My question comes from Bourgain and Demeter's The proof of the $l^2$ Decoupling Conjecture. A related question was asked about a year ago go on MO, but the author was interested in the reverse implication of my question.

Let $P^{n-1}$ be the truncated paraboloid $$P^{n-1}:=\{(\xi_{1},...,\xi_{n-1},\xi_{1}^2+...+\xi_{n-1}^2)\in \mathbb{R}^{n} : |\xi_{i}| \leq \frac{1}{2}\}$$ Let $\mathcal{N}_{\delta}$ be the $\delta$ neighborhood of $P^{n-1}$, and let $\mathcal{P}_{\delta}$ be a finitely overlapping cover of $\mathcal{N}_{\delta}$ 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 $$1_{B_{R}}(x) \leq Cw_{B_{R}}(x) \leq C'(1+\frac{|x-c(B_{R})|}{R})^{-10n}$$

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 $\mathcal{N}_{\delta}$ and each $\delta^{-1}$ ball $B_{\delta^{-1}}$. Here, $f_{\theta}$ denote the Fourier restriction of $f$ to $\theta$. On pg. 20, the authors claim that

$$K^{(1)}_{p,n} \leq C_{p,n}K^{(2)}_{p,n},$$ where $C_{p,n}>0$ is some constant which only depends on $p$ and $n$.

Following the argument in this related question, one has \begin{align*} \|\widehat{gd\sigma}\|_{L^{p}(w_{B_{\delta^{-1}}})}&\leq K_{p,n}^{(2)}(\sum_{\theta\in\mathcal{P}_{\delta}}\|\widehat{\chi_{\theta}}\ast(\widehat{gd\sigma}w_{B_{\delta^{-1}}})\|_{p}^{2})^{1/2}\\ &\leq K_{p,n}^{(2)}(\sum_{\theta\in\mathcal{P}_{\delta}}\sum_{\tau\in\mathcal{P}_{\delta}}\|\widehat{\chi_{\theta}}\ast(\widehat{g_{\tau}d\sigma}w_{B_{\delta^{-1}}})\|_{p}^{2})^{1/2}\\ &=K_{p,n}^{(2)}(\sum_{\theta\in\mathcal{P}_{\delta}}\sum_{\tau\in J(\theta)}\|\widehat{\chi_{\theta}}\ast(\widehat{g_{\tau}d\sigma}w_{B_{\delta^{-1}}})\|_{p}^{2})^{1/2} \tag{1} \end{align*} where we use two applications of Minkowski's inequality. One can show that the cardinality of $J(\theta)=\{\tau : \widehat{\chi_{\theta}}\ast (\widehat{g_{\tau}d\sigma}w_{B_{\delta}^{-1}})\}$ is uniformly bounded in $\theta$. Similarly, for each $\tau$ fixed, $|\{\theta: \tau\in J(\theta)\}|$ is uniformly bounded. My problem is how to get rid of the convolution in (1). If I knew that $\|\widehat{\chi_{\theta}}\|_{1}\lesssim_{n}1$, then I could use Young's inequality and the preceding observations to get \begin{align*} (1)&\leq K_{p,n}^{(2)}(\sum_{\theta\in\mathcal{P}_{\delta}}\sum_{\tau\in J(\theta)}(\|\widehat{\chi_{\theta}}\|_{1}\|\widehat{g_{\tau}d\sigma}w_{B_{\delta^{-1}}}\|_{p})^{2})^{1/2}\\ &\lesssim K_{p,n}^{(2)}(\sum_{\tau\in\mathcal{P}_{\delta}}\|\widehat{g_{\tau}d\sigma}w_{B_{\delta^{-1}}}\|_{p}^{2})^{1/2}\tag{2} \end{align*} But I don't see why this should be the case, given that $\chi_{\theta}$ is a rough frequency projection. If it were a bump function adapted to $\theta$, then there would be no problem; however, this is not the case.