I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that article.
There are some definitions. Let $P^{n-1}$ be the truncated paraboloid. More precisely, $$P^{n-1}=\{(\xi_{1},...,\xi_{n-1},\xi_{1}^2+...+\xi_{n-1}^2)\in \mathbb{R} : |\xi_{i}| \leq \frac{1}{2}\}$$ Let $N_{\delta}$ be the $\delta$ neighborhood of $P^{n-1}$, and let $P_{\delta}$ be a finitely overlapping cover of $N_{\delta}$ with curved regions $\theta$ of the form $$\theta = \{(\xi_{1},...,\xi_{n-1},\eta + \xi_{1}^2+...+\xi_{n-1}^2) : (\xi_{1},...\xi_{n-1}) \in C_{\theta}, \; |\eta| \leq 2\delta \}$$ where $C_{\theta}$ runs over all cubes $c+[-\frac{\delta^{1/2}}{2},\frac{\delta^{1/2}}{2}]^{n-1}$ with $$c \in \frac{\delta^{1/2}}{2}\mathbb{Z}^{n-1} \cap [-\frac{1}{2}, \frac{1}{2}]^{n-1}$$ For a cap $\tau$ on $P^{n-1}$, we let $g_{\tau}=g1_{\tau}$ be the (spatial) restriction of $g$ to $\tau$. Let $w_{B_{R}}(x)$ be weight function such that it is Fourier supported in $B(0,\frac{1}{R})$ and satisfy $$1_{B_{R}}(x) \leq Cw_{B_{R}}(x) \leq C'(1+\frac{|x-c(B_{R})|}{R})^{-10n}$$
The key definitions are followings. Let $K^{(1)}_{p,n}(\delta)$ be the smallest constant such that $$||\widehat{gd\sigma}||_{L^p(w_{B_{\delta^{-1}}})} \leq K^{(1)}_{p,n}(\delta)(\sum_{\theta : \delta^{1/2}-\mathbb{cap}}||\widehat{g_{\theta}d\sigma}||_{L^p(w_{B_{\delta^{-1}}})}^2)^{1/2}$$ for each $g : P^{n-1} \rightarrow \mathbb{C}$ and each $\delta^{-1}$ ball $B_{\delta^{-1}}$.
Let $K^{(2)}_{p,n}(\delta)$ be the smallest constant such that $$||f||_{L^p} \leq K^{(2)}_{p,n}(\delta)(\sum_{\theta \in P_{\delta}}||f_{\theta}||_{L^p}^2)^{1/2}$$
for each $f$ Fourier supported in $N_{\delta}$ and each $\delta^{-1}$ ball $B_{\delta^{-1}}$. Here, $f_{\theta}$ denote the Fourier restriction of $f$ to $\theta$. Now, Bourgain says that
$$C_{p,n}^{-1}K^{(2)}_{p,n}(\delta) \leq K^{(1)}_{p,n} \leq C_{p,n}K^{(2)}_{p,n} $$
It is written at remark 5.2. He did not write the proof but say that the following simple observation justifies the various (entirely routine) localization arguments : If $g$ is supported on $P^{n-1}$ and if $\widehat{w_{B_{R}}}$ is supported in $B(0,R^{-1})$, then $\widehat{(gd\sigma)}w_{B_{R}}$ has Fourier support inside $N_{R^{-1}}$.
I showed right inequality. I got some hints from the Wolff's 2000 paper Local smoothing type estimates on L^p for large $p$.
The proof of $K^{(1)}_{p,n} \leq C_{p,n}K^{(2)}_{p,n} $ :
Write $f(x)=\widehat{gd\sigma}(x)w_{B_{\delta^{-1}}}(x)$. By the Bourgain's observation, \begin{equation} ||\widehat{gd\sigma}||_{L^p(w_{B_{\delta^{-1}}})} =||f||_{L^p} \leq K^{(2)}(\sum_{\theta \in P_{\delta}}||\Theta_{\theta} \ast (\widehat{gd\sigma}w_{B_{\delta^{-1}}})||_{p}^2)^{1/2}=K^{(2)}(\sum_{\theta \in P_{\delta}}||\sum_{\phi}\Theta_{\theta} \ast (\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}})||_{p}^2)^{1/2} \leq K^{(2)}(\sum_{\theta \in P_{\delta}}\sum_{\phi \in J(\theta)}||\Theta_{\theta} \ast (\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}})||_{p}^2)^{1/2} \leq K^{(2)}(\sum_{\phi : \delta^{1/2}-cap}\sum_{\theta \in J(\phi)}||\Theta_{\theta} \ast (\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}})||_{p}^2)^{1/2} \leq K^{(2)}(\sum_{\phi : \delta^{1/2}-cap}||\widehat{gd\sigma}w_{B_{\delta^{-1}}}||_{p}^2)^{1/2} = K^{(2)}(\sum_{\phi : \delta^{1/2}-cap}||\widehat{g_{\theta}d\sigma}||_{L^p(w_{B_{\delta^{-1}}})}^2)^{1/2} \end{equation} Here, $\Theta_{\theta}(x) = \widehat{1_{\theta}}(-x)$ and $J(\theta)=\{\phi : \Theta_{\theta} \ast (\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}}) \neq 0\}$. I used the fact that the cardinality of $J(\theta)$ is uniformly bounded for $\theta$.
I tried to show that $K^{(2)} \leq CK^{(1)}$ but it is extremely hard to me. Any hint or reference are welcomed. Thanks in advance.
