Illinois Journal of Mathematics

The rate of convergence on Schrödinger operator

Zhenbin Cao, Dashan Fan, and Meng Wang

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Abstract

Recently, Du, Guth and Li showed that the Schrödinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.

Article information

Source
Illinois J. Math., Volume 62, Number 1-4 (2018), 365-380.

Dates
Received: 5 December 2018
Revised: 5 December 2018
First available in Project Euclid: 13 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1552442667

Digital Object Identifier
doi:10.1215/ijm/1552442667

Mathematical Reviews number (MathSciNet)
MR3922421

Zentralblatt MATH identifier
07036791

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory

Citation

Cao, Zhenbin; Fan, Dashan; Wang, Meng. The rate of convergence on Schrödinger operator. Illinois J. Math. 62 (2018), no. 1-4, 365--380. doi:10.1215/ijm/1552442667. https://projecteuclid.org/euclid.ijm/1552442667


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References

  • J. Bourgain, On the Schrödinger maximal function in higher dimension, Proc. Steklov Inst. Math. 280 (2013), no. 1, 46–60.
  • J. Bourgain, A note on the Schrödinger maximal function, J. Anal. Math. 130 (2016), no. 1, 393–396.
  • L. Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis, Lecture Notes in Mathematics, vol. 779, 1979, pp. 5–45.
  • B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic analysis, Lecture Notes in Mathematics, vol. 908, 1981, pp. 205–209.
  • C. Demeter and S. Guo, Schrödinger maximal function estimates via the pseudoconformal transformation, 2016; available at \arxivurlarXiv:1608.07640v1 [math.CA].
  • X. Du, L. Guth and X. Li, A sharp Schrödinger maximal estimate in $R^2$, Ann. of Math. 186 (2017), no. 2, 607–640.
  • X. Du, L. Guth, X. Li and R. Zhang, Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates, 2018; available at \arxivurlarXiv:1803.01720.
  • X. Du and R. Zhang, Sharp $L^{2}$ estimate of Schrödinger maximal function in higher dimensions, 2018; available at \arxivurlarXiv:1805.02775V2 [math.CA].
  • D. Fan and F. Zhao, Almost everywhere convergence of Bochner–Riesz means on some Sobolev type spaces, 2016; available at \arxivurlarXiv:1608.01575v1 [math.FA].
  • S. Lee, On pointwise convergence of the solutions to Schrödinger equations in $R^2$, Int. Math. Res. Not. 2006 (2006), 32597.
  • R. Lucà and K. Rogers, An improved necessary condition for the Schrödinger maximal estimate, 2015; available at \arxivurlarXiv:1506.05325v1 [math.CA].
  • R. Lucà and K. Rogers, Coherence on fractals versus pointwise convergence for the Schrödinger equation, Comm. Math. Phys. 351 (2017), no. 1, 341–359.
  • A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not. 1996 (1996), no. 16, 793–815.
  • P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), no. 3, 699–715.
  • P. Sjölin, $L^p$ maximal estimates for solutions to the Schrödinger equation, Math. Scand. 81 (1997), no. 1, 35–68.
  • P. Sjölin, Maximal estimates for solutions to the nonelliptic Schrödinger equation, Bull. Lond. Math. Soc. 39 (2007), no. 3, 404–412.
  • T. Tao and A. Vargas, A bilinear approach to cone multipliers II. Applications, Geom. Funct. Anal. 10 (2000), no. 1, 216–258.
  • L. Vega, Schrödinger equations: Pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), no. 4, 874–878.