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I am currently reading a paper "The proof of the $\ell^2$ decoupling conjecture" by Bourgain and Demeter. I was wondering if someone could explain me one of the arguments in their paper. First I will introduce necessary notations.

Let
$$ P^{n-1}:=\{ (\xi_1, ..., \xi_{n-1}, \xi_1^2 +...+ \xi_{n-1}^2 ) \in \mathbb{R}^n: |\xi| \leq 1/2 \} $$ and 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 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 $c + [- \frac{\delta^{1/2}}{2},\frac{\delta^{1/2}}{2} ]$ with $c = \frac{\delta^{1/2}}{2} \mathbb{Z}^{n-1} \cap [-1/2, 1/2]^{n-1}$.

The part I would like an explanation is the following: Assume $f$ is Fourier supported in $\mathcal{N}_{\delta}$. Then by using a partition of unity write $$ f = \sum_{ \theta \in \mathcal{P}_{\delta} } \tilde{f}_{\theta} $$ with $\tilde{f}_{\theta} = f_{\theta} *K_{\theta}$ Fourier supported in $\frac{9}{10} \theta$ with $\| K_{\theta} \|_1 \ll 1$.

I wasn't sure how this part worked. I would greatly appreciate some explanation. Thank you very much!

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The problem here is technical and you can think of $\tilde{f}_{\theta}$ as being $f_{\theta}$. The set of functions $\{1_{\theta}\}$ don't create a "smooth" partition of unity (in fact $\sum_{\theta}1_{\theta}=2$ away for the boundary of $[-1/2,1/2]^{n-1}$), then they replace $\{1_{\theta}\}$ by a smooth partition, say by function $\eta_{\theta}$ such that $1_{\theta}\eta_{\theta}=\eta_{\theta}$, that's why they are supported in $c\theta$ for $c<1$, they write $\frac{9}{10}$.

Note that $\mathcal{F}^{-1}(\hat{f}1_{\theta}\eta_{\theta})=f_{\theta}*\mathcal{F}^{-1}(\eta_{\theta})$, where $\mathcal{F}$ is Fourier transform and $K_{\theta}=\mathcal{F}^{-1}(\eta_{\theta})$.

There are other ways of decomposing the function in wave packets, see for example Tao or Guth, but in any case the philosophy is the same.

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