Pith. sign in

REVIEW 3 major objections 5 minor 85 references

Characterization of Vanishing Campanato Spaces via Ball Banach Function Spaces and Its Applications

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Vanishing Campanato spaces are unchanged when oscillation is measured in any admissible ball Banach function space.

desk verdict The main theorem is attractive, but the α=1 proof has a genuine gap (k=0 cancellation fails), so the paper needs repair before I'd trust it. read the letter →

arxiv 2508.08536 v1 pith:WGZTG4EU submitted 2025-08-12 math.FA math.APmath.CA

Xing Fu , Yoshihiro Sawano , Jin Tao , Dachun Yang This is my paper · ORCID
classification math.FAmath.APmath.CA MSC 42B3542B2546E30
keywords vanishingCampanatospacesballBanachfunctionmeanoscillationself-improvementsecond-orderdifferencesconvolutionsmoothingcommutatorcompactnessMorrey
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 seeks to prove that the vanishing Campanato spaces—the small-oscillation subspaces of BMO-type function spaces—can be defined using the norm of any ball Banach function space X and still give exactly the same space, provided the Hardy–Littlewood maximal operator is bounded on the associate space X′. The main theorem states that for every α∈[0,∞) and every such X, the classical vanishing spaces YLα and the X-based versions YLα,X coincide for Y = V, X, and C. If true, this is a self-improvement phenomenon: the integrability used to measure oscillation is irrelevant to which functions vanish. The paper also obtains new characterizations of these spaces by pointwise differences and by second-order differences, and shows how the spaces arise in the compactness of fractional integral commutators.

What carries the argument

The load-bearing object is the X-based oscillation Oα,X(f;Q) = |Q|^{-α/n} ||(f - $P_Q^{{(⌊α⌋)}}$ f)1_Q||_X / ||1_Q||_X, which replaces the classical L1-average deviation in the definition of Campanato spaces. The argument runs on two estimates: Oα,X is bounded above by the Lipschitz-type ratio for α∈[0,1), and O1,X is bounded above by the second-order difference sup_{0<|y|≤r} |Δ²_y f(x)|/|y|. The second estimate is obtained by smoothing f through convolution with an even mollifier, expanding in a Taylor polynomial, and controlling the remainder by the second-order difference; the same smoothing underlies the difference characterizations in Proposition 3.6. These bounds convert vanishing of the classical oscillation into vanishing of the X-oscillation.

What would settle it

Take f(x)=x_1 on R^n and any mollifier φ with ∫φ=1; then the Campanato oscillation O1(f;Q) vanishes for every cube, while u0(x,t)=(f∗φ_t)(x)=x_1 does not go to zero even as t→0. Thus the bound |u0(x,t)|+|u1(x,t)| ≲ t O1(f;B(x,t)) used to prove Proposition 3.6(i) fails, and with it the α=1 case of Theorem 1.6, unless a different estimate is supplied.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.6, is that the three vanishing Campanato subspaces—V, X, and C variants—are independent of the ball Banach function space used to measure oscillation. Concretely, if X is a ball Banach function space and the Hardy–Littlewood maximal operator M is bounded on its associate space X′, then YLα = YLα,X with equivalent norms for every α∈[0,∞) and Y∈{V,X,C}. The proof is organized by the smoothness parameter: α=0 uses classical closures of uniformly continuous, smooth, and compactly supported functions; α∈(0,1) uses pointwise Hölder-type differences; α=1 uses second-order differences dominated through convolution smoothing; α>1 is asserted to follow by adapting the α=1 argument. The paper claims the result is new even for α=0 and sharp in that negative α fails.

Load-bearing premise

The proof of the α=1 case assumes that convolving a function with a small mollifier produces an error dominated by the oscillation, but the cancellation that makes this true fails for the lowest-order term, so that estimate is not justified as written.

Editorial extensions

If this is right

  • For every parameter range where M is bounded on X′, the vanishing spaces VLα, XLα, CLα equal their X-based counterparts, so results such as compactness of commutators can be transferred to weighted, variable, mixed-norm, Morrey, Lorentz, and Herz spaces.
  • New characterizations of VMO, XMO, and CMO arise: at α=0 they are recovered from density by uniformly continuous, smooth, and compactly supported functions; at α∈(0,1) they are described by vanishing pointwise Hölder-type ratios.
  • At α=1 and above, vanishing is characterized by the vanishing of second-order differences |Δ²_y f(x)|/|y|, giving a higher-order difference description of the Lipschitz-based vanishing spaces.
  • Compactness of fractional integral commutators [b,Iα] from Morrey spaces into CMO, CMOγ, or mixed-norm Lebesgue spaces follows from the vanishing-space framework.
  • The admissible range α∈[0,∞) is sharp: for negative α the equality fails because Morrey spaces do not self-improve.

Reading between the lines

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

  • If the theorem is right, the vanishing condition is a property of the function itself, not of the norm used to measure oscillation; analogous norm-independence should hold for other vanishing subspaces defined by difference or derivative conditions.
  • The convolution-smoothing mechanism that turns second-order differences into oscillation bounds is a transferable template; it could characterize vanishing conditions in Sobolev-type or BV-type spaces defined through second differences.
  • A testable extension is whether the α=1 step can be repaired by subtracting the first-order polynomial explicitly in the lowest-order estimate; if not, the equality at α=1 may require an extra hypothesis on X.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper defines X-based Campanato seminorms and spaces Lα,X, VLα,X, XLα,X, and CLα,X, where X is a ball Banach function space, and claims Theorem 1.6: for every α∈[0,∞), if the Hardy–Littlewood maximal operator M is bounded on the associate space X′, then YLα = YLα,X for Y∈{V,X,C}. The proof is split into α=0, α∈(0,1), and α=1, with the range α>1 deferred to an unspecified adaptation of the argument in [26]. For α∈(0,1), the proof uses new characterizations of VMOα, XMOα, and CMOα in terms of pointwise difference quotients (Proposition 3.4). For α=1, the proof uses a new characterization in terms of second-order differences (Proposition 3.6), obtained via a convolution smoothing method. The final section derives applications to weighted, variable, mixed-norm, Morrey, grand Besov–Bourgain–Morrey, Lorentz, and Herz spaces, and to compactness of fractional integral commutators in Morrey spaces.

Significance. If Theorem 1.6 were correct, it would provide a unified self-improvement and characterization result for vanishing Campanato spaces in the general framework of ball Banach function spaces, and the applications to compact commutators in Morrey spaces would be of genuine interest. The α∈(0,1) part, based on Proposition 3.4, appears coherent, and the α=0 density arguments are plausible. However, the proof of the α=1 case contains a false estimate in the central convolution argument, and the full range α∈[1,∞) claimed in Theorem 1.6 is not proved. Thus the main theorem is not established as written.

major comments (3)
  1. [§3, Proposition 3.6(i), Eq. (3.15) and (3.18)] The key estimate (3.15) is invoked with k=0 to obtain (3.18). For k=0, the function at defined in the proof equals t^{−1}φ_t, whose integral is t^{−1}∫φ = t^{−1} ≠ 0; the cancellation ∫at = ∫xj at = 0 stated from [26, Lemma 5.20] does not hold for k=0. Consequently the representation of ∂t^0 u0 = u0 as a convolution with a zero-mean kernel is not valid. The asserted bound |u0(x,t)| + |u1(x,t)| ≲ t O1(f;B(x,t)) in (3.18) is in fact false: for affine f, O1(f;B(x,t)) = 0 for every ball while u0(x,t) = f(x), so (3.18) would imply |f(x)| ≲ 0. Since (3.18) is the only control used to bound Δh^2 u0 in (3.19) and hence to reach (3.20), the “only if” direction of Proposition 3.6(i) is not established. As Proposition 3.6(i) feeds directly into the α=1 case of Theorem 1.6, and parts (ii) and (iii) of Proposition 3.6 also reuse (3.18), the main theorem is not proved as written.
  2. [Theorem 1.6 and Remark 1.7(iv)] Theorem 1.6 is stated for α∈[0,∞), but the proof in Section 4.1 treats only α=0, α∈(0,1), and α=1. For α∈(1,∞), Remark 1.7(iv) asserts without proof that the α=1 argument can be adapted using [26, pp. 300–302], and no details are supplied. Since the α=1 argument itself contains the false estimate discussed above, the claimed extension to α>1 is especially unsupported. The authors should either provide a complete proof for α>1 or explicitly restrict the theorem to α∈[0,1]; as it stands, the main theorem's stated range is not established.
  3. [§3, Proposition 3.6(ii)–(iii)] The “only if” directions of Proposition 3.6(ii) and (iii) both rely on the same invalid estimate (3.18): in part (ii) the bound is combined with (3.21), and in part (iii) it is combined with (3.25) to control u0 and u1. Because (3.18) is false for k=0, the claimed characterizations of XL1 and CL1 by second-order differences are not proved. These characterizations are needed in the proof of Theorem 1.6 for α=1, so the defect is load-bearing for the whole α=1 case.
minor comments (5)
  1. [§2, proof of Lemma 2.6] In the proof of (2.1), the word “deifinition” should be “definition.”
  2. [§3, proof of Proposition 3.6, parts (ii) and (iii)] In the displayed estimates for u2, the notation u(x,s) is used where u0(x,s) is intended; this makes the displayed formulas inconsistent with the definition of u2.
  3. [§3, Proposition 3.4] The opening sentence says the equivalences “hold almost everywhere,” but the statement is about equality of subspaces of function spaces, not pointwise equivalences; this wording should be corrected.
  4. [Appendix A.2, proof of Proposition A.4] In the proof of part (i), the sentence “To show (ii), we claim” should read “To show (i), we claim,” and the references to “Proposition A.2(ii)” and “Proposition A.2(iii)” should refer to “Lemma A.2(ii)” and “Lemma A.2(iii).”
  5. [§4.1, proof of Theorem 1.6, case α=0] The expressions “sup_{|Q|≤a}” should be “sup_{Q: ℓ(Q)≤a}” for consistency with the definitions; as written, |Q| is not the natural parameter for cube size in the definitions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 1.6 is derived from independent density and difference characterizations; the cited full-space equivalence is a benchmark, not a disguised version of the vanishing result.

full rationale

The derivation of Theorem 1.6 is self-contained in the sense relevant to circularity. The vanishing inclusions are not assumed; they are established through classical density results for VMO/XMO/CMO (Lemma 3.2), pointwise difference characterizations for alpha in (0,1) (Lemma 3.3 and Proposition 3.4), and second-order difference characterizations for alpha = 1 (Lemma 2.7 and Proposition 3.6). The X-norm enters only through the two-sided oscillation comparisons O_alpha <= O_alpha,X <= C sup-difference terms and through the hypothesis that M is bounded on X'. The cited full-space equivalence L_alpha = L_alpha,X from [43] is used to transfer density approximations into the X-norm for alpha = 0, but it does not contain the vanishing statements and is therefore a normal external citation rather than a load-bearing self-citation. Many other ball-Banach-space references are by the same group, but they serve as boundedness criteria for M on the associate space or as background benchmarks; none of them already implies Theorem 1.6. The paper does contain a genuine proof gap: in Proposition 3.6(i), estimate (3.18) is obtained from (3.15) with k = 0, but the cancellation from [26, Lemma 5.20] fails for k = 0 because a_t = t^{-1} phi_t has nonzero integral, so the proof of the alpha = 1 case is not complete as written and alpha > 1 is only delegated to [26]. This is a correctness defect, not a circular reduction, and it does not affect the circularity score. Accordingly, no circular step is identified and the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted and no new mathematical entities are introduced. The central assumptions are the boundedness of the Hardy-Littlewood maximal operator on the associate space and several prior characterizations (density of smooth functions, Lα=Lα,X). The main unproved load-bearing items are the extension to α>1 and the k=0 convolution estimate, both of which are gaps in the argument rather than independent axioms.

assumptions (5)
  • domain assumption X is a ball Banach function space and M is bounded on its associate space X'.
    This is the standing hypothesis of Theorem 1.6 and controls Lemmas 2.4-2.7 and Proposition 3.1.
  • standard math Classical density characterizations: VMO is the BMO-closure of uniformly continuous functions, XMO of B∞, and CMO of Cc∞.
    Quoted from Sarason, Tao et al., and Uchiyama (Lemma 3.2) and used for the α=0 case.
  • standard math Known equivalence Lα = Lα,X with equivalent norms.
    Proposition 3.1 is cited from [43] and [26] and used as the starting point for the vanishing subspaces.
  • ad hoc to paper The case α∈(1,∞) can be obtained by adapting [26, pp.300-302].
    Asserted in Remark 1.7(iv) without proof; extends Theorem 1.6 beyond the demonstrated range.
  • ad hoc to paper Convolution estimate (3.15) is valid for k=0.
    Used to control u0 in (3.18); the stated cancellation fails for k=0, so the estimate is unsupported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Characterization of Vanishing Campanato Spaces via Ball Banach Function Spaces and Its Applications." pith.science (2026). https://pith.science/paper/WGZTG4EU

@misc{pith2026250808536,
  author       = {Pith},
  title        = {Pith review of: Characterization of Vanishing Campanato Spaces via Ball Banach Function Spaces and Its Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGZTG4EU}},
  note         = {Machine review of arXiv:2508.08536}
}
read the original abstract

In this article, the authors provide some new characterizations of several vanishing Campanato spaces using a type of oscillation defined within the general framework of ball Banach function spaces. This approach yields fresh insights even in the special case of the vanishing BMO space. The characterization reveals a self-improvement phenomenon inherent in vanishing Campanato spaces. A key innovation of this approach lies in using higher-order differences to dominate oscillations. Instead of directly estimating these differences, the authors achieve the domination by smoothing the function via convolution. As additional outcomes, the authors also obtain new characterizations of vanishing Campanato spaces in terms of higher-order differences. Finally, the authors present several examples to show that these vanishing Campanato spaces naturally arise in the study on the compactness of fractional integral commutators in Morrey spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 62 canonical work pages

  1. [26]

    Garc ´ıa-Cuerva and J

    J. Garc ´ıa-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies 116, North-Holland Publishing Co., Amsterdam, 1985

    work page 1985

  2. [1]

    D. R. Adams, Morrey Spaces, Lecture Notes in Applied and Numerical Harmonic Analysis, Birkh¨auser/Springer, Cham, 2015

    2015

  3. [2]

    Alabalik, A

    A. Alabalik, A. Almeida and S. Samko, Preservation of certain vanishing properties of gen- eralized Morrey spaces by some classical operators, Math. Methods Appl. Sci. 43 (2020), 9375–9386

    2020

  4. [3]

    Alabalik, A

    A. Alabalik, A. Almeida and S. Samko, On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators, Banach J. Math. Anal. 14 (2020), 987–1000

    2020

  5. [4]

    Almeida, Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

    A. Almeida, Maximal commutators and commutators of potential operators in new vanishing Morrey spaces. Nonlinear Anal. 192 (2020), Article ID 111684, 12 pp

    2020

  6. [5]

    Almeida and S

    A. Almeida and S. Samko, Approximation in generalized Morrey spaces, Georgian Math. J. 25 (2018), 155–168

    2018

  7. [6]

    Almeida and S

    A. Almeida and S. Samko, Approximation in Morrey spaces, J. Funct. Anal. 272 (2017), 2392–2411

    work page 2017

  8. [7]

    B ´egout and A

    P. B ´egout and A. Vargas, Mass concentration phenomena for the L2-critical nonlinear Schr¨odinger equation, Trans. Amer. Math. Soc. 359 (2007), 5257–5282

    work page 2007

Show all 85 references
  1. [8]

    Benedek and R

    A. Benedek and R. Panzone, The space Lp, with mixed norm, Duke Math. J. 28 (1961), 301–324

    1961

  2. [9]

    Bennett and R

    C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, MA, 1988

    1988

  3. [10]

    Berkovits, J

    L. Berkovits, J. Kinnunen and J. M. Martell, Oscillation estimates, self-improving results and good-λ inequalities, J. Funct. Anal. 270 (2016), 3559–3590. 34 Xing Fu, Yoshihiro Saw ano, Jin Tao and Dachun Yang

    2016

  4. [11]

    Bonami, L

    A. Bonami, L. Liu, D. Yang and W. Yuan, Pointwise multipliers of Zygmund classes on Rn, J. Geom. Anal. 31 (2021), 8879–8902

    2021

  5. [12]

    Bourgain, On the restriction and multiplier problems in R3, in: Geometric Aspects of Functional Analysis (1989–90), pp

    J. Bourgain, On the restriction and multiplier problems in R3, in: Geometric Aspects of Functional Analysis (1989–90), pp. 179–191, Lecture Notes in Mathematics 1469, Springer, Berlin, 1991

    1989

  6. [13]

    Campanato, Propriet `a di una famiglia di spazi funzionali, Ann

    S. Campanato, Propriet `a di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 137–160

    1964

  7. [14]

    Cheung and K.-P

    K. Cheung and K.-P. Ho, Boundedness of Hardy–Littlewood maximal operator on block spaces with variable exponent, Czechoslovak Math. J. 64(139) (2014), 159–171

    2014

  8. [15]

    Chiarenza and M

    F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), 273–279

    1987

  9. [16]

    Cruz-Uribe, Extrapolation and Factorization, arXiv:1706.02620

    D. Cruz-Uribe, Extrapolation and Factorization, arXiv:1706.02620

    arXiv

  10. [17]

    D. V . Cruz-Uribe and A. Fiorenza, Variable Lebesgue Space. Foundations and Harmonic Analysis, Appl. Number. Harmon Aanl., Birkh¨auser/Springer, Heidelberg, 2013

    2013

  11. [18]

    Cruz-Uribe and J

    D. Cruz-Uribe and J. M. Martell, Limited range multilinear extrapolation with applications to the bilinear Hilbert transform, Math. Ann. 371 (2018), 615–653

    2018

  12. [19]

    Cruz-Uribe and L.-A

    D. Cruz-Uribe and L.-A. D. Wang, Extrapolation and weighted norm inequalities in the vari- able Lebesgue spaces, Trans. Amer. Math. Soc. 369 (2017), 1205–1235

    2017

  13. [20]

    F. Dai, X. Lin, D. Yang, W. Yuan and Y . Zhang, Brezis–Van Schaftingen–Yung formulae in ball Banach function spaces with applications to fractional Sobolev and Gagliardo–Nirenberg inequalities, Calc. Var. Partial Differ. Equ. 62 (2023), Paper No. 56, 73 pp

    2023

  14. [21]

    F. Dai, L. Grafakos, Z. Pan, D. Yang, W. Yuan and Y . Zhang, The Bourgain–Brezis– Mironescu formula on ball Banach function spaces, Math. Ann. 388 (2024), 1691–1768

    2024

  15. [22]

    Di Fazio and T

    G. Di Fazio and T. Nguyen, Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations, Rev. Mat. Iberoam. 36 (2020), 1627–1658

    2020

  16. [23]

    Diening, P

    L. Diening, P. Harjulehto, P. H ¨ast¨o and M. R ˚u˘zi˘cka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011

    2017

  17. [24]

    Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, vol

    J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001

    2001

  18. [25]

    H. G. Feichtinger, J. Sun, D. Yang and Wen Yuan, A framework of Besov–Triebel–Lizorkin type spaces via ball quasi-Banach function sequence spaces II: Applications to specific func- tion spaces, Anal. Appl. (Singap.) (2025), https://doi.org/10.1142/S0219530525500228

    2025 doi

  19. [27]

    Greco, T

    L. Greco, T. Iwaniec and C. Sbordone, Inverting thep-harmonic operator, Manuscripta Math. 92 (1997), 249–258

    1997

  20. [28]

    W. Guo, J. He, H. Wu and D. Yang, Boundedness and compactness of commutators associ- ated with Lipschitz functions, Anal. Appl. (Singap.) 20 (2022), 35–71

    2022

  21. [29]

    D. I. Hakim and Y . Sawano, Complex interpolation of variable Morrey spaces, Math. Nachr. 294 (2021), 2140–2150

    2021

  22. [30]

    D. I. Hakim, Y . Sawano and D. Takesako, Compactness of commutators in Morrey spaces, Anal. Math. Phys. 15 (2025), Paper No. 83, 9 pp

    2025

  23. [31]

    Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier trans- forms, J

    C. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier trans- forms, J. Math. Mech. 18 (1968), 283–324

    1968

  24. [32]

    Ho, Rough singular integral operators, spherical maximal functions and maximal Bochner–Riesz operators on grand Morrey spaces, Complex Anal

    K.-P. Ho, Rough singular integral operators, spherical maximal functions and maximal Bochner–Riesz operators on grand Morrey spaces, Complex Anal. Oper. Theory 18 (2024), Paper No. 147, 19 pp

    2024

  25. [33]

    Ho, Fractional integral operators on grand Morrey spaces and grand Hardy–Morrey spaces, J

    K.-P. Ho, Fractional integral operators on grand Morrey spaces and grand Hardy–Morrey spaces, J. Math. Inequal. 18 (2024), 755–774. Vanishing CampanatoSpaces via Ball Banach Function Spaces 35

    2024

  26. [34]

    Ho, Grand Morrey spaces and grand Hardy–Morrey spaces on Euclidean space, J

    K.-P. Ho, Grand Morrey spaces and grand Hardy–Morrey spaces on Euclidean space, J. Geom. Anal. 33 (2023), Paper No. 180, 23 pp

    2023

  27. [35]

    Ho, Atomic decomposition of Hardy–Morrey spaces with variable exponents, Ann

    K.-P. Ho, Atomic decomposition of Hardy–Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), 31–62

    2015

  28. [36]

    Ho, Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces, Anal

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

    2012

  29. [37]

    H ¨ormander, Estimates for translation invariant operators in Lp spaces, Acta Math

    L. H ¨ormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140

    1960

  30. [38]

    Huang, J

    L. Huang, J. Liu, D. Yang and W. Yuan, Atomic and Littlewood–Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications, J. Geom. Anal. 29 (2019), 1991–2067

    2019

  31. [39]

    Huang and D

    L. Huang and D. Yang, On function spaces with mixed norms — a survey, J. Math. Study 54 (2021), 262–336

    2021

  32. [40]

    Iwaniec and C

    T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129–143

    1992

  33. [41]

    Izuki, Another proof of characterization of BMO via Banach function spaces, Rev

    M. Izuki, Another proof of characterization of BMO via Banach function spaces, Rev. Uni´on Mat. Argent. 57 (2016), 103–109

    2016

  34. [42]

    Izuki, T

    M. Izuki, T. Noi and Y . Sawano, The John-Nirenberg inequality in ball Banach function spaces and application to characterization of BMO, J. Inequal. Appl. 2019 (2019), Paper No. 268, 11 pp

    2019

  35. [43]

    Izuki and Y

    M. Izuki and Y . Sawano, Characterization of BMO via ball Banach function spaces, Vestn. St.-Peterbg. Univ., Mat. Meh. Astron. 4 (62) (2017), 78–86

    2017

  36. [44]

    John and L

    F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426

    1961

  37. [45]

    Y . Li, D. Yang and L. Huang, Real-variable theory of Hardy spaces associated with gen- eralized Herz spaces of Rafeiro and Samko, Lecture Notes in Mathematics 2320, Springer, Singapore, 2022

    2022

  38. [46]

    Lin and Q

    C. Lin and Q. Yang, Semigroup characterization of Besov type Morrey spaces and well- posedness of generalized Navier–Stokes equations, J. Di fferential Equations 254 (2013), 804–846

    2013

  39. [47]

    P. I. Lizorkin, Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm. Applications, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 218–247

    1970

  40. [48]

    G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37–55

    1950

  41. [49]

    G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math. 1 (1951), 411–429

    1951

  42. [50]

    Lorist and Z

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

    2024

  43. [51]

    S. Lu, D. Yang and G. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, 2008

    2008

  44. [52]

    Masaki, Two minimization problems on non-scattering solutions to mass-subcritical non- linear Schr¨odinger equation, arXiv: 1605.09234

    S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical non- linear Schr¨odinger equation, arXiv: 1605.09234

    arXiv

  45. [53]

    Masaki and J

    S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the mass- subcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincar ´e C Anal. Non Lin´eaire 35 (2018), 283–326

    2018

  46. [54]

    Masaki and J

    S. Masaki and J. Segata, Refinement of Strichartz estimates for Airy equation in nondiagonal case and its application, SIAM J. Math. Anal. 50 (2018), 2839–2866

    2018

  47. [55]

    Mastyło, Y

    M. Mastyło, Y . Sawano and H. Tanaka, Morrey-type space and its K ¨othe dual space, Bull. Malays. Math. Sci. Soc. 41 (2018), 1181–1198

    2018

  48. [56]

    N. G. Meyers, Mean oscillation over cubes and H ¨older continuity, Proc. Amer. Math. Soc. 15 (1964), 717–721. 36 Xing Fu, Yoshihiro Saw ano, Jin Tao and Dachun Yang

    1964

  49. [57]

    C. B. Morrey, On the solutions of quasi-linear elliptic partial di fferential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166

    1938

  50. [58]

    Moyua, A

    A. Moyua, A. Vargas and L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in R3, Duke Math. J. 96 (1999), 547–574

    1999

  51. [59]

    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

    1972

  52. [60]

    Mudarra and T

    C. Mudarra and T. Oikari, Approximation in H ¨older Spaces, arXiv: 2401.01695

  53. [61]

    Neri, Fractional integration on the space H1 and its dual, Studia Math

    U. Neri, Fractional integration on the space H1 and its dual, Studia Math. 53 (1975), 175– 189

    1975

  54. [62]

    Nieraeth, Extrapolation in general quasi-Banach function spaces, J

    Z. Nieraeth, Extrapolation in general quasi-Banach function spaces, J. Funct. Anal. 285 (2023), Article ID 110130, 109 pp

    2023

  55. [63]

    Rafeiro and S

    H. Rafeiro and S. Samko, Herz spaces meet Morrey type spaces and complementary Morrey type spaces, J. Fourier Anal. Appl. 26 (2020), Paper No. 74, 14 pp

    2020

  56. [64]

    Sarason, Functions of vanishing mean oscillation, Trans

    D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405

    1975

  57. [65]

    Sawano, G

    Y . Sawano, G. Di Fazio and D. I. Hakim, Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, V olume I, Chapman and Hall/CRC, New York, 2020

    2020

  58. [66]

    Sawano, G

    Y . Sawano, G. Di Fazio and D. I. Hakim, Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, V olume II, Chapman and Hall/CRC, New York, 2020

    2020

  59. [67]

    Sawano, K.-P

    Y . Sawano, K.-P. Ho, D. Yang and S. Yang, Hardy spaces for ball quasi-Banach function spaces, Dissertationes Math. (Rozprawy Mat.) 525 (2017), 1–102

    2017

  60. [68]

    Sawano and H

    Y . Sawano and H. Tanaka, The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo 22 (2015), 663–683

    2015

  61. [69]

    Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz do- mains, Amer

    Z. Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz do- mains, Amer. J. Math. 125 (2003), 1079–1115

    2003

  62. [70]

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

    2024

  63. [71]

    J. Tao, Q. Xue, D. Yang and W. Yuan, XMO and weighted compact bilinear commutators, J. Fourier Anal. Appl. 27 (2021), Paper No. 60, 34 pp

    2021

  64. [72]

    J. Tao, Da. Yang and Do. Yang, A new vanishing Lipschitz-type subspace of BMO and bilinear compact commutators, Math. Ann. 386 (2023), 495–531

    2023

  65. [73]

    J. Tao, D. Yang, W. Yuan and Y . Zhang, Compactness characterizations of commutators on ball Banach function spaces, Potential Anal. 58 (2023), 645–679

    2023

  66. [74]

    R. H. Torres and Q. Xue, On compactness of commutators of multiplication and bilinear pesudodifferential operators and a new subspace of BMO, Rev. Mat. Iberoam. 36 (2020), 939–956

    2020

  67. [75]

    Triebel, Theory of Function Spaces, Mathematik und ihre Anwendungen in Physik und Technik, 38, Geest & Portig, Leipzig, 1983

    H. Triebel, Theory of Function Spaces, Mathematik und ihre Anwendungen in Physik und Technik, 38, Geest & Portig, Leipzig, 1983

    1983

  68. [76]

    Uchiyama, On the compactness of operators of Hankel type, Tohoku Math

    A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), 163–171

    1978

  69. [77]

    Y . Wan, D. Yang and Y . Zhao, Grand Triebel–Lizorkin–Bourgain–Morrey Spaces: Non- triviality, Embeddings, and Boundedness of Certain Operators, Bull. Math. Sci. (2025), to appear

    2025

  70. [78]

    X. Yan, Z. He, D. Yang and W. Yuan, Hardy spaces associated with ball quasi-Banach func- tion spaces on spaces of homogeneous type: Littlewood–Paley characterizations with ap- plications to boundedness of Calder ´on–Zygmund operators, Acta Math. Sin. (Engl. Ser.) 38 (2022), 1133–1184

    2022

  71. [79]

    W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010. Vanishing CampanatoSpaces via Ball Banach Function Spaces 37

    2005

  72. [80]

    Zhang, L

    Y . Zhang, L. Huang, D. Yang and W. Yuan, New ball Campanato-type function spaces and their applications, J. Geom. Anal. 32 (2022), Paper No. 99, 42 pp

    2022

  73. [81]

    Zhang, S

    Y . Zhang, S. Wang, D. Yang and W. Yuan, Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calder´on–Zygmund operators, Sci. China Math. 64 (2021), 2007–2064

    2021

  74. [82]

    Y . Zhao, Y . Li, D. Yang, W. Yuan and Y . Zhang, Generalized Frank characterizations of Muckenhoupt weights and homogeneous ball Banach Sobolev spaces, Adv. Math. 458 (2024), Article ID 109957, 63 pp

    2024

  75. [83]

    Y . Zhao, Y . Sawano, J. Tao, D. Yang and W. Yuan, Bourgain–Morrey spaces mixed with structure of Besov spaces, Proc. Steklov Inst. Math. 323 (2023), 244–295

    2023

  76. [84]

    C. Zhu, D. Yang and W. Yuan, Brezis–Seeger–Van Schaftingen–Yung-type characterization of homogeneous ball Banach Sobolev spaces and its applications, Commun. Contemp. Math. 26 (2024), Paper No. 2350041, 48 pp

    2024

  77. [85]

    C. Zhu, D. Yang and W. Yuan, Generalized Brezis–Seeger–Van Schaftingen–Yung formulae and their applications in ball Banach Sobolev spaces, Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234, 76 pp. Xing Fu and Jin Tao (Corresponding author) Hubei Key Laboratory...

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.