Schrödinger equations: pointwise convergence to the initial data
Author:
Luis Vega
Journal:
Proc. Amer. Math. Soc. 102 (1988), 874-878
MSC:
Primary 35J10
DOI:
https://doi.org/10.1090/S0002-9939-1988-0934859-0
MathSciNet review:
934859
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let $u(x,t)$ be the solution of the Schrödinger equation with initial data $f$ in the Sobolev space ${H^s}({{\mathbf {R}}^n})$ with $s > \frac {1}{2}$. The a.e. convergence of $u(x,t)$ to $f(x)$ follows from a weighted estimate of the maximal function $u * (x,t) = {\text {su}}{{\text {p}}_{t > 0}}|u(x,t)|$.
- Anthony Carbery, Radial Fourier multipliers and associated maximal functions, Recent progress in Fourier analysis (El Escorial, 1983) North-Holland Math. Stud., vol. 111, North-Holland, Amsterdam, 1985, pp. 49–56. MR 848141, DOI https://doi.org/10.1016/S0304-0208%2808%2970279-2
- Lennart Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 5–45. MR 576038
- A. Córdoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), no. 3, 505–511. MR 544242
- Michael G. Cowling, Pointwise behavior of solutions to Schrödinger equations, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 83–90. MR 729347, DOI https://doi.org/10.1007/BFb0069152
- Björn E. J. Dahlberg and Carlos E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 205–209. MR 654188
- Carlos E. Kenig and Alberto Ruiz, A strong type $(2,\,2)$ estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 239–246. MR 712258, DOI https://doi.org/10.1090/S0002-9947-1983-0712258-4
- Per Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), no. 3, 699–715. MR 904948, DOI https://doi.org/10.1215/S0012-7094-87-05535-9
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J10
Retrieve articles in all journals with MSC: 35J10
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society