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Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

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Abstract

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.

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References

  1. Abreu, L.D., Dörfler, M.: An inverse problem for localization operators, Inverse Problems 28(11), pp. 115001, 16 (2012)

  2. Berezin F.A.: Wick and anti-wick symbols of operators. Mat. Sb. N. S. 86(128), 578–610 (1971)

    MathSciNet  Google Scholar 

  3. Boggiatto P., Cordero E., Gröchenig K.: Generalized anti-wick operators with symbols in distributional Sobolev spaces. Integral Equ. Oper. Theory 48, 427–442 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boggiatto P., Oliaro A., Wong M.W.: L p boundedness and compactness of localization operators. J. Math. Anal. Appl. 322(1), 193–206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderón A.-P., Vaillancourt R.: On the boundedness of pseudo-differential operators. J. Math. Soc. Japan 23, 374–378 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Cordero E., Gröchenig K.: Symbolic calculus and Fredholm property for localization operators. J. Fourier Anal. Appl 12(4), 371–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cordero E., Okoudjou K.A.: Multilinear localization operators J. Math. Anal. Appl. 325(2), 1103–1116 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cordero E., Pilipović S., Rodino L., Teofanov N.: Localization operators and exponential weights for modulation spaces. Mediterr. J. Math 2(4), 381–394 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cordero E., Pilipović S., Rodino L., Teofanov N.: Quasianalytic Gelfand-Shilov spaces with application to localization operators. Rocky Mt. J. Math 40(4), 1123–1147 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cordero E., Rodino L.: Wick calculus: a time-frequency approach. Osaka J. Math 42(1), 43–63 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Cordero, E., Rodino, L., Gröchenig, K.: Localization operators and time-frequency analysis, in Harmonic, wavelet and p-adic analysis, World Sci. Publ., pp. 83–110 (2007)

  13. Daubechies I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory 34(4), 605–612 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983. and also in Wavelets and Their Applications, Krishna, M., Radha, R., Thangavelu, S. (eds.) Allied Publishers, pp. 99–140 (2003)

  15. Feichtinger, H.G., Nowak, K.: A first survey of Gabor multipliers. In: Advances in Gabor Analysis, Birkhäuser, 99–128 (2003)

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

  17. Gelfand, I.M., Shilov, G.E.: Generalized functions II. Academic Press, New York (1968)

  18. Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser, Boston (2001)

  19. Gröchenig, K.H.: Weight functions in time-frequency analysis. In: Rodino, L., Wong M.W. (eds.) Pseudodifferential operators: partial differential equations and time-frequency analysis. Fields Inst. Comm. 52, 343–366 (2007)

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

    Article  Google Scholar 

  21. Gröchenig, K.H., Heil, C.: Modulation spaces as symbol classes for pseudodifferential operators, In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and their applications, Allied Publishers Private Limited, New Dehli, pp. 151–170 (2003)

  22. Gröchenig K., Zimmermann G.: Hardy’s theorem and the short-time Fourier transform of Schwartz functions. J. Lond. Math. Soc. 63, 205–214 (2001)

    Article  MATH  Google Scholar 

  23. Hörmander, L.: The analysis of linear partial differential Operators, vol. I, Springer, Berlin (1983)

  24. Langenbruch M.: Hermite functions and weighted spaces of generalized functions. Manuscr. Math 119, 269–285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lozanov–Crvenkovic, Z., Perisic, D., Taskovic, M.: Gelfand-Shilov spaces, Structural and Kernel theorems, Preprint. arXiv:0706.2268v2

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ramanathan J., Topiwala P.: Time-frequency localization via the Weyl correspondence. SIAM J. Math. Anal. 24(5), 1378–1393 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin, second edition (2001)

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Slepian D.: Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25(3), 379–393 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  31. Teofanov, N.: Ultradistributions and time-frequency analysis. In: Pseudo-differential operators and related topics, operator theory: advances and applications, Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Birkhäuser, 164,173–191 (2006)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Toft, J.: Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In: Oper. Theory Adv. Appl., 172, 173–206, Birkhäuser (2007)

  35. Toft, J.: Continuity and Schatten properties for Toeplitz operators on modulation spaces. In: Oper. Theory Adv. Appl., 172, 313–328, Birkhäuser (2007)

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Toft, J., Johansson, K., Pilipović, S., Teofanov, N.: Sharp convolution and multiplication estimates in weighted spaces, analysis and applications, accepted for publication (2014)

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wong, M.W.: Weyl transforms, Springer (1998)

  40. Wong, M.W.: Wavelet transforms and localization operators, Birkhäuser Basel (2002)

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

    Nenad Teofanov

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  1. Nenad Teofanov
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Correspondence to Nenad Teofanov.

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This research was supported by MPNTR of Serbia, project No. 174024.

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Teofanov, N. Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces. Mediterr. J. Math. 13, 745–758 (2016). https://doi.org/10.1007/s00009-014-0509-8

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  • DOI: https://doi.org/10.1007/s00009-014-0509-8

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