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Continuity Properties of Multilinear Localization Operators on Modulation Spaces

  • Nenad TeofanovEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We introduce multilinear localization operators in terms of the short-time Fourier transform and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. To obtain such interpretation, we use the kernel theorem for the Gelfand–Shilov space \( {\mathscr {S}}^{( 1)} (\mathbb {R}^d) \) and its dual space of tempered ultra-distributions \( {\mathscr {S}}^{( 1)'} (\mathbb {R}^{2d})\). Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when restricted to the linear case.

Notes

Acknowledgements

This research is supported by MPNTR of Serbia, project numbers 174024 and DS 028 (TIFMOFUS).

References

  1. 1.
    Bényi, Á., Gröchenig, K., Heil, C. , Okoudjou, K. A., Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory 54 (2), 387–399 (2005)Google Scholar
  2. 2.
    Bényi, Á., Okoudjou, K. A., Bilinear pseudodifferential operators on modulation spaces. J. Fourier Anal. Appl. 10 (3), 301–313 (2004)Google Scholar
  3. 3.
    Bényi, Á., Okoudjou, K. A., Modulation space estimates for multilinear pseudodifferential operators. Studia Math. 172 (2), 169–180 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boggiatto, P., Cordero, E., Gröchenig, K., Generalized Anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations Operator Theory 48, 427–442 (2004)Google Scholar
  5. 5.
    Calderón, A.-P. , Vaillancourt, R., On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23, 374–378 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cordero, E., Gröchenig, K., Time-frequency analysis of localization operators, J. Funct. Anal. 205 (1), 107–131 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cordero, E., D’Elia L., Trapasso, S. I., Norm Estimates for \(\tau \)-pseudodifferential operators in Wiener amalgam and modulation spaces, arXiv:1803.07865. Cited 10 Apr 2018
  8. 8.
    Cordero, E., Okoudjou, K. A., Multilinear localization operators, J. Math. Anal. Appl. 325 (2), 1103–1116 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cordero E., Nicola, F., Sharp Integral Bounds for Wigner Distributions, International Mathematics Research Notices, 2018 (6), 1779–1807 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cordero, E., Nicola F., Trapasso, S. I., Almost diagonalization of \(\tau \)-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces. J. Fourier Anal. Appl. (2018) https://doi.org/10.1007/s00041-018-09651-z
  11. 11.
    Cordero, E., Pilipović, S., Rodino L., Teofanov, N., Localization operators and exponential weights for modulation spaces, Mediterr. J. Math. 2(4), 381–394 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cordero, E., Pilipović, S., Rodino L., Teofanov, N., Quasianalytic Gelfand-Shilov spaces with application to localization operators, Rocky Mountain J. Math. 40 (4), 1123–1147 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cordero, E., Tabacco, A., Wahlberg, P., Schrödinger-type propagators, pseudodifferential operators and modulation spaces, Journal of the London Mathematical Society, 88 (2), 375–395 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Daubechies, I. , Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34 (4), 605–612 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Feichtinger, H. G. , Modulation spaces on locally compact abelian groups, Technical Report, University Vienna (1983) and also in Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, 99–140, Allied Publishers, Chennai (2003)Google Scholar
  16. 16.
    Folland, G. B., Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ (1989)Google Scholar
  17. 17.
    de Gosson, M., The Wigner Transform, World Scientific, London (2017)Google Scholar
  18. 18.
    Gelfand, I. M. , Shilov, G. E., Generalized Functions II, Academic Press, New York (1968)Google Scholar
  19. 19.
    Gröchenig, K., Foundations of Time-Frequency Analysis, Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  20. 20.
    Gröchenig, K., Weight functions in time-frequency analysis. In: Rodino, L., Schulze, B.-W., Wong M. W. (eds.) Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, pp. 343–366, Fields Institute Comm., 52 (2007)Google Scholar
  21. 21.
    Gröchenig, K., Heil, C., Modulation spaces and pseudo-differential operators, Integr. Equat. Oper. Th., 34, 439–457 (1999)Google Scholar
  22. 22.
    Gröchenig, K., Heil, C., Modulation spaces as symbol classes for pseudodifferential operators. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, 151–170. Allied Publishers, Chennai (2003)Google Scholar
  23. 23.
    Gröchenig, K., Zimmermann, G., Hardy’s theorem and the short-time Fourier transform of Schwartz functions. J. London Math. Soc. 63, 205–214 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Molahajloo, S., Okoudjou K. A., Pfander, G. E., Boundedness of Multilinear Pseudodifferential Operators on Modulation Spaces, J. Fourier Anal. Appl., 22 (6), 1381-1415 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nicola, F., Rodino, L., Global Pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications 4, Birkhäuser Verlag, (2010)Google Scholar
  26. 26.
    Pilipović, S., Teofanov, N., Toft, J., Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl., 1, 341–376 (2010)Google Scholar
  27. 27.
    Prangoski, B., Pseudodifferential operators of infinite order in spaces of tempered ultradistributions, J. Pseudo-Differ. Oper. Appl., 4 (4), 495-549 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shubin, M. A., Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, second edition (2001)CrossRefGoogle Scholar
  29. 29.
    Sjöstrand, J., An algebra of pseudodifferential operators, Math. Res. Lett., 1, 185–192 (1994)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Teofanov, N., Ultradistributions and time-frequency analysis. In: Boggiatto, P., Rodino, L., Toft, J., Wong, M. W. (eds.) Oper. Theory Adv. Appl., 164, 173–191, Birkhäuser, Verlag (2006)Google Scholar
  31. 31.
    Teofanov, N., Gelfand-Shilov spaces and localization operators, Funct. Anal. Approx. Comput. 7 (2), 135–158 (2015)Google Scholar
  32. 32.
    Teofanov, N., Continuity and Schatten-von Neumann properties for localization operators on modulation spaces, Mediterr. J. Math. 13 (2), 745–758 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Teofanov, N., Bilinear Localization Operators on Modulation Spaces, Journal of Function Spaces (2018)  https://doi.org/10.1155/2018/7560870MathSciNetCrossRefGoogle Scholar
  34. 34.
    Toft, J., Continuity properties for modulation spaces with applications to pseudo-differential calculus, I, J. Funct. Anal., 207, 399–429 (2004)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Toft, J., Continuity properties for modulation spaces with applications to pseudo-differential calculus, II, Ann. Global Anal. Geom., 26, 73–106 (2004)Google Scholar
  36. 36.
    Toft, J., Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In: Toft, J., Wong, M. W., Zhu, H. (eds.) Oper. Theory Adv. Appl. 172, 173–206, Birkhäuser, Verlag (2007)Google Scholar
  37. 37.
    Toft, J., The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., 3 (2), 145–227 (2012)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Toft, J., Johansson, K., Pilipović, S., Teofanov, N., Sharp convolution and multiplication estimates in weighted spaces, Analysis and Applications, 13 (5), 457–480 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Toft, J., Khrennikov, A., Nilsson, B., Nordebo, S., Decompositions of Gelfand-Shilov kernels into kernels of similar class, J. Math. Anal. Appl. 396 (1), 315–322 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Treves, F., Topological Vector Spaces, Distributions and Kernels, Academic Press, New York (1967)Google Scholar
  41. 41.
    Wong, M. W. , Weyl Transforms, Springer-Verlag, 1998.Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia

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