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arXiv · 2111.00897

$L^2$ Schr\"{o}dinger maximal estimates associated with finite type phases in $\mathbb{R}^2$

Zhuoran Li·Junyan Zhao·Tengfei Zhao
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Abstract

In this paper, we establish Schr\"{o}dinger maximal estimates associated with the finite type phases \begin{equation*} \phi(\xi_1,\xi_2):=\xi^m_1+\xi^m_2,\;(\xi_1,\xi_2)\in [0,1]^2, \end{equation*} where $m \geq 4$ is an even number. Following [12], we prove an $L^2$ fractal restriction estimate associated with the surfaces \begin{equation*} F^2_m:=\{(\xi_1,\xi_2,\phi(\xi_1,\xi_2)):\;(\xi_1,\xi_2)\in [0,1]^2\} \end{equation*} as the main result, which also gives results on the average Fourier decay of fractal measures associated with these surfaces. The key ingredients of the proof include the rescaling technique from [16], Bourgain-Demeter's $\ell^2$ decoupling inequality, the reduction of dimension arguments from [17] and induction on scales.

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Zhuoran Li, Junyan Zhao, Tengfei Zhao. 2021-10-29. $L^2$ Schr\"{o}dinger maximal estimates associated with finite type phases in $\mathbb{R}^2$. https://arxiv.org/abs/2111.00897

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