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arXiv · 2109.11998

$\ell^2$ decoupling for certain surfaces of finite type in $\mathbb{R}^3$

Zhuoran Li·Jiqiang Zheng
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Abstract

In this article, we establish an $\ell^2$ decoupling inequality for the surface $$F_4^2:=\Big\{(ξ_1,ξ_2,ξ_1^4+ξ_2^4): (ξ_1,ξ_2) \in [0,1]^2\Big\}$$ associated with the decomposition adapted to finite type geometry from our previous work. The key ingredients of the proof include the so-called generalized rescaling technique, an $\ell^2$ decoupling inequality for the surfaces $$\Big\{(ξ_1,ξ_2,ϕ_1(ξ_1)+ξ_2^4): (ξ_1,ξ_2) \in [0,1]^2\Big\}$$ with $ϕ_1$ being non-degenerate, reduction of dimension arguments and induction on scales.

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Zhuoran Li, Jiqiang Zheng. 2021-09-24. $\ell^2$ decoupling for certain surfaces of finite type in $\mathbb{R}^3$. https://arxiv.org/abs/2109.11998

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