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.

$$ P^{n-1}:=\{ (\xi_1, ..., \xi_{n-1}, \xi_1^2 +...+ \xi_{n-1}^2 ) \in \mathbb{R}^n: |\xi| \leq 1/2 \} $$ 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!


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.