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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
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Ying Wang
The Graduate School of China Academy of Engineering Physics
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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.
25 Jun 2022Submitted to Mathematical Methods in the Applied Sciences
25 Jun 2022Assigned to Editor
25 Jun 2022Submission Checks Completed
05 Jul 2022Reviewer(s) Assigned