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

TL;DR

This paper proves $L^2$ Schr"odinger maximal estimates for finite type phase functions in two dimensions, extending fractal restriction estimates and Fourier decay results for surfaces defined by polynomial phases.

Contribution

It introduces new $L^2$ maximal estimates for finite type phases with $m \\geq 4$, utilizing advanced decoupling and rescaling techniques.

Findings

Established $L^2$ fractal restriction estimates for surfaces defined by polynomial phases.

Derived results on the average Fourier decay of fractal measures associated with these surfaces.

Extended the understanding of Schr"odinger maximal estimates in the context of finite type phases.

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.