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A counterexample on the global $L^3$ Schr\”{o}dinger maximal estimate in $\mathbb{R}^2$
  • Zhuoran Li,
  • Ying Wang
Zhuoran Li
The graduate school of China Academy of engineering Physics

Corresponding Author:[email protected]

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Ying Wang
The Graduate School of China Academy of Engineering Physics
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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.