My question comes from Bourgain and Demeter's [<i>The proof of the $l^2$ Decoupling Conjecture</i>][1]. A related [question][2] 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][2], 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.

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Following Prof. Tao's suggestion, introduce a smooth partition of unity $\{\widehat{K_{\theta}}\}_{\theta\in\mathcal{P}_{\delta}}$ of Fourier space such that $\mathrm{supp}(K_{\theta})\subset \frac{9}{10}\theta$, and $\|K_{\theta}\|_{1}\lesssim_{n} 1$ (see pg. 18 of the arXiv version). We introduce the notation $\tilde{f}_{\theta}:=f\ast K_{\theta}$. Note that $\tilde{f}_{\theta}=f_{\theta}\ast K_{\theta}$. Similarly, for $g: P^{n-1}\rightarrow\mathbb{C}$, we write $\tilde{g}_{\theta}:=g\widehat{K_{\theta}}$. For $\delta$ sufficiently small, $\mathrm{supp}((\tilde{g}_{\theta}d\sigma)\ast w_{B_{\delta^{-1}}}^{\vee})\subset\theta$.

Now write
$$\mathcal{P}_{\delta}=\bigcup_{j=1}^{N}\mathcal{P}_{\delta}^{j}$$
where the $\mathcal{P}_{\delta}^{j}$ are $\sim\delta^{1/2}$-separated subcollections of caps in $\mathcal{P}_{\delta}$. Note that $N\lesssim_{n}1$. If we choose the separation appropriately, then for any fixed $\tau\in\mathcal{P}_{\delta}$,
$$|I_{j}(\tau)|:=|\{\theta\in\mathcal{P}_{\delta}^{j}: \chi_{\tau}((\tilde{g}_{\theta}d\sigma)\ast w_{B_{\delta^{-1}}}^{\vee}\neq 0\}|\leq 1$$

Set $h_{j}:=\sum_{j=1}^{N}\sum_{\theta\in\mathcal{P}_{\delta}^{j}}\tilde{g}_{\theta}$. By linearity and the triangle inequality,
$$\|\widehat{gd\sigma} w_{B_{\delta^{-1}}}\|_{p}\leq\sum_{j=1}^{N}\|\widehat{h_{j}d\sigma} w_{B_{\delta^{-1}}}\|_{p}$$
So without loss of generality, we may and will assume that $g=h_{j}$ for $j=1$. Since $\chi_{\theta}=1$ on the support of $(\tilde{g}_{\theta}d\sigma)\ast w_{B_{\delta^{-1}}}^{\vee}$, applying the decoupling inequality gives
\begin{align*}
\|\widehat{gd\sigma} w_{B_{\delta^{-1}}}\|_{p}&\leq K_{p,n}^{(2)}(\delta)(\sum_{\tau\in\mathcal{P}_{\delta}}\|\widehat{\chi_{\tau}}\ast(\widehat{gd\sigma}w_{B_{\delta^{-1}}})\|_{p}^{2})^{1/2}\\
&=K_{p,n}^{(2)}(\delta)(\sum_{\tau\in\mathcal{P}_{10\delta}}\|\sum_{\theta\in I(\tau)}\widehat{\tilde{g}_{\theta}d\sigma}w_{B_{\delta^{-1}}})\|_{p}^{2})^{1/2}\\
&\leq K_{p,n}^{(2)}(\delta)(\sum_{\theta\in\mathcal{P}_{\delta}^{1}}\|\widehat{\tilde{g}_{\theta}d\sigma}w_{B_{\delta^{-1}}}\|_{p}^{2})^{1/2}\\
&=K_{p,n}^{(2)}(\delta)(\sum_{\theta\in\mathcal{P}_{\delta}^{1}}\|(\widehat{g_{\theta}d\sigma}\ast K_{\theta})w_{B_{\delta^{-1}}}\|_{p}^{2})^{1/2} \tag{3}
\end{align*}
To get rid of $K_{\theta}$, we argue as follows. By Holder's inequality and Fubini,
\begin{align*}
\|(\widehat{g_{\theta}d\sigma}\ast K_{\theta})w_{B_{\delta^{-1}}}\|_{p}^{p}&\leq\int_{\mathbb{R}^{n}}(\int_{\mathbb{R}^{n}}|\widehat{g_{\theta}d\sigma}(y)|^{p}|K_{\theta}(x-y)|dy)\|K_{\theta}(x-\cdot)\|_{1}^{p/p'}w_{B_{\delta}^{-1}}(x)dx\\
&\lesssim_{n,p}\int_{\mathbb{R}^{n}}|\widehat{g_{\theta}d\sigma}(y)|^{p}(\int_{\mathbb{R}^{n}}|K_{\theta}(x-y)w_{B_{\delta^{-1}}}(x)dx)dy\\
&=:\|\widehat{g_{\theta}d\sigma}\|_{L^{p}(\tilde{w}_{B_{\delta}^{-1}})}^{p}\
\end{align*}
If we construct the $K_{\theta}$ appropriately, then $\tilde{w}_{B_{\delta^{-1}}}$ is another weight, though not necessarily Fourier supported in $B(0,\delta)$. I'm not sure how to fix this issue.


  [1]: http://arxiv.org/pdf/1403.5335v3.pdf
  [2]: http://mathoverflow.net/questions/211585/localization-arguments-in-the-paper-the-proof-of-l2-decoupling-conjecture