A counterexample on the global $L^3$
Schr\”{o}dinger maximal estimate in
$\mathbb{R}^2$
Abstract
In this paper, we give an elementary counterexample to show that the
global $L^3$ Schr\“{o}dinger maximal estimate
\begin{align*} \big\Vert
\sup_{0<{\vert t
\vert}\leq 1} \vert
e^{it\Delta}f \vert
\big\Vert_{L^3(\mathbb{R}^2)}
\leq C \Vert f
\Vert_{H^{s}(\mathbb{R}^2)},\;\;\forall
\,f\in
H^{s}(\mathbb{R}^2)
\end{align*} fails if $s<
\frac{1}{3}$. The argument also adapts to the case
of 2D fractional Schr\”{o}dinger operators, and does
not rely on any facts from number theory.
