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Triebel–Lizorkin–Morrey spaces associated to Hermite operators

  • Nguyen Ngoc Trong
  • Le Xuan TruongEmail author
  • Tran Tri Dung
  • Hanh Nguyen Vo
Article
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Abstract

The aim of this article is to establish molecular decomposition of homogeneous and inhomogeneous Triebel–Lizorkin–Morrey spaces associated to the Hermite operator \(\mathbb {H} \equiv -\Delta +|x|^2\) on the Euclidean space \(\mathbb {R}^n\). As applications of the molecular decomposition theory, we show the Triebel–Lizorkin–Morrey boundedness of Riesz potential, Bessel potential and spectral multipliers associated to the operator \({\mathbb {H}}\). These results generalize the corresponding results in Bui and Duong (J Fourier Anal Appl 21:405–448, 2015).

Keywords

Hermite operator Triebel–Lizorkin–Morrey space Molecular decomposition 

Mathematics Subject Classification

42B35 42B20 

Notes

Acknowledgements

The authors are grateful to referee for his/her valuable suggestions which improved this paper.

References

  1. 1.
    Auscher, P., Ben Ali, B.: Maximal inequalities and Riesz transform estimates on \(L^p\) spaces for Schrödinger operators with non-negative potentials. Annales de I’Institut Fourier 57(6), 1975–2013 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bui, H.Q., Duong, X.T., Yan, L.: Calderón reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229(4), 2449–2502 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bui, T.A., Duong, X.T.: Besov and Triebel–Lizorkin spaces associated to Hermite operators. J. Fourier Anal. Appl. 21, 405–448 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coulhon, T., Duong, X.T.: Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv. Differ. Equ. 5(1–3), 343–368 (2000)zbMATHGoogle Scholar
  5. 5.
    Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fu, J., Xu, J.: Characterizations of Morrey type Besov and Triebel–Lizorkin spaces with variable exponents. J. Math. Anal. Appl. 381, 280–298 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Petrushev, P., Xu, Y.: Decomposition of spaces of distributions induced by Hermite expansion. J. Fourier Anal. Appl. 14(3), 372–414 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sawano, Y., Tanaka, H.: Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces. Math. Z. 257(4), 871–905 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sawano, Y.: Wavelet characterization of Besov–Morrey and Triebel–Lizorkin–Morrey spaces. Funct. Approx. Comment. Math. 38, 93–108 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sawano, Y., Tanaka, H.: Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces for nondoubling measures. Math. Nachr. 282(12), 1788–1810 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sawano, Y.: A note on Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces. Acta. Math. Sin. Engl. Ser. 25(8), 1223–1242 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sawano, Y.: Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces on domains. Math. Nachr. 283(10), 1456–1487 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278, 904–917 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, H.: Decomposition for Morrey type Besov–Triebel spaces. Math. Nachr. 282(5), 774–787 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Xu, J.: A characterization of Morrey type Besov and Triebel–Lizorkin spaces. Vietnam J. Math. 33(4), 369–379 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  • Nguyen Ngoc Trong
    • 1
    • 2
  • Le Xuan Truong
    • 3
    • 4
    Email author
  • Tran Tri Dung
    • 5
  • Hanh Nguyen Vo
    • 6
  1. 1.Faculty of Mathematics and Computer ScienceVUNHCM - University of ScienceHo Chi Minh CityVietnam
  2. 2.Department of Primary EducationHoChiMinh City University of EducationHo Chi Minh CityVietnam
  3. 3.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  5. 5.Department of MathematicsHoChiMinh City University of EducationHo Chi Minh CityVietnam
  6. 6.Department of MathematicsMacquarie UniversitySydneyAustralia

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