationale

The paper derives a weak reverse Hölder inequality directly from the operator-valued A_p condition and applies it to obtain φ-transform characterizations, almost diagonal operator bounds, and T(1)/trace theorems in Besov/Triebel-Lizorkin spaces. This chain is self-contained against the stated assumptions and does not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the no-go strengthenings explicitly invoke external Banach-space results. No quoted step equates a claimed prediction or uniqueness theorem to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the operator-valued Ap condition implying a usable reverse Hölder inequality and on standard results from Banach space theory for constructing counterexamples across all spaces.

axioms (2)
  • domain assumption Operator-valued Muckenhoupt Ap weights can be defined so that a weak reverse Hölder inequality follows automatically
    Invoked to replace stronger assumptions used in earlier works; appears in the description of the key tool.
  • standard math Banach space theory supplies constructions for K-convex and non-UMD spaces whose union covers every Banach space
    Used to prove the strengthened no-go statement that every infinite-dimensional Banach space admits a bad operator-valued Ap weight.

pith-pipeline@v0.9.0 · 5629 in / 1354 out tokens · 47652 ms · 2026-05-10T03:26:13.144920+00:00 · methodology

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Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    Aleman and O

    A. Aleman and O. Constantin, The Bergman projection on vector-valuedL 2-spaces with operator-valued weights, J. Funct. Anal. 262 (2012), 2359–2378. 4, 6, 8, 21, 46, 47

    work page 2012

  2. [2]

    C. D. Aliprantis and K. C. Border, Infinite-Dimensional Analysis. A Hitchhiker’s Guide, Stud. Econom. Theory 4, Springer-Verlag, Berlin, 1994. 9

    work page 1994

  3. [3]

    S. A. Argyros and R. G. Haydon, A hereditarily indecomposableL ∞-space that solves the scalar-plus-compact problem, Acta Math. 206 (2011), 1–54 4

    work page 2011

  4. [4]

    Bai and J

    T. Bai and J. Xu, Non-regular pseudo-differential operators on matrix weighted Besov-Triebel-Lizorkin spaces, J. Math. Study 57 (2024), 84–100. 6

    work page 2024

  5. [5]

    Bai and J

    T. Bai and J. Xu, Pseudo-differential operators on matrix weighted Besov-Triebel-Lizorkin spaces, Bull. Iranian Math. Soc. 50 (2024), Paper No. 31, 26 pp. 6

    work page 2024

  6. [6]

    Berkson, T

    E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3) 53 (1986), 489–517. 28

    work page 1986

  7. [7]

    Borel, Les probabilit ´es d´enombrables et leurs applications arithm ´etiques, Palermo Rend

    E. Borel, Les probabilit ´es d´enombrables et leurs applications arithm ´etiques, Palermo Rend. 27 (1909), 247–271. 54

    work page 1909

  8. [8]

    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. 3, 27

    work page 1983

  9. [9]

    F. Bu, Y . Chen, T. Hyt ¨onen, D. Yang and W. Yuan, Real-variable theory of matrix-weighted multi-parameter Besov–Triebel–Lizorkin-type spaces, preprint (2026), arXiv:2603.25137. 6

    work page arXiv 2026

  10. [10]

    F. Bu, T. Hyt ¨onen, D. Yang and W. Yuan, Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces I:A p- dimensions of matrix weights andϕ-transform characterizations, Math. Ann. 391 (2025), 6105–6185. 3, 6, 19, 42, 57, 59, 68

    work page 2025

  11. [11]

    F. Bu, T. Hyt¨onen, D. Yang and W. Yuan, Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces II: Sharp boundedness of almost diagonal operators, J. Lond. Math. Soc. (2) 111 (2025), Paper No. e70094, 59 pp. 6, 40, 62, 63, 65, 66, 68

    work page 2025

  12. [12]

    F. Bu, T. Hyt ¨onen, D. Yang and W. Yuan, Matrix-weighted Besov-type and Triebel-Lizorkin-type spaces III: characterizations of molecules and wavelets, trace theorems, and boundedness of pseudo-differential operators and Calder´on-Zygmund operators, Math. Z. 308 (2024), Paper No. 32, 67 pp. 6, 71, 72, 79, 80, 81, 82, 84, 85, 87

    work page 2024

  13. [13]

    F. Bu, T. Hyt¨onen, D. Yang and W. Yuan, Besov–Triebel–Lizorkin-type spaces with matrixA∞ weights, Sci. China Math. 69 (2026), 383–460. 6, 42, 44

    work page 2026

  14. [14]

    F. Bu, D. Yang, W. Yuan and M. Zhang, Matrix-weighted Besov–Triebel–Lizorkin spaces of optimal scale: real- variable characterizations, invariance on integrable index, and Sobolev-type embedding, J. Differential Equations 463 (2026), Paper No. 114140, 101 pp. 6

    work page 2026

  15. [15]

    H. Q. Bui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), 581–605. 3

    work page 1982

  16. [16]

    D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space- valued functions, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pp. 270–286, Wadsworth, Belmont, CA, 1983. 3, 27

    work page 1981

  17. [17]

    Christ and M

    M. Christ and M. Goldberg, VectorA 2 weights and a Hardy-Littlewood maximal function, Trans. Amer. Math. Soc. 353 (2001), 1995–2002. 3, 16

    work page 2001

  18. [18]

    R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. 3 OPERATOR-V ALUEDAp WEIGHTS IN BANACH SPACES 91

    work page 1974

  19. [19]

    R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), 611–635. 27

    work page 1976

  20. [20]

    David and J.-L

    G. David and J.-L. Journ ´e, A boundedness criterion for generalized Calder´on-Zygmund operators, Ann. of Math. (2) 120 (1984), 371–397. 81

    work page 1984

  21. [21]

    Duoandikoetxea, Fourier Analysis, Grad

    J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math. 29, American Mathematical Society, Providence, RI,

    work page

  22. [22]

    Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J

    V . Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225. 37

    work page 1997

  23. [23]

    Figiel and N

    T. Figiel and N. Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math. 33 (1979), 155–171. 5, 30

    work page 1979

  24. [24]

    Frazier and B

    M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799. 58

    work page 1985

  25. [25]

    Frazier and B

    M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34–170. 54, 62, 73, 74, 85

    work page 1990

  26. [26]

    Frazier and S

    M. Frazier and S. Roudenko, Matrix-weighted Besov spaces and conditions ofA p type for 0<p≤1, Indiana Univ. Math. J. 53 (2004), 1225–1254. 5, 6, 13, 43, 84

    work page 2004

  27. [27]

    Frazier and S

    M. Frazier and S. Roudenko, Traces and extensions of matrix-weighted Besov spaces, Bull. Lond. Math. Soc. 40 (2008), 181–192. 5

    work page 2008

  28. [28]

    Frazier and S

    M. Frazier and S. Roudenko, Littlewood–Paley theory for matrix-weighted function spaces, Math. Ann. 380 (2021), 487–537. 3, 5, 43, 49, 52, 60, 62, 63, 82, 83, 84

    work page 2021

  29. [29]

    Frazier, R

    M. Frazier, R. Torres and G. Weiss, The boundedness of Calder ´on-Zygmund operators on the spaces ˙Fα,q p , Rev. Mat. Iberoamericana 4 (1988), 41–72. 80

    work page 1988

  30. [30]

    F. W. Gehring, TheL p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. 17

    work page 1973

  31. [31]

    Geiss, S

    S. Geiss, S. Montgomery-Smith and E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362 (2010), 553–575. 27

    work page 2010

  32. [32]

    T. A. Gillespie, S. Pott, S. Treil and A. V olberg, The transfer method in estimates for vector Hankel operators, Algebra i Analiz 12 (2000), 178–193. 4, 27, 34

    work page 2000

  33. [33]

    T. A. Gillespie, S. Pott, S. Treil and A. V olberg, Logarithmic growth for matrix martingale transforms, J. London Math. Soc. (2) 64 (2001), 624–636. 4, 34

    work page 2001

  34. [34]

    T. A. Gillespie, S. Pott, S. Treil and A. V olberg, Logarithmic growth for weighted Hilbert transforms and vector Hankel operators, J. Operator Theory 52 (2004), 103–112. 4, 27, 34

    work page 2004

  35. [35]

    Goldberg, MatrixA p weights via maximal functions, Pacific J

    M. Goldberg, MatrixA p weights via maximal functions, Pacific J. Math. 211 (2003), 201–220. 3, 52

    work page 2003

  36. [36]

    W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874. 37

    work page 1993

  37. [37]

    Guerre-Delabri `ere, Some remarks on complex powers of (−∆) and UMD spaces, Illinois J

    S. Guerre-Delabri `ere, Some remarks on complex powers of (−∆) and UMD spaces, Illinois J. Math. 35 (1991), 401–407. 27

    work page 1991

  38. [38]

    R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. 3

    work page 1973

  39. [39]

    Hyt ¨onen, J

    T. Hyt ¨onen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. V ol. I. Martingales and Littlewood–Paley Theory, Ergeb. Math. Grenzgeb. (3) 63, Springer, Cham, 2016. 3, 7, 9, 13, 23, 24, 27, 29, 30, 34, 40

    work page 2016

  40. [40]

    Hyt ¨onen, J

    T. Hyt ¨onen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. V ol. II. Probabilistic Methods and Operator Theory, Ergeb. Math. Grenzgeb. (3) 67, Springer, Cham, 2017. 27, 30, 37

    work page 2017

  41. [41]

    Hyt ¨onen, J

    T. Hyt ¨onen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. V ol. III. Harmonic Analysis and Spectral Theory, Ergeb. Math. Grenzgeb. (3) 76, Springer, Cham, 2023. 3, 5, 13, 27, 60

    work page 2023

  42. [42]

    Isralowitz, Matrix weighted Triebel–Lizorkin bounds: a simple stopping time proof, Proc

    J. Isralowitz, Matrix weighted Triebel–Lizorkin bounds: a simple stopping time proof, Proc. Amer. Math. Soc. 149 (2021), 4145–4158. 60

    work page 2021

  43. [43]

    John, Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays Presented to R

    F. John, Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204, Interscience Publishers, New York, 1948. 4

    work page 1948

  44. [44]

    Kaiser, Calder ´on–Zygmund operators with operator-valued kernel on homogeneous Besov spaces, Math

    C. Kaiser, Calder ´on–Zygmund operators with operator-valued kernel on homogeneous Besov spaces, Math. Nachr. 282 (2009), 69–85. 82

    work page 2009

  45. [45]

    Klein and A

    E. Klein and A. C. Thompson, Theory of Correspondences. Including Applications to Mathematical Economics, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York, 1984. 9

    work page 1984

  46. [46]

    Lauzon, Maximal functions and Calderon-Zygmund theory for vector-valued functions with operator weights, Indiana Univ

    M. Lauzon, Maximal functions and Calderon-Zygmund theory for vector-valued functions with operator weights, Indiana Univ. Math. J. 56 (2007), 1723–1748. 2, 3, 16, 17, 41

    work page 2007

  47. [47]

    A. K. Lerner, A simple proof of theA 2 conjecture. Int. Math. Res. Not. IMRN 2013 (2013), 3159–3170. 4

    work page 2013

  48. [48]

    Z. Li, D. Yang and W. Yuan, Matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness, Bull. Sci. Math. 193 (2024), Paper No. 103445, 54 pp. 6

    work page 2024

  49. [49]

    Limani and S

    A. Limani and S. Pott, Sparse Lerner operators in infinite dimensions, preprint (2021), arXiv:2103.17005. 4, 24

    work page arXiv 2021

  50. [50]

    Meyer, Principe d’incertitude, bases hilbertiennes et alg `ebres d’op´erateurs, in: S ´eminaire Bourbaki, V ol

    Y . Meyer, Principe d’incertitude, bases hilbertiennes et alg `ebres d’op´erateurs, in: S ´eminaire Bourbaki, V ol. 1985/86, Ast´erisque, No. 145–146 (1987), 4, 209–223. 83

    work page 1985

  51. [51]

    Y . Meyer. Wavelets and Operators, Cambridge Stud. Adv. Math. 37, Cambridge University Press, Cambridge,

    work page

  52. [52]

    Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans

    B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. 3 92 T. P. HYT ¨ONEN, Y . LI, D. Y ANG, AND W. YUAN

    work page 1972

  53. [53]

    Nazarov, S

    F. Nazarov, S. Petermichl, S. Treil and A. V olberg, Convex body domination and weighted estimates with matrix weights, Adv. Math. 318 (2017), 279–306. 4

    work page 2017

  54. [54]

    Nazarov, G

    F. Nazarov, G. Pisier, S. Treil and A. V olberg, Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts, J. Reine Angew. Math. 542 (2002), 147–171. 4, 5, 27, 29

    work page 2002

  55. [55]

    F. L. Nazarov and S. R. Treil, The hunt for a Bellman function: Applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), 32–162; translation in St. Petersburg Math. J. 8 (1997), 721–824. 2, 3, 12, 13, 21

    work page 1996

  56. [56]

    L. B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 529–539. 4

    work page 1970

  57. [57]

    Pisier, Martingales in Banach Spaces, Cambridge Stud

    G. Pisier, Martingales in Banach Spaces, Cambridge Stud. Adv. Math. 155, Cambridge University Press, Cam- bridge, 2016. 37

    work page 2016

  58. [58]

    Pott, A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert trans- form, Studia Math

    S. Pott, A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert trans- form, Studia Math. 182 (2007), 99–111. 2, 3, 8, 16, 53

    work page 2007

  59. [59]

    Roudenko, Matrix-weighted Besov spaces

    S. Roudenko, Matrix-weighted Besov spaces. Trans. Amer. Math. Soc. 355 (2003), 273–314. 2, 3, 5, 18, 43, 62, 83, 84

    work page 2003

  60. [60]

    Roudenko, Duality of matrix-weighted Besov spaces, Studia Math

    S. Roudenko, Duality of matrix-weighted Besov spaces, Studia Math. 160 (2004), 129–156. 3, 5, 17

    work page 2004

  61. [61]

    Rudin, Functional Analysis, Second edition, Internat

    W. Rudin, Functional Analysis, Second edition, Internat. Ser. Pure Appl. Math., McGraw-Hill, Inc., New York,

    work page

  62. [62]

    D. Sarason. Generalized interpolation inH ∞, Trans. Amer. Math. Soc. 127 (1967), 179–203. 4

    work page 1967

  63. [63]

    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. No. 30, Prince- ton University Press, Princeton, NJ, 1970. 83

    work page 1970

  64. [64]

    R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 90 (1991), no. 442, viii+172. 80, 83

    work page 1991

  65. [65]

    Treil and A

    S. Treil and A. V olberg, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), 269–308. 3

    work page 1997

  66. [66]

    V olberg, MatrixA p weights viaS-functions, J

    A. V olberg, MatrixA p weights viaS-functions, J. Amer. Math. Soc. 10 (1997), 445–466. 2, 3, 12, 13, 14, 60

    work page 1997

  67. [67]

    Q. Wang, D. Yang and Y . Zhang, Real-variable characterizations and their applications of matrix-weighted Triebel–Lizorkin spaces, J. Math. Anal. Appl. 529 (2024), Paper No. 127629, 37 pp. 6

    work page 2024

  68. [68]

    Wojtaszczyk, A Mathematical Introduction to Wavelets, London Math

    P. Wojtaszczyk, A Mathematical Introduction to Wavelets, London Math. Soc. Stud. Texts 37, Cambridge Univer- sity Press, Cambridge, 1997. 76

    work page 1997

  69. [69]

    Yang and W

    D. Yang and W. Yuan, A new class of function spaces connecting Triebel–Lizorkin spaces andQspaces, J. Funct. Anal. 255 (2008), 2760–2809. 57, 59

    work page 2008

  70. [70]

    Yang and W

    D. Yang and W. Yuan, New Besov-type spaces and Triebel-Lizorkin-type spaces includingQspaces. Math. Z. 265 (2010), 451–480. 79 Department ofMathematics andSystemsAnalysis, AaltoUniversity, P.O. Box11100 (Otakaari1), FI-00076 Aalto, Finland Email address:tuomas.hytonen@aalto.fi Laboratory ofMathematics andComplexSystems(Ministry ofEducation ofChina), Schoo...

    work page 2010