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Multiplication properties in pseudo-differential calculus with small regularity on the symbols

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Abstract

We consider modulation space and spaces of Schatten–von Neumann symbols where corresponding pseudo-differential operators map one Hilbert space to another. We prove Hölder–Young and Young type results for such spaces under dilated convolutions and multiplications. We also prove continuity properties for such spaces under the twisted convolution, and the Weyl product. These results lead to continuity properties for twisted convolutions on Lebesgue spaces, e.g. \({L^p_{(\omega )}}\) is a twisted convolution algebra when 1 ≤ p ≤ 2 and ω is an appropriate weight.

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References

  1. Bergh J., Löfström J.: Interpolation Spaces, An Introduction. Springer, Berlin (1976)

    MATH  Google Scholar 

  2. Birman M.S., Solomyak M.Z.: Estimates for the singular numbers of integral operators (Russian). Usbehi Mat. Nauk. 32, 17–84 (1977)

    MATH  Google Scholar 

  3. Bony J.M., Chemin J.Y.: Espaces functionnels associés au calcul de Weyl-Hörmander. Bull. Soc. math. France 122, 77–118 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Boulkhemair A.: L 2 estimates for pseudodifferential operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(4), 155–183 (1995)

    MATH  MathSciNet  Google Scholar 

  5. Boulkhemair A.: L 2 estimates for Weyl quantization. J. Funct. Anal. 165, 173–204 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cordero E., Gröchenig K.: Time–frequency analysis of localization operators. J. Funct. Anal. 205(1), 107–131 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cordero E., Gröchenig K.: Symbolic calculus and Fredholm property for localization operators. J. Fourier Anal. Appl. 12, 371–392 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cordero E., Okoudjou K.: Multilinear localization operators. J. Math. Anal. Appl. 325, 1103–1116 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feichtinger, H.G.: Banach spaces of distributions of Wiener’s type and interpolation. In: Butzer, P., Nagy, B.Sz., Görlich, E. (eds.) Proceedings of Conference Oberwolfach, Functional Analysis and Approximation, August 1980, vol. 69, pp. 153–165. Int. Ser. Num. Math. Birkhäuser Verlag, Basel (1981)

  10. Feichtinger, H.G.: Banach convolution algebras of Wiener’s type. In: Proceedings of Functions, Series, Operators in Budapest, Colloquia Math. Soc. J. Bolyai. North Holland Publication Co., Amsterdam (1980)

  11. Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and their Applications, Technical Report. University of Vienna, Vienna, 1983, pp. 99–140. Allied Publishers Private Limited, New Dehli (2003)

  12. Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. In: Proceedings of Conference on Constructive Function Theory, Rocky Mountain J. Math., vol. 19, pp. 113–126 (1989)

  13. Feichtinger H.G.: Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process. 5, 109–140 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Feichtinger H.G., Gröchenig K.H.: Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86, 307–340 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Feichtinger H.G., Gröchenig K.H.: Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108, 129–148 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Feichtinger H.G., Gröchenig K.H.: Gabor frames and time-frequency analysis of distributions. J. Functional Anal. 146, 464–495 (1997)

    Article  MATH  Google Scholar 

  17. Folland G.B.: Harmonic analysis in phase space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  18. Gröchenig K.H.: Foundations of Time–Frequency Analysis. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  19. Gröchenig K.H.: Composition and spectral invariance of pseudodifferential operators on modulation spaces. J. Anal. Math. 98, 65–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gröchenig K.H., Heil C.: Modulation spaces and pseudo-differential operators. Integr. Equ. Operator Theory 34(4), 439–457 (1999)

    Article  Google Scholar 

  21. Gröchenig, K., Toft, J.: Localization operator representation of modulation spaces (preprint, 2009). arXiv:0905.4954

  22. He Z., Wong M.W.: Localization operators associated to square integrable group representations. Panamer. Math. J. 6, 93–104 (1996)

    MATH  MathSciNet  Google Scholar 

  23. Holst A., Toft J., Wahlberg P.: Weyl product algebras and modulation spaces. J. Funct. Anal. 251, 463–491 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I, III. Springer, Berlin (1983, 1985)

  25. Labate D.: Pseudodifferential operators on modulation spaces. J. Math. Anal. Appl. 262, 242–255 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lieb E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31, 594–599 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Muramatu, T.: Estimates for the norm of pseudodifferential operators by means of Besov spaces. In: Pseudodifferential Operators, Proceedings of the Conference Held in Oberwolfach, 2–8 February 1986, Lecture Notes in Mathematics, vol. 1256. Springer, Berlin (1987)

  28. Schatten R.: Norm ideals of completely continuous operators. Springer, Berlin (1960)

    MATH  Google Scholar 

  29. Schulze, B.W., Tarkhanov, N.N.: Pseudodifferential operators with operator-valued symbols. Israel Math. Conf. Proc. 16 (2003)

  30. Shubin M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987)

    MATH  Google Scholar 

  31. Simon B.: Trace ideals and their applications I, Lecture Note Series. London Mathematical Society. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  32. Sjöstrand J.: An algebra of pseudodifferential operators. Math. Res. L. 1, 185–192 (1994)

    MATH  Google Scholar 

  33. Sugimoto M.: L p boundedness of pseudodifferential operators satisfying Besov estimates I. J. Math. Soc. Jpn. 40, 105–122 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Teofanov, N.: Ultramodulation spaces and pseudodifferential operators. Endowment Andrejević, Beograd (2003)

  35. Teofanov N.: Modulation spaces, Gelfand–Shilov spaces and pseudodifferential operators. Sampl. Theory Signal Image Process 5, 225–242 (2006)

    MATH  MathSciNet  Google Scholar 

  36. Toft, J.: Continuity and Positivity Problems in Pseudo-Differential Calculus, Thesis. Department of Mathematics, University of Lund, Lund (1996)

  37. Toft J.: Subalgebras to a Wiener type algebra of pseudo-differential operators. Ann. Inst. Fourier 51, 1347–1383 (2001)

    MATH  MathSciNet  Google Scholar 

  38. Toft J.: Continuity properties for non-commutative convolution algebras with applications in pseudo-differential calculus. Bull. Sci. Math. 126 2, 115–142 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Toft J.: Positivity properties for non-commutative convolution algebras with applications in pseudo-differential calculus. Bull. Sci. Math. 127(2), 101–132 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  40. Toft J.: Continuity properties for modulation spaces with applications to pseudo-differential calculus, I. J. Funct. Anal. 207, 399–429 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  41. Toft J.: Continuity properties for modulation spaces with applications to pseudo-differential calculus, II. Ann. Global Anal. Geom. 26, 73–106 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  42. Toft, J.: Convolution and embeddings for weighted modulation spaces. In: Boggiatto, P., Ashino, R., Wong, M.W., (eds.) Advances in Pseudo-Differential Operators, Operator Theory: Advances and Applications, vol. 155, pp. 165–186. Birkhäuser Verlag, Basel (2004)

  43. Toft, J.: Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In: Toft, J. Wong, M.W., Zhu, H. (eds.) Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications, pp. 173–206. Birkhäuser Verlag, Basel (2007)

  44. Toft, J.: Continuity and Schatten properties for Toeplitz operators on modulation spaces. In: Toft, J., Wong, M.W., Zhu, H. (eds). Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications, pp. 313–328. Birkhäuser Verlag, Basel (2007)

  45. Toft J., Boggiatto P.: Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces. Adv. Math. 217, 305–333 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Systems Engineering, Växjö University, Växjö, Sweden

    Joachim Toft

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  1. Joachim Toft
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Toft, J. Multiplication properties in pseudo-differential calculus with small regularity on the symbols. J. Pseudo-Differ. Oper. Appl. 1, 101–138 (2010). https://doi.org/10.1007/s11868-010-0007-0

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  • DOI: https://doi.org/10.1007/s11868-010-0007-0

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