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Fractional time-dependent Schrödinger equation on the Heisenberg group

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Abstract

Let Δ be the Kohn sublaplacian on the Heisenberg group \({\mathbb{H}}^n, L =(-\Delta)^\alpha\), \(\frac{2}{3} < \alpha < 1\). In this paper we estimate the L 2-norm of the local maximal function of the unitary group of operators generated by L, by the Sobolev W γ,ε -norm for some γ > 0 and for all ε > 0.

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References

  1. Bourgain, J.: A remark on Schrödinger operators. Isr. J. Math. 77, 1–16 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carbery, A., Christ, M., Wright, J.: Multidimensional van der Corput and sublevel set estimates. J. Am. Math. Soc. 12(4), 981–1015 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Deshouillers, J.M.: Probléme de Waring avec exposants non entiers. (French) Bull. Soc. Math. France 101, 285–295 (1973)

    MATH  MathSciNet  Google Scholar 

  4. Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums, London Mathematical Society Lecture Note Series 126

  5. Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. 2nd edn. Springer Study Edition. Springer, Berlin (1990)

  6. Müller, D.: A restriction theorem for the Heisenberg group. Ann. of Math. (2) 131(3), 567–587 (1990)

    Article  MathSciNet  Google Scholar 

  7. Vaughan, R.C.: The Hardy-Littlewood method. In: Cambridge Tracts in Mathematics, vol. 125, 2nd edn. Cambridge University Press, Cambridge (1997)

  8. Zienkiewicz, J.: Initial value problem for the time dependent Schrödinger equation on the Heisenberg group. Studia Math. 122(1), 15–37 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Zienkiewicz, J.: Schrödinger equation on the Heisenberg group. Studia Math. 161(2), 99–111 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384, Wroclaw, Poland

    Roman Urban & Jacek Zienkiewicz

Authors
  1. Roman Urban
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  2. Jacek Zienkiewicz
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Correspondence to Roman Urban.

Additional information

Research supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. The first author was also supported by the MNiSW research grant N201 012 31/1020.

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Urban, R., Zienkiewicz, J. Fractional time-dependent Schrödinger equation on the Heisenberg group. Math. Z. 260, 931–948 (2008). https://doi.org/10.1007/s00209-008-0308-7

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  • DOI: https://doi.org/10.1007/s00209-008-0308-7

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