A Restriction Estimate for a Certain Surface of Finite Type in $${\mathbb {R}}^3$$

Decoupling for finite type phases in higher dimensions

“…From now on, we focus on the case that m ≥ 4 is an even number. We adopt the notations as in the previous work [18]. At first, given R ≫ 1, we divide [0, 1] into…”

“…where 1) , j = 1, 2, ..., n − 1 . We refer to [18] for the detailed decomposition (1.3). Buschenhenke [9] utilized the analogous decomposition to study the restriction estimates for certain conic surfaces of finite type.…”

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In this article, we establish an ℓ 2 decoupling inequality for the hypersurface, where m ≥ 4 is an even number, associated with the decomposition adapted to finite type hypersurfaces from our previous work [18]. The key ingredients of the proof include an ℓ 2 decoupling inequality for the hypersurfaces (ξ 1 , ..., ξ n−1 , φ 1 (ξ 1 )+...+φs(ξs)+ξ m s+1 +...+ξ m n−1 ) : (ξ 1 , ..., ξ n−1 ) ∈ [0, 1] n−1 , 0 ≤ s ≤ n − 1, with φ 1 , ..., φs being non-degenerate.

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$L^2$ Schrödinger maximal estimates associated with finite type phases in $\mathbb{R}^2$

“…By the argument in [6] as well as the generalized rescaling techniques from [10], we have Proposition 2.2.…”

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

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In this article, we establish an ℓ 2 decoupling inequality for the surface2 associated with the decomposition adapted to finite type geometry from our previous work [11]. The key ingredients of the proof include the so-called generalized rescaling technique, an ℓ 2 decoupling inequality for the surfaces (ξ 1 , ξ 2 , φ 1 (ξ 1 ) + ξ 42 ) : (ξ 1 , ξ 2 ) ∈ [0, 1] 2 with φ 1 being non-degenerate, reduction of dimension arguments and induction on scales.

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