2. Proof of Theorem 1·3

Let us begin with the equivalent formulation of the restriction conjecture.

Proposition 2·1 ([ Reference Guth, Maldague and OhGMO24 , Proposition 1·5]). Let $d \geq 2$ and $\vec{\alpha}=(1,\ldots,1)$ . If Conjecture 1·1 is true for some d, then for $2 \leq p \leq 2+{2}/({d-1})$ and $\epsilon\gt0$ , we have

(2·1) \begin{equation} \|F\|_{L^p(\mathbb{R}^d)} \leq C_{p,\epsilon} R^{(d-1)(\frac12-\frac1p)+\epsilon} \left( \sum_{\gamma \in \Gamma_{\vec{\alpha}}(R^{-1}) }\| \mathcal{P}_{\gamma}F \|_{L^p(\mathbb{R}^d)}^p \right)^{\frac1p}\end{equation}

for all functions $F: \mathbb{R}^d \rightarrow \mathbb{C}$ whose Fourier transforms are supported on $\mathcal{N}_{\mathcal{H}_{\vec{\alpha}}^{d-1}}(R^{-1})$ .

The proof of Theorem 1·3 relies on an iterative argument. This argument is inspired by [ Reference Fan, Staffilani, Wang and WilsonFSWW18 ]. Consider the hyperbolic paraboloid $\mathcal{H}_{\vec{v}}^{2d-2}$ in $\mathbb{R}^{2d-1}$ associated with $\vec{v}=(v_1,\ldots,v_{2d-2}) \in \{-1,1\}^{2d-2}$ . Our aim is to prove that for $ 2 \leq p \leq 2+ {4}/({2d-2})$ , $\vec{\alpha}=(1/2,\ldots,1/2)$ , and $\epsilon\gt0$ ,

(2·2) \begin{equation} \|F\|_{L^p(\mathbb{R}^{2d-1})} \leq C_{p,\epsilon} R^{\frac12(2d-2)(\frac12-\frac1p)+\epsilon} \left( \sum_{\theta \in {\Gamma}_{\vec{\alpha}}(R^{-1}) }\|F_{\theta}\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p} \end{equation}

for all functions $F: \mathbb{R}^{2d-1} \rightarrow \mathbb{C}$ whose Fourier supports are in $\mathcal{N}_{\mathcal{H}_{\vec{\alpha}}^{2d-2}}(R^{-1})$ . For the rest of the proof, assume that $2 \leq p \leq 2+{4}/({2d-2})$ .

Lemma 2·2. Suppose that $0 \leq a_2 \leq a_1 \leq 1/2$ . Define two vectors

(2·3) \begin{equation} \begin{split} &\vec{\alpha}\;:\!=\;(a_1,\ldots,a_1,a_2,\ldots,a_2) \in [0,1]^{d-1} \times [0,1]^{d-1} \\&\vec{\beta}\;:\!=\;(a_1,\ldots,a_1,2a_1-a_2,\ldots,2a_1-a_2) \in [0,1]^{d-1} \times [0,1]^{d-1}. \end{split}\end{equation}

Under the assumption of Conjecture 1·1 for the hyperbolic paraboloid in $\mathbb{R}^d$ for some d, for given $\tau \in \Gamma_{ \vec{\alpha} }(R^{-1})$ and $\epsilon'\gt0$ , we have

(2·4) \begin{equation} \|F\|_{L^p(\mathbb{R}^{2d-1})} \leq C_{p,\epsilon'} R^{{(2a_1-2a_2)(d-1)}(\frac12-\frac1p)+\epsilon'} \left( \sum_{\theta \in {\Gamma}_{\vec{\beta}}(R^{-1}) }\|\mathcal{P}_{\theta}F\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p} \end{equation}

for all functions F whose Fourier supports are contained in $\tau$ . The constant $C_{p,\epsilon'}$ is independent of the choice of $\tau, a_1, a_2$ .

Let us assume the lemma and continue the proof. Fix $\epsilon\gt0$ . By the assumption on the Fourier support of F, we may write

(2·5) \begin{equation} F=\sum_{\theta_1 \in {\Gamma}_{\vec{\alpha_1}}(R^{-1}) }\mathcal{P}_{\theta_1}F,\end{equation}

where $\vec{\alpha_1}\;:\!=\;({\epsilon},\ldots,{\epsilon},0,\ldots,0) \in [0,1]^{d-1} \times [0,1]^{d-1}$ . Since the cardinality of the collection ${\Gamma}_{\vec{\alpha_1}}(R^{-1}) $ is comparable to $R^{{(d-1)\epsilon} }$ , by the triangle inequality and Hölder’s inequality, we have

(2·6) \begin{equation} \|F\|_{L^p(\mathbb{R}^{2d-1})} \leq C_p R^{(d-1)\epsilon (1-\frac1p) } \left( \sum_{\theta_1 \in {\Gamma}_{\vec{\alpha_1}}(R^{-1}) }\|\mathcal{P}_{\theta_1}F\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p}.\end{equation}

To simplify the notation, let us introduce $C\;:\!=\;(d-1)(1-1/p)$ .

Define

(2·7) \begin{equation} \vec{\alpha}_n\;:\!=\; \begin{cases} {\epsilon}( n-1 ,\ldots,n-1, n,\ldots, n ) \;\;\; \mathrm{if} \, n \, \mathrm{is \, even} \\ {\epsilon}( n ,\ldots,n, n-1,\ldots, n-1 ) \;\;\; \mathrm{if} \, n \, \mathrm{is \, odd}. \end{cases} \end{equation}

By applying Lemma 2·2 to the right-hand side of (2·6) with $\epsilon'=\epsilon^2$ , $a_1=\epsilon$ , and $a_2=0$ , we have

\begin{equation*} \|F\|_{L^p(\mathbb{R}^{2d-1})} \leq C_{p,\epsilon^2}C_p R^{C\epsilon} R^{2\epsilon(d-1)(\frac12-\frac1p)+\epsilon^2}\left( \sum_{\theta_2 \in {\Gamma}_{\vec{\alpha_2}}(R^{-1}) }\|\mathcal{P}_{\theta_2}F\|_{L^p(\mathbb{R}^{2d-1})}^p\right)^{\frac1p}.\end{equation*}

Note that we can apply Lemma 2·2 again to the right-hand side of the above inequality with $\epsilon'=\epsilon^2$ , $a_1=2\epsilon$ , and $a_2=\epsilon$ (when we apply the lemma, we switch $\xi_{d},\ldots,\xi_{2d-2}$ variables and $\xi_1,\ldots,\xi_{d-1}$ variables). By repeating this process $[\epsilon^{-1}/2]$ -times, we have

\begin{equation*}\begin{split} \|F\|_{L^p(\mathbb{R}^{2d-1})} &\leq (C_{p,\epsilon^2})^{[\epsilon^{-1}/2]}C_p R^{C\epsilon} R^{2\epsilon [\epsilon^{-1}/2](d-1)(\frac12-\frac1p)+\frac{\epsilon}{2}} \\ &\quad{} \times \left( \sum_{\theta \in {\Gamma}_{\vec{\alpha}_{[\epsilon^{-1}/2]}}(R^{-1}) }\|\mathcal{P}_{\theta }F\|_{L^p(\mathbb{R}^{2d-1})}^p\right)^{\frac1p}.\end{split}\end{equation*}

By the triangle inequality, we finally obtain

(2·8) \begin{equation} \|F\|_{L^p(\mathbb{R}^{2d-1})} \leq \widetilde{C}_{p,\epsilon} R^{2C\epsilon+\frac{\epsilon}{2}} (R)^{(d-1)(\frac12-\frac1p)}\left( \sum_{\theta \in {\Gamma}_{\vec{\alpha}}(R^{-1}) }\|\mathcal{P}_{\theta }F\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p},\end{equation}

where $\vec{\alpha}=(1/2,\ldots,1/2)$ and $\widetilde{C}_{p,\epsilon}=(C_{p,\epsilon^2})^{[\epsilon^{-1}/2]}C_p$ . This is the desired inequality. It remains to prove the lemma.

Proof of Lemma 2·2. Let us fix $\tau \in \Gamma_{\vec{\alpha}}(R^{-1})$ and suppose that the Fourier support of F is contained in $\tau$ . We need to prove (2·4). We write

(2·9) \begin{equation} \tau= \big(\big((c_1,\ldots,c_{2d-2})+ ([0,R^{-a_1}]^{n-1} \times [0,R^{-a_{2}}]^{n-1}) \big) \times \mathbb{R} \big) \cap \mathcal{N}_{\mathcal{H}_{\vec{v}}^{2d-2}}(R^{-1}).\end{equation}

To simplify the notation, we introduce

(2·10) \begin{equation} \begin{split} &c\;:\!=\;(c_1,\ldots,c_{2d-2}), \;\; \;\; \xi=(\xi',\xi_{2d-1}) \in \mathbb{R}^{2d-2} \times \mathbb{R}, \\&P(\xi')\;:\!=\;v_1\xi_1^2+\cdots+v_{2d-2}\xi_{2d-2}^2. \end{split}\end{equation}

Define the affine transformation

(2·11) \begin{equation} L(\xi)\;:\!=\;(\xi'+c,\xi_{2d-1}+P(c) + \xi' \cdot \nabla P(c)), \;\;\;\;\; \xi \in\mathbb{R}^{2d-1}\end{equation}

and the function $G_0$ satisfying $ \hat{G}_0(\xi)\;:\!=\;\hat{F}(L\xi)$ . Then the Fourier support of $G_0$ is contained in the following set

(2·12) \begin{equation} \big( [0,R^{-a_1}]^{d-1} \times [0,R^{-a_2}]^{d-1} \times \mathbb{R} \big) \cap \mathcal{N}_{\mathcal{H}_{\vec{v}}^{2d-2}}(16R^{-1}). \end{equation}

Recall that $a_2 \leq a_1$ . Define the linear mapping $L_1$ and the function G so that

(2·13) \begin{equation} L_1(\xi)\;:\!=\;(R^{-a_2}\xi',R^{-2a_2}\xi_{2d-1}), \;\;\; \hat{G}(\xi)\;:\!=\;\hat{G_0}(L_1(\xi))=\hat{F}(L_1L(\xi)). \end{equation}

Then the Fourier support of G is contained in

(2·14) \begin{equation} \big( [0,R^{a_2-a_1}]^{d-1} \times [0,1]^{d-1} \times \mathbb{R} \big) \cap \mathcal{N}_{\mathcal{H}_{\vec{v}}^{2d-2}}(16R^{2a_2-1}). \end{equation}

Note that $R^{2a_2-1} \leq R^{-2a_1+2a_2}$ . By performing the linear change of variables (2·13), the desired inequality (2·4) follows from

\begin{equation*} \|G\|_{L^p(\mathbb{R}^{2d-1})} \leq C_{p,\epsilon'} {R^{(2a_1-2a_2)(d-1){(\frac12-\frac1p)+\epsilon'}}} \left( \sum_{\theta \in {\Gamma}_{\vec{\beta_0}}(\widetilde{R}^{-1}) }\|\mathcal{P}_{\theta}G\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p},\end{equation*}

where $\widetilde{R}^{-1}\;:\!=\;16R^{-2a_1+2a_2}$ and $\vec{\beta_0}\;:\!=\;(1/2,\ldots,1/2,1,\ldots,1) \in \mathbb{R}^{d-1} \times \mathbb{R}^{d-1}$ . To prove the above inequality, let us first observe that the set (2·14) is contained in

(2·15) \begin{equation} [0,R^{a_2-a_1}]^{d-1} \times \mathcal{N}_{\mathcal{H}_{\vec{v_0}}^{d-1}}(d\widetilde{R}^{-1}),\end{equation}

where $\vec{v}_0=(v_d,\ldots,v_{2d-2})$ . To prove this, suppose that $\xi=(\xi_1,\ldots,\xi_{2d-1})$ is an element of (2·14). We need to verify that

(2·16) \begin{equation} \left|\xi_{2d-1}-\sum_{i=d}^{2d-2}v_i\xi_i^2\right| \leq d\widetilde{R}^{-1}.\end{equation}

This is true because

(2·17) \begin{equation}\begin{split} &\left|\sum_{i=1}^{2d-2}v_i\xi_i^2- \sum_{i=d}^{2d-2}v_i\xi_i^2\right| \leq \sum_{i=1}^{d-1}|\xi_i|^2 \leq (d-1)R^{-2a_1+2a_2}, \\ & \left| \xi_{2d-1} - \sum_{i=1}^{2d-2}v_i\xi_i^2 \right| \leq 10R^{2a_2-1} \leq 10R^{-2a_1+2a_2}.\end{split}\end{equation}

Since the Fourier support of G is contained in (2·15), to get the desired inequality, it suffices to decompose frequencies for $\xi_d,\ldots,\xi_{2d-2}$ -variables. We apply Proposition 2·1 for the last d-variables with $R^{-1}$ replaced by $d\widetilde{R}^{-1}$ , and obtain

\begin{equation*} \|G\|_{L^p(\mathbb{R}^{2d-1})} \leq C_{p,\epsilon'} {R^{(2a_1-2a_2)(d-1){(\frac12-\frac1p)+\epsilon'}}} \left( \sum_{\theta \in {\Gamma}_{\vec{\beta_0}}(d\widetilde{R}^{-1}) }\|\mathcal{P}_{\theta}G\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p}.\end{equation*}

It suffices to apply the triangle inequality and obtain

\begin{equation*} \|G\|_{L^p(\mathbb{R}^{2d-1})} \leq {C}_{p,\epsilon'}' {R^{(2a_1-2a_2)(d-1){(\frac12-\frac1p)+\epsilon'}}} \left( \sum_{\theta \in {\Gamma}_{\vec{\beta_0}}(\widetilde{R}^{-1}) }\|\mathcal{P}_{\theta}G\|_{L^p(\mathbb{R}^{2d-1})}^p \right)^{\frac1p}.\end{equation*}

This completes the proof.