Certain averaging operators on Lebesgue spaces

  • Junyan ZhaoEmail author
  • Dashan Fan
Original Paper


In the paper, we study some multiplier operator \(\mu _{\gamma ,\alpha }\) raised from studying the \(L^{p}\)-approximation of the spherical mean \(S_{t}^{\gamma }.\) We obtain the optimal range of exponents \( (\alpha ,\gamma ,p)\) such that \(\mu _{\gamma ,\alpha }\) is an \(L^{p}\) multiplier. As an application, we obtain the convergence rate for \( S_{t}^{\gamma }\left( f\right) \ \)in the \(L^{p}\) spaces.


Spherical mean Sobolev spaces \(L^p\) norm convergence Bessel function Wave operator 

Mathematics Subject Classification

42B15 41A35 46E35 47B38 



The authors would like to thank the referees for carefully reading the manuscript and for making several helpful suggestions. The research was supported by National Natural Science Foundation of China (Grant nos. 11771388, 11371316, 11471288, 11601456).


  1. 1.
    Chen, J., Fan, D., Zhao, F.: On the rate of almost everywhere convergence of combinations and multivariate averages Potential Anal. (2018). MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dai, F., Ditzian, Z.: Combinations of multivariate averages. J. Approx. Theory 131(2), 268–283 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fan, D., Lou, Z., Wang, Z.: A note on iterated spherical average on Lebesgue spaces. Nonlinear Anal. 180, 170–183 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fan, D., Zhao, F.: Approximation properties of combination of multivariate averages on Hardy spaces. J. Approx. Theory 223, 77–95 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fan, D., Zhao, F.: Block-Sobolev spaces and the rate of almost everywhere convergence of Bochner-Riesz Means. Constr. Approx. 45(3), 391–405 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gelfand, I., Moiseevich, S., Georgi, E.: Generalized Functions. Vol. I: Properties and Operations, Translated by Eugene Saletan. Academic, New York (1964)Google Scholar
  7. 7.
    Miyachi, A.: On some singular Fourier multipliers. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(2), 267–315 (1981)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Juan, C.: Peral, \(L^{p}\) estimates for the wave equation. J. Funct. Anal. 36(1), 114–145 (1980)CrossRefGoogle Scholar
  9. 9.
    Stein, E.M.: Maximal functions I: Spherical means. Proc. Natl. Acad. Sci. USA 73(7), 2174–2175 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)Google Scholar
  11. 11.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, N.J (1971)zbMATHGoogle Scholar
  12. 12.
    Strichartz, R.S.: Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060 (1967)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Strichartz, R., S.: \(H^p\) Sobolev spaces, Colloq. Math. 60/61(1), 129–139 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

Personalised recommendations