Advertisement

Pseudodifferential operators with symbols in the Hörmander class \(S^0_{\alpha ,\alpha }\) on \(\alpha \)-modulation spaces

  • Tomoya KatoEmail author
  • Naohito Tomita
Article
  • 19 Downloads

Abstract

In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class \(S^0_{\rho ,\rho }\) on \(\alpha \)-modulation spaces \(M_{p,q}^{s,\alpha }\), and consider the relation between \(\alpha \) and \(\rho \). In particular, we show that pseudodifferential operators with symbols in \(S^0_{\alpha ,\alpha }\) are bounded on all \(\alpha \)-modulation spaces \(M^{s,\alpha }_{p,q}\), for arbitrary \(s\in \mathbb {R}\) and for the whole range of exponents \(0 < p,q \le \infty \).

Keywords

Pseudodifferential operators Hörmander class \(\alpha \)-Modulation spaces 

Mathematics Subject Classification

35S05 42B35 

Notes

Acknowledgements

The authors sincerely express deep gratitude to the anonymous referees for their careful reading and giving fruitful suggestions and comments. The first author is supported by Grant-in-Aid for JSPS Research Fellow (No. 17J00359). The second author is partially supported by Grant-in-aid for Scientific Research from JSPS (No. 16K05201).

References

  1. 1.
    Borup, L.: Pseudodifferential operators on \(\alpha \)-modulation spaces. J. Funct. Spaces Appl. 2, 107–123 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borup, L., Nielsen, M.: Banach frames for multivariate \(\alpha \)-modulation spaces. J. Math. Anal. Appl. 321, 880–895 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borup, L., Nielsen, M.: Boundedness for pseudodifferential operators on multivariate \(\alpha \)-modulation spaces. Ark. Mat. 44, 241–259 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borup, L., Nielsen, M.: Nonlinear approximation in \(\alpha \)-modulation spaces. Math. Nachr. 279, 101–120 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borup, L., Nielsen, M.: On anisotropic Triebel–Lizorkin type spaces, with applications to the study of pseudo-differential operators. J. Funct. Spaces Appl. 6(2), 107–154 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourdaud, G.: \(L^p\) estimates for certain nonregular pseudodifferential operators. Commun. Part. Differ. Equ. 7, 1023–1033 (1982)CrossRefzbMATHGoogle Scholar
  7. 7.
    Calderón, A.-P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Natl. Acad. Sci. USA 69, 1185–1187 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups. Technical Report, University of Vienna (1983)Google Scholar
  9. 9.
    Gibbons, G.: Opérateurs pseudo-différentiels et espaces de Besov. C. R. Acad. Sci. Paris Sér. A-B 286, A895–A897 (1978)zbMATHGoogle Scholar
  10. 10.
    Gröbner, P.: Banachräume glatter Funktionen und Zerlegungsmethoden. Ph.D. thesis, University of Vienna (1992)Google Scholar
  11. 11.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gröchenig, K., Heil, C.: Modulation spaces and pseudo differential operators. Integral Equ. Oper. Theory 34, 439–457 (1999)CrossRefGoogle Scholar
  13. 13.
    Han, J., Wang, B.: \(\alpha \)-modulation spaces (I) scaling, embedding and algebraic properties. J. Math. Soc. Jpn. 66, 1315–1373 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kato, T.: The inclusion relations between \(\alpha \)-modulation spaces and \(L^p\)-Sobolev spaces or local Hardy spaces. J. Funct. Anal. 272, 1340–1405 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kobayashi, M.: Modulation spaces \(M^{p, q}\) for \(0 < p, q \le \infty \). J. Funct. Spaces Appl. 4, 329–341 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kobayashi, M.: Dual of modulation spaces. J. Funct. Spaces Appl. 5, 1–8 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kumano-go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981)zbMATHGoogle Scholar
  18. 18.
    Stein, E.M.: Harmonic Analysis. Princeton University Press (1993)Google Scholar
  19. 19.
    Sugimoto, M.: Pseudo-differential operators on Besov spaces. Tsukuba J. Math. 12, 43–63 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sugimoto, M., Tomita, N.: A counterexample for boundedness of pseudo-differential operators on modulation spaces. Proc. Am. Math. Soc. 136, 1681–1690 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sugimoto, M., Tomita, N.: Boundedness properties of pseudo-differential and Calderón–Zygmund operators on modulation spaces. J. Fourier Anal. Appl. 14, 124–143 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tachizawa, K.: The boundedness of pseudodifferential operators on modulation spaces. Math. Nachr. 168, 263–277 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I. J. Funct. Anal. 207, 399–429 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Toft, J., Wahlberg, P.: Embeddings of \(\alpha \)-modulation spaces. Pliska Stud. Math. Bulgar. 21, 25–46 (2012)MathSciNetGoogle Scholar
  25. 25.
    Triebel, H.: Theory of Function Spaces. Birkhäuser, Boston (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, B., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ. 232, 36–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

Personalised recommendations