Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces

Abstract

The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of degenerate cases. A major obstacle in establishing sharp restriction estimates is the failure of rescaling invariance, which is crucial for induction on scale to be effective. Motivated by the work of Guo and Oh (2022), we introduce a method, building on an iterative variant of the broad-narrow analysis, that does not heavily rely on induction on scale. To obtain suitable transversality conditions for this analysis and to derive desirable bounds for the broad part, we define a generalized notion of Jacobian, and establish its structural properties. These properties are proved using tools and techniques from both algebra and graph theory.

Figures (10)

• Figure 1: $\mathbf{Q}(\xi) = (\xi_1\xi_2, \xi_2\xi_3, \xi_3\xi_1, \xi_2\xi_4, \xi_4\xi_5, \xi_5\xi_6, \xi_5\xi_4, \xi_7\xi_8)$ with $d=n=8$. The left component has two cycles, so $J_\mathbf{Q}(\xi) \equiv 0$.
• Figure 2: $\mathbf{Q}(\xi) = (\xi_1\xi_2, \xi_2\xi_3, \xi_3\xi_1, \xi_2\xi_4, \xi_4^2, \xi_5\xi_6, \xi_5\xi_7, \xi_8\xi_9, \xi_8\xi_{10})$ with $d=10$ and $n=9$. The left component has a cycle and a loop, so $J_\mathbf{Q}(\xi;i) \equiv 0$ for any $i$.
• Figure 3: $\mathbf{Q}(\xi) = (\xi_1\xi_2, \xi_2\xi_3, \xi_3\xi_4, \xi_4\xi_1, \xi_1\xi_5, \xi_6\xi_7, \xi_6\xi_8)$ with $d=8$ and $n=7$. The left component has a cycle of even length, so $J_\mathbf{Q}(\xi;i) \equiv 0$ for any $i$.
• Figure 4: $\mathbf{Q}(\xi) = (\xi_1\xi_2, \xi_2\xi_3, \xi_3\xi_1, \xi_2\xi_4, \xi_5\xi_6, \xi_6^2, \xi_7\xi_8, \xi_7\xi_9)$ with $d=9$ and $n=8$. Note that the left component is of Type (i) with a cycle of odd length, the middle component is of Type (ii), and the right component is of Type (iii). It is easy to check that $J_\mathbf{Q}(\xi;i)$ has the form (\ref{['eq:linear_factor']}) if and only if $i \in \{7,8,9\}$.
• Figure 5: $\mathbf{Q}'(\xi) = (\xi_1\xi_2, \xi_2\xi_3, \xi_2\xi_4, \xi_5\xi_6, \xi_6^2, \xi_7\xi_8, \xi_7\xi_9)$ with $d'=9$ and $n'=7$. The left component becomes Type (iii), and $J_{\mathbf{Q}'}(\xi;i_1',i_2') \not\equiv 0$ if and only if $i_1' \in \{1,2,3,4\}$ and $i_2' \in \{7,8,9\}$, or $i_1' \in \{7,8,9\}$ and $i_2' \in \{1,2,3,4\}$.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.1
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:Jacobian']}
• Theorem 2.2
• proof
• Definition 3.1
• Theorem 3.1
• Lemma 3.2: $\ell^p$ decoupling for hypersurfaces with nonzero Gaussian curvature, BD17
• Lemma 3.3: Flat decoupling, BD17