For an initial datum f ∈ L2 (Rn ), consider the linear Schrödinger equation
?
iut − ∆x u = 0,
(x, t) ∈ Rn × R.
u(x, 0) = f (x)
In 1980, Carleson asked which additional conditions on f guarantee
(⋆)
lim u(x, t) = f (x)
for almost every x ∈ Rn .
t→0
More precisely, what is the minimal Sobolev regularity index s such that (⋆) holds
whenever f ∈ H s (Rn) ?
Whilst the n = 1 case was fully understood by the early 1980s, in higher
dimensions the situation is much more nuanced. Nevertheless, a recent series of
dramatic developments brought about an almost complete resolution of the
problem. First Bourgain 2016 produced a subtle counterexample demonstrating that
pointwise convergence can fail for certain f ∈ H s (Rn ) with s < n/(2n+2)
. Complementing
this, convergence was then shown to hold for s > n/(2n+2)
when n = 2 in a landmark
2(n+1)paper of Du, Guth and Li 2017 and later in all dimensions in equally important work
of Du and Zhang 2019.
This seminar will explore the positive result of Du and Zhang 2019. The argument
combines sophisticated modern machinery from harmonic analysis such as the
multilinear Strichartz estimates of Bennett, Carbery and Tao 2006 and the l2
decoupling theory of Bourgain and Demeter 2015. However, equally important are
a variety of elementary guiding principles, rooted in Fourier analysis, which govern
the behaviour of solutions to the Schrödinger equation. The talk will focus on these
basic Fourier analytic principles, building intuition and presenting a powerful toolbox
for tackling problems in modern PDE and harmonic analysis.
