Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ESTIMATES FOR SCHRÖDINGER MAXIMAL OPERATORSALONG CURVE WITH COMPLEX TIME

  • Niu, Yaoming (Faculty of Mathematics Baotou Teachers' College of Inner Mongolia University of Science and Technology) ;
  • Xue, Ying (Faculty of Mathematics Baotou Teachers' College of Inner Mongolia University of Science and Technology)
  • Received : 2018.08.14
  • Accepted : 2019.10.23
  • Published : 2019.12.30
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Abstract

In the present paper, we give some characterization of the L2 maximal estimate for the operator Pta,γf(Γ(x, t)) along curve with complex time, which is defined by $$P^t_{a,{\gamma}}f({\Gamma}(x,t))={\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}}}}\;e^{i{\Gamma}(x,t){\xi}}e^{it{\mid}{\xi}{\mid}^a}e^{-t^{\gamma}{\mid}{\xi}{\mid}^a}{\hat{f}}({\xi})d{\xi}$$, where t, γ > 0 and a ≥ 2, curve Γ is a function such that Γ : ℝ×[0, 1] → ℝ, and satisfies Hölder's condition of order σ and bilipschitz conditions. The authors extend the results of the Schrödinger type with complex time of Bailey [1] and Cho, Lee and Vargas [3] to Schrödinger operators along the curves.

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References

  1. A. D. Bailey, Boundedness of maximal operators of Schrodinger type with complex time, Rev. Mat. Iberoam. 29 (2013), no. 2, 531-546. https://doi.org/10.4171/RMI/729
  2. L. Carleson, Some analytic problems related to statistical mechanics, in Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), 5-45, Lecture Notes in Math., 779, Springer, Berlin, 1980.
  3. C.-H. Cho, S. Lee, and A. Vargas, Problems on pointwise convergence of solutions to the Schrodinger equation, J. Fourier Anal. Appl. 18 (2012), no. 5, 972-994. https://doi.org/10.1007/s00041-012-9229-2
  4. B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrodinger equation, in Harmonic analysis (Minneapolis, Minn., 1981), 205-209, Lecture Notes in Math., 908, Springer, Berlin.
  5. Y. Ding and Y. Niu, Weighted maximal estimates along curve associated with dispersive equations, Anal. Appl. (Singap.) 15 (2017), no. 2, 225-240. https://doi.org/10.1142/S021953051550027X
  6. S. Lee and K. M. Rogers, The Schrodinger equation along curves and the quantum harmonic oscillator, Adv. Math. 229 (2012), no. 3, 1359-1379. https://doi.org/10.1016/j.aim.2011.10.023
  7. P. Sjolin, Global maximal estimates for solutions to the Schrodinger equation, Studia Math. 110 (1994), no. 2, 105-114. https://doi.org/10.4064/sm-110-2-105-114
  8. P. Sjolin, Maximal operators of Schrodinger type with a complex parameter, Math. Scand. 105 (2009), no. 1, 121-133. https://doi.org/10.7146/math.scand.a-15109
  9. P. Sjogren and P. Sjolin, Convergence properties for the time-dependent Schrodinger equation, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 1, 13-25. https://doi.org/10.5186/aasfm.1989.1428
  10. P. Sjolin and F. Soria, A note on Schrodinger maximal operators with a complex parameter, J. Aust. Math. Soc. 88 (2010), no. 3, 405-412. https://doi.org/10.1017/s1446788710000170
  11. E. M. Stein, Oscillatory integrals in Fourier analysis, in Beijing lectures in harmonic analysis (Beijing, 1984), 307-355, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986.
  12. L. Vega, El Multiplicador de Schrodinger: la Funcion Maximal y los Operadores de Restriccion (The Schrodinger Multiplier: the Maximal Function and the Restriction Operators - in Spanish), PhD thesis, Universidad Autonoma de Madrid, 1988.