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Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calder\'{o}n--Zygmund Operators

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Pith's one-line read The paper establishes boundedness of pseudo-differential, trace, and Calderón–Zygmund operators on generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces, including a trace theorem that is an equivalence.

desk verdict A solid, honest technical continuation; the operator boundedness results are real and new, but all six theorems lean on the companion paper's almost-diagonal theorem, so peer review should happen with that dependency visible. read the letter →

arxiv 2504.19060 v2 pith:RB2NA2ME submitted 2025-04-27 math.FA math.CA

Dachun Yang , Wen Yuan , Mingdong Zhang This is my paper · ORCID
classification math.FAmath.CA MSC 42B2046E3535S0547A5646E4042B35
keywords matrixweightA∞generalizedBesov–Triebel–Lizorkin-typespacepseudo-differentialoperatortraceCalderón–Zygmundalmostdiagonalmolecularandwaveletcharacterization
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

This paper aims to show that the main operator classes of harmonic analysis—pseudo-differential operators with homogeneous symbols, trace and extension operators across a hyperplane, and Calderón–Zygmund singular integrals—are bounded on the generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces $\dot A^{s,\upsilon}_{p,q}(W)$ introduced in the companion paper [13]. The weights are matrix-valued $A_{p,\infty}$ weights, the current broadest matrix-weight class, and the growth function $\upsilon$ lets the results cover Morrey-type, classical, and weighted scales at once. Every theorem is proved by the same route: replace the operator by an almost diagonal matrix on the associated sequence space, apply the boundedness criterion from [13, Theorem 3.6], and reassemble the output through synthesis molecules. The trace result is an iff statement, so it identifies exactly which matrix-weight comparison is needed for a bounded boundary trace. If the proofs are correct, one uniform framework yields all these operator bounds, recovering and in several cases improving earlier scalar and matrix-weighted results.

What carries the argument

The load-bearing mechanism is the boundedness of almost diagonal operators on the matrix-weighted sequence spaces $\dot a^{s,\upsilon}_{p,q}(W)$. An infinite matrix $U=\{u_{Q,R}\}$ is $(D,E,F)$-almost diagonal when $|u_{Q,R}|$ is controlled by $[1+|x_Q-x_R|/(\ell(Q)\vee\ell(R))]^{-D}$ times powers of the scale ratio $\ell(Q)/\ell(R)$, and Theorem 3.6 gives explicit thresholds on $D,E,F$ in terms of the smoothness $s$, the growth function $\upsilon$, and the $A_{p,\infty}$-lower and upper dimensions of $W$. This single criterion powers the molecular, $\phi$-transform, and wavelet characterizations from [13]; each operator is converted into its coefficient matrix between analysis and synthesis molecules, the matrix is shown to be almost diagonal, and boundedness follows. Pseudo-differential operators enter through the lemma that $|Q|^{\eta/n}T_\theta(\psi_Q)$ is a molecule, trace and extension operators enter through compactly supported wavelets and comparisons of reducing operators, and Calderón–Zygmund operators enter through the atom-to-molecule proposition of the paper.

What would settle it

In the scalar unweighted case ($m=1$, $W\equiv 1$, $\upsilon(Q)=|Q|^\tau$) the thresholds in Theorem 3.6 must reduce to the known sharp almost-diagonal indices for classical Besov–Triebel–Lizorkin-type spaces; a mismatch for any choice of $p,q,\tau$ would falsify the criterion and pull down all six theorems. Alternatively, for $n=1$ and $p=2$, choose explicit matrix weights $W,V$ for which the comparison (3.21) fails yet $s$ lies above the stated threshold; if the trace operator is still bounded from $\dot B^{s,\upsilon(2)}_{2,q}(W)$ to $\dot B^{s-\gamma/2,\upsilon(1)}_{2,q}(V)$, the equivalence in Theorem 3.24 is false.

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Extended reading notes

Core claim

The central claim is that the full set of operator-boundedness phenomena survives on $\dot A^{s,\upsilon}_{p,q}(W)$ with matrix $A_{p,\infty}$ weights. Theorem 3.10 constructs an extension $\widetilde T_\theta$ of a pseudo-differential operator with symbol $\theta \in \dot S^\eta_{1,1}$ and proves it is bounded from $\dot A^{s+\eta,\upsilon}_{p,q}(W)$ to $\dot A^{s,\upsilon}_{p,q}(W)$ whenever $F_{\dot a^{s,\upsilon}_{p,q}(W)} < n/2$, with a vanishing-moment condition covering the critical case. Theorem 3.24 proves that the trace operator extends to a bounded map from $\dot B^{s,\upsilon(n+1)}_{p,q}(W)$ on $\mathbb R^{n+1}$ to $\dot B^{s-\gamma/p,\upsilon(n)}_{p,q}(V)$ on $\mathbb R^n$ if and only if, for every dyadic cube $Q'$ and vector $z$, $\int_{Q'} |V^{1/p}(x)z|^p\,dx \le C 2^{j_{Q'}\gamma}\int_{P(Q',0)} |W^{1/p}(x)z|^p\,dx$; Theorems 3.30, 3.32, and 3.35 give the companion extension operators and the Triebel–Lizorkin version. Theorem 3.51 proves that Calderón–Zygmund operators in the class $CZO_\sigma(E,F,G,H)$ are bounded whenever the indices satisfy (3.62), by showing they map sufficiently regular atoms to synthesis molecules. Specializing $\upsilon(Q)=|Q|^\tau$ recovers matrix-weighted Besov–Triebel–Lizorkin-type spaces, so the results are new already for matrix-weighted Besov–Triebel–Lizorkin spaces with matrix $A_{p,\infty}$ weights.

Load-bearing premise

The load-bearing premise is that the companion paper's boundedness criterion for almost diagonal matrices on the sequence spaces—together with the molecular and wavelet characterizations built on it—is correct; every main theorem is only that criterion applied after checking that the operator's coefficient matrix is almost diagonal.

Editorial extensions

If this is right

  • Homogeneous pseudo-differential operators of order $\eta$ shift the smoothness scale: they boundedly map $\dot A^{s+\eta,\upsilon}_{p,q}(W)$ into $\dot A^{s,\upsilon}_{p,q}(W)$, exactly as in classical Besov–Triebel–Lizorkin theory.
  • The trace and extension theorems make $\dot B^{s,\upsilon(n+1)}_{p,q}(W)$ and $\dot B^{s-\gamma/p,\upsilon(n)}_{p,q}(V)$ exact boundary counterparts, with $Tr \circ Ext$ the identity, precisely under the matrix-weight comparison (3.21) or its reverse.
  • Calderón–Zygmund operators are bounded on every space of this family whose indices satisfy (3.62), so the result covers matrix-weighted Besov–Triebel–Lizorkin spaces $\dot A^s_{p,q}(W)$ with $W\in A_{p,\infty}$—cases that are new even without the Morrey-type parameter.
  • Because $\upsilon(Q)=|Q|^\tau$ and $\upsilon(Q)=\int_Q w$ are included, the same proofs deliver the results simultaneously for matrix-weighted Besov–Triebel–Lizorkin, Besov–Triebel–Lizorkin-type, and weighted spaces.
  • In the scalar unweighted limit the theorems match or improve prior Besov–Triebel–Lizorkin-type results, in particular removing the upper bound on $\tau$ that earlier pseudo-differential and trace theorems required.

Reading between the lines

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

  • The iff trace theorem suggests treating (3.21) as the definition of a traceability class for pairs of matrix weights $(W,V)$ across a hyperplane; classifying which pairs satisfy it for given $p,\gamma$ would be a natural next step the paper does not isolate.
  • The extra term $d^{upper}_{p,\infty}(W)/p$ in the index thresholds appears only because the weights are $A_{p,\infty}$ rather than $A_p$; since the paper observes it vanishes for $p\le 1$, testing diagonal-matrix examples for $p>1$ should show whether the term is necessary or an artefact of the reducing-operator estimate.
  • The same molecule-and-wavelet scheme is directly portable: any setting with an almost-diagonal boundedness theorem and an atom-to-molecule lemma (mixed norms, product spaces, inhomogeneous spaces) should inherit matching pseudo-differential, trace, and Calderón–Zygmund boundedness results.
  • Since all proofs are reductions to almost diagonal matrices, the constants in the operator bounds are controlled by the same quantities as the sequence-space criterion; extracting explicit dependence on the $A_{p,\infty}$ characteristic would make the results quantitative.
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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

2 major / 5 minor

Summary. The paper proves boundedness results for pseudodifferential, trace, and Calderón–Zygmund operators on the generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces Ȧ^{s,υ}_{p,q}(W) with matrix A_{p,∞} weights, which were introduced in the companion paper [13]. The main results are Theorem 3.10 (pseudodifferential operators), Theorems 3.24, 3.30, 3.32, and 3.35 (trace and extension operators), and Theorem 3.51 (Calderón–Zygmund operators). The proofs are structured as reductions: using the ϕ-transform, molecular, and wavelet characterizations from [13], each operator is shown to map suitable molecules or atoms to synthesis molecules, with the admissible index ranges controlled by the almost-diagonal-operator thresholds D_ȧ, E_ȧ, F_ȧ from Theorem 3.6. The trace theorem is stated as an if-and-only-if result in terms of a two-weight matrix inequality (3.21).

Significance. If the companion results in [13] are sound, the paper delivers a broad and coherent framework for operator boundedness on matrix-weighted Besov–Triebel–Lizorkin-type spaces with A_{p,∞} weights. It subsumes or extends earlier A_p results of Bu–Hytönen–Yang–Yuan when p∈(0,1], and in the scalar unweighted case it recovers or improves several Besov-type and Triebel–Lizorkin-type results of Sawano–Yang–Yuan. The proofs are detailed and the reduction strategy is transparent, and the authors are explicitly honest about the comparison with prior work, including the non-sharpness issue for p∈(1,∞) recorded in Remark 3.7. The main limitation is the heavy reliance on the companion preprint [13], whose proofs are not included in the manuscript.

major comments (2)
  1. [Theorem 3.6 and Lemmas 3.16, 3.23 (used in Theorems 3.10, 3.24, 3.30, 3.32, 3.35, 3.51)] All six main theorems are proved by reducing operator boundedness to the almost diagonal operator theorem and the molecular/wavelet characterizations restated from the companion preprint [13] as Theorem 3.6, Lemma 3.16, and Lemma 3.23. The index thresholds D_ȧ, E_ȧ, F_ȧ in Theorem 3.6 determine the exact hypotheses of every main theorem, and the constants involving d_lower and d_upper are load-bearing for the stated parameter ranges. Since [13] is a submitted preprint and its proofs are not reproduced here, the present manuscript cannot be fully certified on its own: any correction or revision of those index formulas would shift the ranges in all six theorems. Please either include complete proofs of the supporting results (at least of the specific consequences used), or condition the paper on the acceptance of [13] and state this dependence explicitly in the introduction.
  2. [Theorem 3.24 and Theorem 3.32 (necessity direction)] The necessity direction of the trace theorems tests boundedness on a single function built from one wavelet coefficient, namely the sequence (3.22). This is a valid and efficient testing strategy, but the argument implicitly relies on the wavelet characterization in Lemma 3.23 to identify the norm of that test function, and Lemma 3.23 is itself one of the imported results from [13]. This is not a separate error, but it reinforces the first major comment: the if-and-only-if statements are only as reliable as the companion results, and the dependence should be made visible in the statement of the theorems rather than only in the proofs.
minor comments (5)
  1. [Title and page 2] There are typos in the title and first lines: “Optima l Scale” and “defintion” should read “Optimal Scale” and “definition”; please proofread the manuscript.
  2. [Remark 3.26] The notation A := {1}_{Q∈D(Rn)} is ambiguous; it should be A := {1_Q}_{Q∈D(Rn)} or explicitly described as the sequence of identity matrices on the scalar-valued setting.
  3. [Theorem 3.51 proof] In the proof of Theorem 3.51, the abbreviation “WPB” appears where “WBP” (the weak boundedness property defined earlier) is meant.
  4. [Theorem 3.51(ii)] The deduction that H ≥ 0 and G ≥ 0 imply T(1)=0 and T*(1)=0 via Definition 3.40(iv) is only implicit; it should be stated explicitly because it is exactly what makes [19, Theorem 1] applicable.
  5. [Title and Remark 3.7] The phrase “optimal scale” in the title and abstract refers to the scale of spaces and growth functions, but Remark 3.7 shows that the operator-index ranges are not optimal for p∈(1,∞). Please add a clarifying sentence in the introduction or in Remark 3.7 so that readers do not interpret the operator bounds as sharp in the parameter ranges.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: all six operator theorems are proved from the companion almost-diagonal/molecular/wavelet characterizations of [13] used as explicit premises; the dependency is a verification caveat, not a reduction by construction.

full rationale

The paper is transparent about its derivation chain: Section 3 states that the main tools are the molecular and wavelet characterizations established in [13], and each theorem is proved by reducing operator boundedness to Theorem 3.6 ("exactly [13, Theorem 5.6]"), Lemma 3.16, Lemma 3.19, and Lemma 3.23. This is a dependency on a companion paper by overlapping authors, and the proof of Theorem 3.6 is not reproduced here; thus the index ranges in Theorems 3.10, 3.24, 3.30, 3.32, 3.35, and 3.51 are only as reliable as that external result. However, this is not circular reasoning: Theorem 3.6 and the imported lemmas are stated as premises with their own hypotheses, and none of those hypotheses includes the target boundedness conclusions. The pseudo-differential extension in (3.6) is a Calderón reproducing formula rather than a definition of the conclusion; the trace theorem's condition (3.21) is an independent weight inequality, obtained in the necessity direction and assumed in sufficiency; the Calderón--Zygmund result reduces to Proposition 3.49, a published result, plus the same companion characterizations. No fitted parameter is renamed as a prediction, and no target statement is assumed in the proofs. The score of 2 reflects the verification caveat from the load-bearing self-citation to the unproved companion result, not a finding of circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper does not introduce free parameters or invented entities. Its central claims rest on a sequence of imported theorems: especially Theorem 3.6 (almost diagonal operator boundedness) and the molecular/wavelet characterizations from the companion preprint [13], plus the reducing operator estimates from [11]. These are domain assumptions in the sense that the present paper assumes their correctness without proof. They are not circular in the sense of assuming the conclusion, but they are load-bearing external inputs.

assumptions (7)
  • domain assumption Almost diagonal operator boundedness on the sequence spaces a^{s,υ}_{p,q}(W) (Theorem 3.6, imported from [13, Theorem 5.6]).
    Used repeatedly, for example in Lemma 3.43 and in the molecular characterizations, and not proved in this paper.
  • domain assumption Molecular and wavelet characterizations of A^{s,υ}_{p,q}(W) (Lemmas 3.16 and 3.23 here, from [13, Theorems 5.17 and 5.20]).
    These are the main tools; every main theorem relies on them to transfer sequence bounds to function-space bounds.
  • domain assumption Sharp estimate for reducing operators generated by matrix A_{p,∞} weights (Lemma 3.4, from [11, Lemma 6.8(i)]).
    Used in the trace and extension proofs, for example in (3.27) and (3.36), to control the change of reducing operators across cubes.
  • domain assumption Existence of reducing operators for matrix weights for all p in (0,∞).
    Invoked in Definition 3.2 and in the proofs via sequences A_Q; the existence is cited from [33, Proposition 1.2] and [26, p. 1237].
  • standard math Standard Littlewood-Paley and Calderon reproducing formula machinery, including Lemmas 3.18 and 3.19.
    Classical background needed to define the spaces and to pass between function and sequence spaces.
  • standard math Existence of Daubechies wavelets with arbitrary finite smoothness and vanishing moments (Lemma 3.21 and (3.13)).
    This is the basis of the wavelet characterization used in the trace and extension theorems.
  • domain assumption The matrix A_{p,∞} class and the Ap,∞-dimension bounds d_lower and d_upper are well defined and finite (Definitions 2.1 and 3.3, citing [11]).
    These dimensions enter the index formulas in Theorem 3.6 and hence all later theorems.

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Pith. "Pith review of Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calder\'{o}n--Zygmund Operators." pith.science (2026). https://pith.science/paper/RB2NA2ME

@misc{pith2026250419060,
  author       = {Pith},
  title        = {Pith review of: Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calder\'on--Zygmund Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RB2NA2ME}},
  note         = {Machine review of arXiv:2504.19060}
}
abstract

This article is a continuation of our work on generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. In this article, we establish the boundedness of pseudo-differential, trace, and Calder\'{o}n--Zygmund operators on these spaces. The main tools involved in this article are the molecular and the wavelet characterizations of these spaces. Since generalized matrix-weighted Besov--Triebel--Lizorkin-type spaces include many classical function spaces such as matrix-weighted Besov--Triebel--Lizorkin spaces, all the results in this article are of wide generality.

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Forward citations

Cited by 2 Pith papers

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Works this paper leans on

76 extracted references · 42 canonical work pages · cited by 2 Pith papers

  1. [13]

    F. Bu, D. Y ang, W. Y uan and M. Zhang, Matrix-weighted Besov–Triebel–Lizorkin spaces of optimal scale: Real-variable characterizations, invaria nce on integrable index, and Sobolev- type embedding, Submitted or arXiv: 2505.02136

    arXiv

  2. [9]

    F. Bu, T. Hyt¨ onen, D. Y ang and W. Y uan, Matrix-weighted Besov-type and Triebel–Lizorkin- type spaces II: Sharp boundedness of almost diagonal operat ors, J. Lond. Math. Soc. (2) 111 (2025), Paper No. e70094, 59 pp

    work page 2025

  3. [7]

    F. Bu, T. Hyt¨ onen, D. Y ang and W. Y uan, Matrix-weighted Besov-type and Triebel–Lizorkin- type spaces III: Characterizations of molecules and wavele ts, trace theorems, and bound- edness of pseudo-di fferential operators and Calder´ on–Zygmund operators, Math . Z. 308 (2024), Paper No. 32, 67 pp

    work page 2024

  4. [1]

    Bai and J

    T. Bai and J. Xu, Pseudo-di fferential operators on matrix weighted Besov–Triebel–Lizo rkin spaces, Bull. Iranian Math. Soc. 50 (2024), Paper No. 31, 26 p p

    work page 2024

  5. [2]

    Bai and J

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

    work page 2024

  6. [3]

    Bai and J

    T. Bai and J. Xu, Precompactness in matrix weighted Bourg ain–Morrey spaces, Filomat (to appear) or arXiv: 2406.11531

    arXiv

  7. [4]

    Bownik, Atomic and molecular decompositions of aniso tropic Besov spaces, Math

    M. Bownik, Atomic and molecular decompositions of aniso tropic Besov spaces, Math. Z. 250 (2005), 539–571

    work page 2005

  8. [5]

    Bownik, Anisotropic Triebel–Lizorkin spaces with do ubling measures, J

    M. Bownik, Anisotropic Triebel–Lizorkin spaces with do ubling measures, J. Geom. Anal. 17 (2007), 387–424

    work page 2007

Show all 76 references
  1. [6]

    Bownik and K.-P

    M. Bownik and K.-P . Ho, Atomic and molecular decompositi ons of anisotropic Triebel– Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), 1469–1 510

    2006

  2. [8]

    F. Bu, T. Hyt¨ onen, D. Y ang and W. Y uan, Matrix-weighted Besov-type and Triebel–Lizorkin- type spaces I: Ap-dimensions of matrix weights and ϕ-transform characterizations, Math. Ann. 391 (2025), 6105–6185

    2025

  3. [10]

    F. Bu, T. Hyt¨ onen, D. Y ang and W. Y uan, Besov–Triebel–L izorkin-type spaces with matrix A∞ weights, Sci. China Math. (2025), https: //doi.org/10.1007/s11425-024-2385-x

    2025 doi

  4. [11]

    F. Bu, T. Hyt¨ onen, D. Y ang and W. Y uan, New characterizations and properties of matrix A∞ weights, arXiv: 2311.05974

    arXiv

  5. [12]

    F. Bu, D. Y ang and W. Y uan, Real-variable characterizations and their applications of matrix- weighted Besov spaces on spaces of homogeneous type, Math. Z . 305 (2023), 81 pp

    2023

  6. [14]

    F. Bu, D. Y ang, W. Y uan and Y . Zhao, Matrix weights and matrix-weighted Besov–Triebel– Lizorkin-type spaces — a survey, Submitted

  7. [15]

    Cleanthous, A

    G. Cleanthous, A. G. Georgiadis and M. Nielsen, Molecul ar decomposition of anisotropic homogeneous mixed-norm spaces with applications to the bou ndedness of operators, Appl. Comput. Harmon. Anal. 47 (2019), 447–480

    2019

  8. [16]

    Dafni and J

    G. Dafni and J. Xiao, Some new tent spaces and duality the orems for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004), 377–422

    2004

  9. [17]

    Daubechies, Orthonormal bases of compactly support ed wavelets, Comm

    I. Daubechies, Orthonormal bases of compactly support ed wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996

    1988

  10. [18]

    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regi onal Conference Series in Ap- plied Mathematics 61, Society for Industrial and Applied Ma thematics (SIAM), Philadel- phia, PA, 1992

    1992

  11. [19]

    David and J.-L

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

    1984

  12. [20]

    Di Plinio, T

    F. Di Plinio, T. Hyt¨ onen and K. Li, Sparse bounds for max imal rough singular integrals via the Fourier transform, Ann. Inst. Fourier (Grenoble) 70 (20 20), 1871–1902

    1902

  13. [21]

    Fe fferman and B

    C. Fe fferman and B. Muckenhoupt, Two nonequivalent conditions forweight functions, Proc. Amer. Math. Soc. 45 (1974), 99–104. 36 Dachun Yang, Wen Yuan and Mingdong Zhang

    1974

  14. [22]

    G. B. Folland, Real Analysis, Modern Techniques and The ir Applications, Second edition, Pure and Applied Mathematics, John Wiley & Sons, Inc., New Y o rk, 1999

    1999

  15. [23]

    Frazier, Decomposition and traces of weighted mixed -norm Besov spaces, Matematica 3 (2024), 1320–1359

    M. Frazier, Decomposition and traces of weighted mixed -norm Besov spaces, Matematica 3 (2024), 1320–1359

    2024

  16. [24]

    Frazier and B

    M. Frazier and B. Jawerth, A discrete transform and deco mpositions of distribution spaces, J. Funct. Anal. 93 (1990), 34–170

    1990

  17. [25]

    Frazier, B

    M. Frazier, B. Jawerth and G. Weiss, Littlewood–Paley T heory and The Study of Function Spaces, CBMS Regional Conference Series in Mathematics 79, Published for the Confer- ence Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providenc...

    1991

  18. [26]

    Frazier and S

    M. Frazier and S. Roudenko, Matrix-weighted Besov spac es and conditions of Ap type for 0< p ≤ 1, Indiana Univ. Math. J. 53 (2004), 1225–1254

    2004

  19. [27]

    Frazier and S

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

    2008

  20. [28]

    Frazier and S

    M. Frazier and S. Roudenko, Littlewood–Paley theory fo r matrix-weighted function spaces, Math. Ann. 380 (2021), 487–537

    2021

  21. [29]

    Frazier, R

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

    1988

  22. [30]

    A. G. Georgiadis, J. Johnsen and M. Nielsen, Wavelet tra nsforms for homogeneous mixed- norm Triebel–Lizorkin spaces, Monatsh. Math. 183 (2017), 5 87–624

    2017

  23. [31]

    A. G. Georgiadis, G. Kyriazis and P . Petrushev, Product Besov and Triebel–Lizorkin spaces with application to nonlinear approximation, Constr. Appr ox. 53 (2021), 39–83

    2021

  24. [32]

    A. G. Georgiadis and M. Nielsen, Pseudodi fferential operators on mixed-norm Besov and Triebel–Lizorkin spaces, Math. Nachr. 289 (2016), 2019–20 36

    2016

  25. [33]

    Goldberg, Matrix Ap weights via maximal functions, Pacific J

    M. Goldberg, Matrix Ap weights via maximal functions, Pacific J. Math. 211 (2003), 2 01– 220

    2003

  26. [34]

    Grafakos, Classical Fourier Analysis, Third editio n, Graduate Texts in Mathematics 249, Springer, New Y ork, 2014

    L. Grafakos, Classical Fourier Analysis, Third editio n, Graduate Texts in Mathematics 249, Springer, New Y ork, 2014

    2014

  27. [35]

    Grafakos, Modern Fourier Analysis, Third edition, G raduate Texts in Mathematics 250, Springer, New Y ork, 2014

    L. Grafakos, Modern Fourier Analysis, Third edition, G raduate Texts in Mathematics 250, Springer, New Y ork, 2014

    2014

  28. [36]

    Grafakos and R

    L. Grafakos and R. H. Torres, Pseudodi fferential operators with homogeneous symbols, Michigan Math. J. 46 (1999), 261–269

    1999

  29. [37]

    R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, C orrected reprint of the 1991 original, Cambridge University Press, Cambridge, 1994

    1991

  30. [38]

    R. A. Horn and C. R. Johnson, Matrix Analysis, Second edi tion, Cambridge University Press, Cambridge, 2013

    2013

  31. [39]

    D. D. Haroske, S. D. Moura and L. Skrzypczak, Wavelet dec omposition and embeddings of generalised Besov–Morrey spaces, Nonlinear Anal. 214 (202 2), Paper No. 112590, 26 pp

  32. [40]

    Haroske and P

    D.D. Haroske and P . Skandera, Embeddings of doubling we ighted Besov spaces, Function spaces X, 105–119, Banach Center Publ., 102, Polish Acad. Sc i. Inst. Math., Warsaw, 2014

    2014

  33. [41]

    D. D. Haroske and L. Skrzypczak, Continuous embeddings of Besov–Morrey function spaces, Acta Math. Sin. (Engl. Ser.) 28 (2012), 1307–1328

    2012

  34. [42]

    D. D. Haroske and L. Skrzypczak, Embeddings of Besov–Mo rrey spaces on bounded do- mains, Studia Math. 218 (2013), 119–144

    2013

  35. [43]

    Jawerth, Some observations on Besov and Lizorkin–Tr iebel spaces, Math

    B. Jawerth, Some observations on Besov and Lizorkin–Tr iebel spaces, Math. Scand. 40 (1977), 94–104

    1977

  36. [44]

    Kozono and M

    H. Kozono and M. Y amazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014. Ma trix-Weighted Besov–T riebel–L izorkin Spaces of Optimal Scale 37

    1994

  37. [45]

    Lerner, K

    A. Lerner, K. Li, S. Ombrosi and I. Rivera-R´ ıos, On the s harpness of some quantitative Muckenhoupt–Wheeden inequalities, C. R. Math. Acad. Sci. P aris 362 (2024), 1253–1261

    2024

  38. [46]

    Lerner, K

    A. Lerner, K. Li, S. Ombrosi and I. Rivera-R´ ıos, On some improved weighted weak type inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2024) , https: //doi.org/10.2422/2036- 2145.202407 012

    2024

  39. [47]

    P . Li, J. Xiao and Q. Y ang, Global mild solutions to modifi ed Navier–Stokes equations with small initial data in critical Besov- Q spaces, Electron. J. Di fferential Equations 2014, No. 185, 37 pp

    2014

  40. [48]

    Li and Q

    P . Li and Q. Y ang, Wavelets and the well-posedness of inc ompressible magneto-hydro- dynamic equations in Besov type Q-space, J. Math. Anal. Appl. 405 (2013), 661–686

    2013

  41. [49]

    Z. Li, D. Y ang and W. Y uan, Matrix-weighted Besov–Triebel–Lizorkin spaces with logarith- mic smoothness, Bull. Sci. Math. 193 (2024), Paper No. 10344 5, 54 pp

    2024

  42. [50]

    Z. Li, D. Y ang and W. Y uan, Matrix-weighted Poincar´ e-type inequalities with applications to logarithmic Hajłasz–Besov spaces on spaces of homogeneo us type, Submitted

  43. [51]

    A. L. Mazzucato, Besov–Morrey spaces: Function space t heory and applications to non- linear PDE, Trans. Amer. Math. Soc. 355 (2003), 1297–1364

    2003

  44. [52]

    Meyer, Wavelets and Operators, Translated from the 1 990 French original by D

    Y . Meyer, Wavelets and Operators, Translated from the 1 990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics 37, Ca mbridge University Press, Cambridge, 1992

    1992

  45. [53]

    F. L. Nazarov and S. R. Tre˘ ıl, The hunt for a Bellman func tion: Applications to estimates for singular integral operators and to other classical problem s of harmonic analysis, (Russian) Algebra i Analiz 8(5) (1996), 32–162; translation in St. Pet ersburg Math. J. 8 (1997),...

    1996

  46. [54]

    Nielsen, Bandlimited multipliers on matrix-weight ed Lp-spaces, J

    M. Nielsen, Bandlimited multipliers on matrix-weight ed Lp-spaces, J. Fourier Anal. Appl. 31 (2025), Paper No. 3, 10 pp

    2025

  47. [55]

    Roudenko, Matrix-weighted Besov spaces, Trans

    S. Roudenko, Matrix-weighted Besov spaces, Trans. Ame r. Math. Soc. 355 (2003), 273–314

    2003

  48. [56]

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

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

    2004

  49. [57]

    Sawano, D

    Y . Sawano, D. Y ang and W. Y uan, New applications of Besov -type and Triebel–Lizorkin- type spaces, J. Math. Anal. Appl. 363 (2010), 73–85

    2010

  50. [58]

    J. Sun, D. Y ang and W. Y uan, A framework of Besov–Triebel –Lizorkin type spaces via ball quasi-Banach function sequence spaces I: Real-variable ch aracterizations, Math. Ann. 390 (2024), 4283–4360

    2024

  51. [59]

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

    1991

  52. [60]

    Treil and A

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

    1997

  53. [61]

    Triebel, Theory of Function Spaces, Monographs in Ma thematics 78, Birkh¨ auser, Basel, 1983

    H. Triebel, Theory of Function Spaces, Monographs in Ma thematics 78, Birkh¨ auser, Basel, 1983

    1983

  54. [62]

    Triebel, Theory of Function Spaces

    H. Triebel, Theory of Function Spaces. II, Monographs i n Mathematics 84, Birkh¨ auser, Basel, 1992

    1992

  55. [63]

    Triebel, Theory of Function Spaces

    H. Triebel, Theory of Function Spaces. III, Monographs in Mathematics 100, Birkh¨ auser V erlag, Basel, 2006

    2006

  56. [64]

    V olberg, Matrix Ap weights via S -functions, J

    A. V olberg, Matrix Ap weights via S -functions, J. Amer. Math. Soc. 10 (1997), 445–466

    1997

  57. [65]

    S. Wang, P . Guo and J. Xu, Embedding and duality of matrix -weighted modulation spaces, Taiwanese J. Math. 29 (2025), 171–187

    2025

  58. [66]

    S. Wang, P . Guo and J. Xu, Precompact sets in matrix weigh ted Lebesgue spaces with vari- able exponent, Georgian Math. J. (2025), https: //doi.org/10.1515/gmj-2025-2029

    2025 doi

  59. [67]

    Q. Wang, D. Y ang and Y . Zhang, Real-variable characteri zations and their applications of matrix-weighted Triebel–Lizorkin spaces, J. Math. Anal . Appl. 529 (2024), Paper No. 127629, 37 pp. 38 Dachun Yang, Wen Yuan and Mingdong Zhang

    2024

  60. [68]

    Wiener and P

    N. Wiener and P . Masani, The prediction theory of multiv ariate stochastic processes. II. The linear predictor, Acta Math. 99 (1958), 93–137

    1958

  61. [69]

    Y ang and W

    D. Y ang and W. Y uan, A new class of function spaces connec ting Triebel–Lizorkin spaces and Q spaces, J. Funct. Anal. 255 (2008), 2760–2809

    2008

  62. [70]

    Y ang and W

    D. Y ang and W. Y uan, New Besov-type spaces and Triebel–L izorkin-type spaces including Q spaces, Math. Z. 265 (2010), 451–480

    2010

  63. [71]

    Y ang and W

    D. Y ang and W. Y uan, Relations among Besov-type spaces, Triebel–Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. 92 (20 13), 549–561

  64. [72]

    Y ang, W

    D. Y ang, W. Y uan and C. Zhuo, Musielak–Orlicz Besov-typ e and Triebel–Lizorkin-type spaces, Rev. Mat. Complut. 27 (2014), 93–157

    2014

  65. [73]

    Y uan, D

    W. Y uan, D. D. Haroske, S. D. Moura, L. Skrzypczak and D. Y ang, Limiting embeddings in smoothness Morrey spaces, continuity envelopes and appl ications, J. Approx. Theory 192 (2015), 306–335

    2015

  66. [74]

    Y uan, D

    W. Y uan, D. D. Haroske, L. Skrzypczak and D. Y ang, Embedd ing properties of Besov-type spaces, Appl. Anal. 94 (2015), 319–341

    2015

  67. [75]

    Y uan, D

    W. Y uan, D. D. Haroske, L. Skrzypczak and D. Y ang, Embedd ing properties of weighted Besov-type spaces, Anal. Appl. (Singap.) 13 (2015), 507–55 3

    2015

  68. [76]

    Y uan, W

    W. Y uan, W. Sickel and D. Y ang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-V erlag, Berlin, 2010. Dachun Y ang (Corresponding author), Wen Y uan and Mingdong Zhang. Laboratory of Mathematics and Complex Systems (Ministr...

    2005

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