Weak Type Estimates of Singular Integral Operators on Morrey–Banach Spaces

  • Kwok-Pun HoEmail author


We establish the weak type estimates of singular integral operators on Morrey spaces built on Banach function space. In particular, we have these weak type estimates for Morrey spaces with variable exponent when the infimum of the exponent function equals to 1.


Weak type estimates Singular integral operators Morrey spaces Banach function space variable exponent 

Mathematics Subject Classification

Primary 42B20 42B25 Secondary 42B35 46E30 



  1. 1.
    Almedia, M., Ferreira, L.: On the Navier–Stokes equations in the half-space with initial and boundary rough data in Morrey space. J. Differ. Equ. 254, 1548–1570 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almeida, A., Samko, S.: Approximation in Morrey space. J. Funct. Anal. 272, 2392–2411 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alvarez, J., Pérez, C.: Estimates with \(A_{\infty }\) weights for various singular integral operators. Boll. Un. Mat. Ital. A (7) 8, 123–133 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alvarez, J.: Continuity of Calderón–Zygmund type operators on the predual of a Morrey space. In: Ryan, J. (ed.) Clifford Algebras in Analysis and Related Topics, pp. 309–319. CRC Press, Boca Raton (1996)Google Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  6. 6.
    Calderón, A., Zygmund, A.: On singular integral. Am. J. Math. 78, 289–309 (1956)CrossRefGoogle Scholar
  7. 7.
    Christ, M.: Weak type (1,1) bounds for rough operators. Ann. Math. 128, 19–42 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chanillo, S., Christ, M.: Weak (1,1) bounds for oscillatory singular integrals. Duke Math. J. 55, 141–155 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coifman, R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque 57, 1–210 (1978)zbMATHGoogle Scholar
  10. 10.
    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: The maximal function on variable \(L_{p}\) spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cruz-Uribe, D., Fiorenza, S.F.O,A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^{p}\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cruz-Uribe, D., Martell, J., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advance and Applications, vol. 215. Birkhäuser, Basel (2011)zbMATHGoogle Scholar
  13. 13.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  14. 14.
    Curbera, G., García-Cuerva, J., Martell, J., Pérez, C.: Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203, 256–318 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin (2011)CrossRefGoogle Scholar
  16. 16.
    Ferreira, L.: On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier–Stokes equations. J. Math. Pures Appl. (9) 105, 228–247 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Folland, G.: Real Analysis, Modern Techniques and Their Applications. Wiley, Hoboken (1984)zbMATHGoogle Scholar
  18. 18.
    Giga, Y., Miyakawa, T.: Navier–Stokes flow in \({\mathbb{R}}^{3}\) with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14, 577–618 (1989)CrossRefGoogle Scholar
  19. 19.
    Grafakos, L.: Modern Fourier Analysis. Springer (2009)Google Scholar
  20. 20.
    Guliyev, V.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009, 503948 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ho, K.-P.: Atomic decomposition of Hardy spaces and characterization of \(BMO\) via Banach function spaces. Anal. Math. 38, 173–185 (2012). Ho, K.-P.: Correction of “Atomic decomposition of Hardy spaces and characterization of \(BMO\) via Banach function spaces. Anal. Math. (to appear)Google Scholar
  22. 22.
    Ho, K.-P.: Vector-valued singular integral operators on Morrey type spaces and variable Triebel–Lizorkin–Morrey spaces. Ann. Acad. Sci. Fenn. Math. 37, 375–406 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ho, K.-P.: Sobolev–Jawerth embedding of Triebel–Lizorkin–Morrey–Lorentz spaces and fractional integral operators on Hardy type spaces. Math. Nachr. 287, 1674–1686 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ho, K.-P.: Vector-valued operators with singular kernel and Triebel–Lizorkin-block spaces with variable exponents. Kyoto J. Math. 56, 97–124 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ho, K.-P.: Extrapolation, John–Nirenberg inequalities and characterizations of \(BMO\) in terms of Morrey type spaces. Rev. Mat. Complut. 30, 487–505 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ho, K.-P.: Fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents. J. Math. Soc. Jpn. 69, 1059–1077 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ho, K.-P.: Singular integral operators with rough kernel on Morrey type spaces. Studia Math. 244, 217–243 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ho, K.-P.: Definability of singular integral operators on Morrey–Banach spaces J. Math. Soc. Jpn. (published online)Google Scholar
  29. 29.
    Kalton, N., Peck, N., Roberts, J.: An \(F\)-Space Sampler, London Mathematical Society, Lecture Note Series, vol. 89. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
  30. 30.
    Kato, T.: Strong solutions of the Navier–Stokes equation in Morrey spaces. Bol. Soc. Bras. Mat. 22, 127–155 (1992)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kokilashvili, V., Meskhi, A.: Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armen. J. Math. 1, 18–28 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Maio, C.-X., Yuan, B.-Q.: Weak Morrey spaces and strong solutions to the Navier–Stokes equations. Sci. China Ser. A 50, 1401–1417 (2007)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Meyer, Y., Coifman, R.: Wavelets: Calderón–Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  34. 34.
    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–104 (1994)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Nakai, E.: Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Studia Math. 188, 193–221 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  37. 37.
    Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  38. 38.
    Taylor, M.E.: Analysis on Morrey spaces and appplications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Information TechnologyThe Education University of Hong KongTai PoChina

Personalised recommendations