On Pointwise Convergence for Schrödinger Operator in a Convex Domain

  • Jiqiang ZhengEmail author


In this paper, we prove that the maximal inequality
$$\begin{aligned} \big \Vert \sup _{|t|<1}|e^{it\Delta _D}f(x,y)|\big \Vert _{L^2_{\mathrm{loc}}(\Omega )}\le C\Vert f\Vert _{H^s_D(\Omega )},\quad \forall ~f\in H^s_D(\Omega ) \end{aligned}$$
holds for any \(s>\tfrac{1}{2}\) with \(\Omega =\{(x,y)\in \mathbb {R}^2\mid x>0\}\) and \(\Delta _D=\partial _x^2+(1+x)\partial _y^2\). As a direct application, we obtain the pointwise convergence for the free Schrödinger equation \(i\partial _tu+\Delta _D u=0\) with initial data \(u(0)=f\) inside strictly convex domain.


Schrödinger operator Pointwise convergence Airy function 

Mathematics Subject Classification

35Q55 33C10 42B25 



The author would like to express his gratitude to the anonymous referees for their invaluable comments and suggestions. The author would like to thank Fabrice Planchon for his helpful discussions and encouragement. The author was also partly supported by the ANR-16-TERC-0006-01, ANADEL.


  1. 1.
    Adams, A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Barcelo, J.A., Bennett, J., Carbery, A., Rogers, K.M.: On the dimension of divergence sets of dispersive equations. Math. Ann. 349, 599–622 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bahouri, H., Chemin, Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J.: A remark on Schrödinger operators. Isreal J. Math. 77, 1–16 (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Some new estimates on osillatory integrals. In: Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton, NJ 1991. Princeton Mathematical Series, vol. 42, pp. 83–112. Princeton University Press, New Jersey (1995)Google Scholar
  6. 6.
    Bourgain, J.: On the Schrödinger maximal function in higher dimensions. Proc. Steklov Inst. Math. 280(1), 46–60 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J.: A note on the Schrödinger maximal function. J. Anal. Math. 130, 393–396 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carleson, L.: Some analytical problems related to statistical mechanics. Euclidean Harmonic Analysisi. Lecture Notes in Mathematics, vol. 779, pp. 5–45, Springer, Berlin (1979)Google Scholar
  9. 9.
    Cho, C., Lee, S., Vargas, A.: Problems on pointwise convergence of solutions to the Schrödinger equation. J. Fourier Anal. Appl. 18, 972–994 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cowling, M.: Pointwise behavior of solutions to Schrödinger equations. In: Harmonic Analysis (Cortona, 1982). Lecture Notes in Mathematics, vol. 992, pp. 83–90. Springer, Berlin (1983)Google Scholar
  11. 11.
    Dahlberg, B.E.J., Kenig, C.E.: A note on the almost everywhere behavior of solutions to the Schrödinger equation. In: Proceedings of Italo-American Symposium in Harmonic Analysis, University of Minnesota. Lecture Notes in Mathematics, vol. 908, pp. 205–208. Springer, Berlin (1982)Google Scholar
  12. 12.
    Demeter, C., Guo, S.: Schrödinger maximal function estimates via the pseudoconformal transformation. arXiv: 1608.07640
  13. 13.
    Du, X., Guth, L., Li, X.: A sharp Schrödinger maximal estimate in \({\mathbb{R}}^{2}\). Ann. Math. 188, 607–640 (2017)CrossRefzbMATHGoogle Scholar
  14. 14.
    Du, X., Zhang, R.: Sharp \(L^2\) estimate of Schrödinger maximal function in higher dimensions. arXiv:1805.02775
  15. 15.
    Gigante, G., Soria, F.: On the the boundedness in \(H^{1/4}\) of the maximal square function associated with the Schrödinger equation. J. Lond. Math. Soc. 77, 51–68 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guth, L., Katz, N.: On the Erdös distinct distance problem in the plane. Ann. Math. 181, 155–190 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ivanovici, O., Lebeau, G., Planchon, F.: Dispersion for the wave equation inside strictly convex domain I: the Friedlander model case. Ann. Math. 180, 323–380 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ivanovici, O.: Counterexamples to Strichartz estimates for the wave equation in domains. Math. Anna. 347, 627–673 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lee, S.: On pointwise convergence of the solutions to Schrödinger equation in \({\mathbb{R}}^{2}\). IMRN 2006, 32597 (2006)zbMATHGoogle Scholar
  20. 20.
    Luca, R., Rogers, M.: An improved neccessary condition for Schrödinger maximal estimate. arXiv: 1506.05325
  21. 21.
    Luca, R., Rogers, M.: Coherence on fractals versus pointwise convergence for the Schrödinger equation. Commun. Math. Phys. 351, 341–359 (2017)CrossRefzbMATHGoogle Scholar
  22. 22.
    Melrose, R., Taylor, M.: Boundary problems for the wave equations with grazing and gliding rays.
  23. 23.
    Miao, C., Yang, J., Zheng, J.: An improved maximal inequality for 2D fractional order Schrödinger operators. Stud. Math. 230, 121–165 (2015)zbMATHGoogle Scholar
  24. 24.
    Miao, C., Zhang, J., Zheng, J.: Maximal estimates for Schrödinger equation with inverse-square potential. Pac. J. Math. 273, 1–19 (2015)CrossRefzbMATHGoogle Scholar
  25. 25.
    Moyua, A., Vargas, A., Vega, L.: Schrödinger maximal function and restriction properties of the Fourier transform. IMRN 1996, 793–815 (1996)CrossRefzbMATHGoogle Scholar
  26. 26.
    Olver, F.W.J.: Asymptotics and special functions. In: The Computer Science and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974)Google Scholar
  27. 27.
    Rogers, K., Vargas, A., Vega, L.: Pointwise convergence of solutions to the nonelliptic Schrödinger equation. Indiana Univ. Math. J. 55(6), 1893–1906 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rogers, K., Villarroya, P.: Sharp estimates for maximal operators associated to the wave equation. Ark. Mat. 46, 143–151 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55(3), 699–715 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shao, S.: On localization of the Schrödinger maximal operator. arXiv: 1006.2787v1
  31. 31.
    Tao, T.: A sharp bilinear restriction estimate for parabloids. Geom. Funct. Anal. 13(6), 1359–1384 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tao, T., Vargas, A.: A bilinear approach to cone multipliers. II. Appl. Geom. Funct. Anal. 10(1), 216–258 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vega, L.: Schrödinger equations: pointwise convergence to the initial data. Proc. Am. Math. Soc. 102(4), 874–878 (1988)zbMATHGoogle Scholar
  34. 34.
    Walther, G.: Some \(L^p (L^\infty )\)- and \(L^2(L^2)\)-estimates for oscillatory Fourier transforms. In: Analysis of Divergence (Orono, ME, 1997), Applied and Numerical Harmonic Analysis, pp. 213–231. Birkhäuser, Boston, MA (1999)Google Scholar

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Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.LJADUniversité Côte d’AzurNiceFrance

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