Pointwise convergence of solutions of the Schrödinger equation to the initial datum

DATE: 17-21 June 2019 (5 sessions)
TIME: 10:00 - 12:00 (a total of 10 hours)
LOCATION: BCAM Seminar room

We will discuss the techniques used in [DGL17], [DZ18] to (almost) solve the problem of pointwise convergence of the Schrödinger equation to the initial data.

PROGRAMME:
Denoting with eit∆ f the solution of the linear Schr¨0dinger equation on Rn with initial datum f, a very delicate problem is to identify the smallest regularity s ≥ 0 such that:

One can show by elementary methods that this convergence holds at any x ∈ Rn when f ∈ Hs and s > n/2, so the problem is interesting when s ≤ n/2. This low regularity regime has been first studied by Carleson [Car80], who proved the sufficiency of the condition s ≥ 1/4 in one dimension. Using a very simple counterexample, namely a superposition of wave packets, Dahlberg-Kenig [DK82] proved that for any s < 1/4 there are initial data f ∈ Hs for which the solution eit∆ f does not converge pointwise to f on a set of full Lebesgue measure. In particular, s ≥ 1/4 is necessary and sufficient for (1) in one dimension. The situation in higher dimensions is far more complicated. For instance, in dimensions n ≥ 3, no improvements to the sufficient condition s > 1/2, independently obtained by Vega [Veg88] and Sjo ̈lin [Sjo ̈87], have been obtained for a long time. However, recently Bourgain [Bou16] showed that s ≥ n/(2n + 2) is necessary and then s > n/(2n + 2) has been proved to be also sufficient in two dimensions by Du-GuthLi [DGL17] and in higher dimensions by Du-Zhang [DZ18], so that the problem is solved (modulo endpoints).
We will give an introduction to the higher dimensional techniques introduced in [DGL17], [DZ18]. These techniques find applications also in related problems in Harmonic Analysis.


REFERENCES:
[Bou16] J. Bourgain. A note on the Schrödinger maximal function. J. Anal. Math., 130:393-396, 2016.
[Car80] L. Carleson. Some analytic problems related to statistical mechanics. In Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), volume 779 of Lecture Notes in Math., pages 5-45. Springer, Berlin, 1980.
[DGL17] X. Du, L. Guth, and X. Li. A sharp Schrödinger maximal estimate in R2. Ann. of Math. (2), 186(2):607-640, 2017.
[DK82] B. E. J. Dahlberg and C. E. Kenig. A note on the almost everywhere behavior of solutions to the Schrödinger equation. In Harmonic analysis (Minneapolis, Minn., 1981), volume 908 of Lecture Notes in Math., pages 205-209. Springer, Berlin-New York, 1982.
[DZ18] X. Du and R. Zhang. Sharp l2 estimate of Schrödinger maximal function in higher dimensions, 2018, arXiv:1805.02775.
[Sjo ̈87] P. Sjo ̈lin. Regularity of solutions to the Schrödinger equation. Duke Math. J., 55(3):699-715, 1987.
[Veg88] L. Vega. Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 102(4):874-878, 1988.

*Registration is free, but mandatory before June 14th: To sign-up go to https://bit.ly/2Gzyfeu and fill the registration form. Student grants are available. Please, let us know if you need support for travel and accommodation expenses when you fill the form.

• BCAMCourse_2019_06_17