On the non-tangential convergence of Poisson and modified Poisson semigroups at the smoothness points of \(L_{p}\)-functions

  • Simten BayrakciEmail author
  • M. F. Shafiev
  • Ilham A. Aliev


The high-dimensional version of Fatou’s classical theorem asserts that the Poisson semigroup of a function \(f\in L_{p}(\mathbb {R}^{n}), \ 1\le p \le \infty \), converges to f non-tangentially at Lebesque points. In this paper we investigate the rate of non-tangential convergence of Poisson and metaharmonic semigroups at \(\mu \)-smoothness points of f.


Poisson semigroup Metaharmonic semigroup Non-tangential convergence Smoothness point Rate of convergence 

Mathematics Subject Classification

31A05 31A20 35J05 42B99 



The authors would like to thank the editor and the anonymous referees for their carefully reading the manuscript and useful comments and suggestions, which improved this paper.


  1. 1.
    I.A. Aliev, On the Bochner–Riesz summability and restoration of \(\mu \)-smooth functions by means of their Fourier transforms. Fract. Calc. Appl. Anal. 2(3), 265–277 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    I.A. Aliev, M. Shafiev, S. Sezer, On order of nontangential convergence of the metaharmonic semigroup. Dokl. Akad. Nauk Azerb. LXI 1, 10–15 (2005). (Russian)Google Scholar
  3. 3.
    I.A. Aliev, S. Çobanoğlu, The rate of convergence of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroup. Integral Transforms Spec. Funct. 25(12), 943–954 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I.A. Aliev, M. Eryiğit, On a rate of convergence of trancated hypersingular integrals associated to Riesz and Bessel potentials. J. Math. Anal. Appl. 406, 352–359 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    I.A. Aliev, B. Rubin, Wavelet-like transforms for admissible semi-groups; inversion formulas for potentials and Radon transforms. J. Fourier Anal. Appl. 11(3), 333–352 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. Johansson, A counterexample on nontangential convergence for oscillatory integrals. Publ. Inst. Math. (Beograd) (N.S.) 87, 129–137 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.E. Littlewood, On a theorem of Fatou. J. Lond. Math. Soc. 2, 172–176 (1927)CrossRefGoogle Scholar
  8. 8.
    P.I. Lizorkin, The functions of Hirschman type and relations between the spaces \(B_{p}^{r}(E_{n})\) and \(L_{p}^{r}(E_{n})\). Mat. Sb. (N.S.) 63(4), 505–535 (1964). (Russian)MathSciNetGoogle Scholar
  9. 9.
    A. Nagel, E.M. Stein, On certain maximal functions and approach regions. Adv. Math. 54, 83–106 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Pineda, W.R. Urbina, Non tangential convergence for the Ornstein–Uhlenbeck semigroup. Divulg. Math. 16(1), 107–124 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    B. Rubin, Fractional Integrals and Potentials (Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82 (Longman, Harlow, 1996)Google Scholar
  12. 12.
    S. Sezer, I.A. Aliev, On the Gauss–Weierstrass summability of multiple trigonometric series at \(\mu \)-smoothness points. Acta Math. Sin. (Engl. Ser.) 27(4), 741–746 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)zbMATHGoogle Scholar
  14. 14.
    A. Zygmund, On a theorem of Littlewood. Summa Brasil. Math. 2, 1–7 (1949)MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2020

Authors and Affiliations

  • Simten Bayrakci
    • 1
    Email author
  • M. F. Shafiev
    • 2
  • Ilham A. Aliev
    • 1
  1. 1.Department of MathematicsAkdeniz UniversityAntalyaTurkey
  2. 2.Azerbaijan National Aviation AcademyBakuAzerbaijan

Personalised recommendations