Journal of Fourier Analysis and Applications

, Volume 14, Issue 1, pp 124–143 | Cite as

Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

Article
First Online:
Received:

Abstract

In this article, we study the boundedness of pseudo-differential operators with symbols in S ρ,δ m on the modulation spaces M p,q. We discuss the order m for the boundedness Op(S ρ,δ m )⊂ℒ(M p,q) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space M p,q with q≠2. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on M p,q whose symbols are of the class S 1,δ 0 with 0<δ<1.

Keywords

Calderón-Zygmund operators Modulation spaces Pseudo-differential operators 

Mathematics Subject Classification (2000)

42B20 42B35 47G30 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Calderón, A.P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. 69, 1185–1187 (1972) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    David, G., Journé, J.L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. Math. 120, 371–397 (1984) CrossRefGoogle Scholar
  3. 3.
    Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, vol. 29. Amer. Math. Soc., Providence (2001) MATHGoogle Scholar
  4. 4.
    Fefferman, C.: L p bounds for pseudo-differential operators. Israel J. Math. 14, 413–417 (1973) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Feichtinger, H.G.: Banach spaces of distributions of Wiener’s type and interpolation. In: Butzer, P., Nagy, B.Sz., Görlich, E. (eds.) Proc. Conf. Oberwolfach, Functional Analysis and Approximation, August 1980. Int. Ser. Num. Math., vol. 60, pp. 153–165. Birkhäuser, Basel (1981) Google Scholar
  6. 6.
    Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and their Applications. pp. 99–140. Chennai, India, Allied Publishers, New Delhi (2003). Updated version of a Technical report, University of Vienna (1983) Google Scholar
  7. 7.
    Feichtinger, H.G.: Modulation spaces: Looking back and ahead. Sampl. Theory Signal Image Process. 5, 109–140 (2006) MathSciNetMATHGoogle Scholar
  8. 8.
    Frazier, M., Torres, R., Weiss, G.: The boundedness of Calderón-Zygmund operators on the spaces \(\dot{F}_{p}^{\alpha,q}\) . Rev. Mat. Iberoamericana 4, 41–72 (1988) MATHMathSciNetGoogle Scholar
  9. 9.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001) MATHGoogle Scholar
  10. 10.
    Gröchenig, K.: Time-Frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoamericana 22, 703–724 (2006) MATHGoogle Scholar
  11. 11.
    Gröchenig, K., Heil, C.: Modulation spaces and pseudodifferential operators. Integral Equations Operator Theory 34, 439–457 (1999) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kumano-go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981) Google Scholar
  13. 13.
    Lemarié, P.G.: Continuité sur les espaces de Besov des opérateurs définis par des intégrales singulières. Ann. Inst. Fourier (Grenoble) 35, 175–187 (1985) MATHMathSciNetGoogle Scholar
  14. 14.
    Meyer, Y., Coifman, R.: Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  15. 15.
    Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1, 185–192 (1994) MATHMathSciNetGoogle Scholar
  16. 16.
    Stein, E.M.: Harmonic Analysis, Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  17. 17.
    Sugimoto, M., Tomita, N.: The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 248, 79–106 (2007) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sugimoto, M., Tomita, N.: A counterexample for boundedness of pseudo-differential operators on modulation spaces. Proc. Amer. Math. Soc. (2008, to appear) Google Scholar
  19. 19.
    Tachizawa, K.: The boundedness of pseudodifferential operators on modulation spaces. Math. Nachr. 168, 263–277 (1994) MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus-I. J. Funct. Anal. 207, 399–429 (2004) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Torres, R.: Boundedness results for operators with singular kernels on distribution spaces. Mem. Amer. Math. Soc. 442 (1991) Google Scholar
  22. 22.
    Triebel, H.: Modulation spaces on the Euclidean n-space. Z. Anal. Anwendungen 2, 443–457 (1983) MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

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

Personalised recommendations