Skip to main content
SpringerLink
Log in

On the Predual of a Morrey–Lorentz Space and Its Applications to the Linear Calderón–Zygmund Operators

  • Research Article
  • Published:
Frontiers of Mathematics Aims and scope Submit manuscript

Abstract

Our main purpose in this paper is to construct a predual of Morrey–Lorentz space via the block spaces, defined in [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1999, 28(1): 31–40]. As a direct application of duality, we obtain the Morrey–Lorentz boundedness of linear Calderón–Zygmund operators. Moreover, we study a weak Hardy factorization in terms of linear Calderón–Zygmund operators in Morrey–Lorentz spaces. As a consequence of this result, we obtain a characterization of functions in BMO(ℝn) through the boundedness of commutator [b, T], where T is a homogeneous Calderón–Zygmund operator. Finally, we prove a Morrey–Lorentz compactness characterization of [b, T]. Precisely, [b, T] is a compact operator on Morrey–Lorentz spaces if and only if b ∈ CMO(ℝn).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams D.R., A note on Riesz potentials. Duke Math. J., 1975, 42(4): 765–778

    Article  MathSciNet  Google Scholar 

  2. Adams D.R., Lewis J.L., On Morrey–Besov inequalities. Studia Math., 1982, 74(2): 169–182

    Article  MathSciNet  Google Scholar 

  3. Adams D.R., Xiao J., Morrey spaces in harmonic analysis. Ark. Mat., 2012, 50(2): 201–230

    Article  MathSciNet  Google Scholar 

  4. Alvarez J., Continuity of Calderón–Zygmund type operators on the predual of a Morrey space. In: Clifford Algebras in Analysis and Related Topics, Stud. Adv. Math., Boca Raton, FL: CRC Press, 1996, 309–319

    Google Scholar 

  5. Beatrous F., Li S.Y., On the boundedness and compactness of operators of Hankel type. J. Funct. Anal., 1993, 111(2): 350–379

    Article  MathSciNet  Google Scholar 

  6. Blasco O., Ruiz A., Vega L., Non-interpolation in Morrey-Campanato and block spaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1999, 28(1): 31–40

    MathSciNet  Google Scholar 

  7. Brudnyi Y., Compactness criteria for spaces of measurable functions. St. Petersburg Math. J., 2015, 26(1): 49–68

    Article  MathSciNet  Google Scholar 

  8. Campanato S., Theoremi di interpolazione per trasformazioni che applicano Lp in Ch,α. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 1964, 18: 345–360

    MathSciNet  Google Scholar 

  9. Coifman R.R., Rochberg R., Weiss G., Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2), 1976, 103(3): 611–635

    Article  MathSciNet  Google Scholar 

  10. Coifman R.R., Weiss G., Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 1977, 83(4): 569–645

    Article  MathSciNet  Google Scholar 

  11. Dao N.A., Duong X.T., Ha L.K., Commutators of Cauchy–Fantappié type integrals on generalized Morrey spaces on complex ellipsoids. J. Geom. Anal., 2021, 31(7): 7538–7567

    Article  MathSciNet  Google Scholar 

  12. Dao N.A., Krantz S.G., Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type. Nonlinear Anal., 2021, 203: Paper No. 112162, 22 pp.

  13. Dao N.A., Krantz S.G., Lam N., Cauchy integral commutators and Hardy factorization on Lorentz spaces. J. Math. Anal. Appl., 2021, 498(1): Paper No. 124926, 22 pp.

  14. Dao N.A., Wick B.D., Hardy factorization in terms of multilinear Calderón–Zygmund operators using Morrey spaces. Potential Anal., 2023, 59(1): 41–64

    Article  MathSciNet  Google Scholar 

  15. David G., Journe J.L., A boundedness criterion for generalized Calderón–Zygmund operators. Ann. of Math. (2), 1984, 120(2): 371–397

    Article  MathSciNet  Google Scholar 

  16. Di Fazio G., Ragusa M.A., Commutators and Morrey spaces. Boll. Un. Mat. Ital. A (7), 1991, 5(3): 323–332

    MathSciNet  Google Scholar 

  17. Di Fazio G., Ragusa M.A., Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal., 1993, 112(2): 241–256

    Article  MathSciNet  Google Scholar 

  18. Ding Y., Mei T., Boundedness and compactness for the commutators of bilinear operators on Morrey spaces. Potential Anal., 2015, 42(3): 717–748

    Article  MathSciNet  Google Scholar 

  19. Duong X.T., Gong R.M., Kuffner M.J.S., Li J., Wick B.D., Yang D.Y., Two weight commutators on spaces of homogeneous type and applications. J. Geom. Anal., 2021, 31(1): 980–1038

    Article  MathSciNet  Google Scholar 

  20. Duong X.T., Li J., Wick B.D., Yang D.Y., Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting. Indiana Univ. Math. J., 2017, 66(4): 1081–1106

    Article  MathSciNet  Google Scholar 

  21. Fefferman C., Stein E.M., Some maximal inequalities. Amer. J. Math., 1971, 93: 107–115

    Article  MathSciNet  Google Scholar 

  22. Fefferman C., Stein E.M., Hp spaces of several variables. Acta Math., 1972, 129(3–4): 137–193

    Article  MathSciNet  Google Scholar 

  23. Ferguson S.H., Lacey M.T., A characterization of product BMO by commutators. Acta Math., 2002, 189(2): 143–160

    Article  MathSciNet  Google Scholar 

  24. Goldberg D., A local version of real Hardy spaces. Duke Math. J., 1979, 46(1): 27–42

    Article  MathSciNet  Google Scholar 

  25. Grafakos L., Classical Fourier Analysis, 3rd Edition. Graduate Texts in Mathematics, Vol. 249, New York: Springer, 2014

    Book  Google Scholar 

  26. Komori Y., Calderón–Zygmund operators on the predual of a Morrey space. Acta Math. Sin. (Engl. Ser.), 2003, 19(2): 297–302

    Article  MathSciNet  Google Scholar 

  27. Komori Y., Mizuhara T., Factorization of functions in H1(ℝn) and generalized Morrey spaces. Math. Nachr., 2006, 279(5–6): 619–624

    Article  MathSciNet  Google Scholar 

  28. Krantz S.G., Li S.Y., Boundedness and compactness of integral operators on spaces of homogeneous type and applications, I. J. Math. Anal. Appl., 2001, 258(2): 629–641

    Article  MathSciNet  Google Scholar 

  29. Lerner A.K., Weighted norm inequalities for the local sharp maximal function. J. Fourier Anal. Appl., 2004, 10(5): 465–474

    Article  MathSciNet  Google Scholar 

  30. Li J., Nguyen T.T.T., Ward L.A., Wick B.D., The Cauchy integral, bounded and compact commutators. Studia Math., 2020, 250(2): 193–216

    Article  MathSciNet  Google Scholar 

  31. Li J., Wick B.D., Characterizations of \(H_{{\Delta _N}}^1({\mathbb{R}^n})\) and \({\rm{BM}}{{\rm{O}}_{{\Delta _N}}}({\mathbb{R}^n})\) via weak factorizations and commutators. J. Funct. Anal., 2017, 272(12): 5384–5416

    Article  MathSciNet  Google Scholar 

  32. Mazzucato A.L., Besov–Morrey spaces: function space theory and applications to nonlinear PDE. Trans. Amer. Math. Soc., 2003, 355(4): 1297–1364

    Article  MathSciNet  Google Scholar 

  33. Mingione G., Gradient estimates below the duality exponent. Math. Ann., 2010, 346(3): 571–627

    Article  MathSciNet  Google Scholar 

  34. Morrey C.B.Jr., Multiple integral problems in the calculus of variations and related topics. Univ. California Publ. Math. (N.S.), 1943, 1: 1–130

    MathSciNet  Google Scholar 

  35. Morrey C.B.Jr., Second order elliptic equations in several variables and Hölder continuity. Math. Z., 1959/1960, 72: 146–164

    Article  MathSciNet  Google Scholar 

  36. Peetre J., On the theory of Lp,λ spaces. J. Functional Analysis, 1969, 4: 71–87

    Article  MathSciNet  Google Scholar 

  37. Phuc N.C., Morrey global bounds and quasilinear Riccati type equations below the natural exponent. J. Math. Pures Appl. (9), 2014, 102(1): 99–123

    Article  MathSciNet  Google Scholar 

  38. Sawano Y., El-Shabrawy S.R., Weak Morrey spaces with applications. Math. Nachr., 2018, 291(1): 178–186

    Article  MathSciNet  Google Scholar 

  39. Sawano Y., Tanaka H., The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo, 2015, 22(3): 663–683

    MathSciNet  Google Scholar 

  40. Taibleson M.H., Weiss G., The molecular characterization of certain Hardy spaces. In: Representation Theorems for Hardy Spaces, Astérisque, Vol. 77, Paris: Soc. Math. France, 1980, 67–149

    Google Scholar 

  41. Tao J., Yang D.C., Yang D.Y., Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces. Math. Methods Appl. Sci., 2019, 42(5): 1631–1651

    Article  MathSciNet  Google Scholar 

  42. Torchinsky A., Real-variable Methods in Harmonic Analysis. Pure and Applied Mathematics, Vol. 123, Orlando, FL: Academic Press, Inc., 1986

    Google Scholar 

  43. Uchiyama A., On the compactness of operators of Hankel type. Tohoku Math. J. (2), 1978, 30(1): 163–171

    Article  MathSciNet  Google Scholar 

  44. Uchiyama A., The factorization of Hp on the space of homogeneous type. Pacific J. Math., 1981, 92(2): 453–468

    Article  MathSciNet  Google Scholar 

  45. Zorko C.T., Morrey space. Proc. Amer. Math. Soc., 1986, 98(4): 586–592

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author is funded by University of Economics Ho Chi Minh City (UEH), Vietnam.

Author information

Authors and Affiliations

  1. School of Economic Mathematics and Statistics, University of Economics Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Anh Dao & Steven G. Krantz

  2. Department of Mathematics, Washington University in St. Louis, St. Louis, MO, 63130, USA

    Nguyen Anh Dao & Steven G. Krantz

Authors
  1. Nguyen Anh Dao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven G. Krantz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steven G. Krantz.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dao, N.A., Krantz, S.G. On the Predual of a Morrey–Lorentz Space and Its Applications to the Linear Calderón–Zygmund Operators. Front. Math 19, 385–418 (2024). https://doi.org/10.1007/s11464-022-0124-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-022-0124-0

Keywords

MSC2020

Search

Navigation