Fourier support condition in the paper 'A study guide for the $l^2$ decoupling theorem'

Question

I'm currently reading Bourgain and Demeter's study guide for the $l^2$ decoupling theorem (https://arxiv.org/pdf/1604.06032.pdf). I have some trouble with understanding the proof of Proposition 8.4.

Let me describe the setting. We define the truncated paraboloid $$ \mathbb{P}^{n-1} = \{(\xi_1,\dotsc,\xi_{n-1},\xi_1^2 + \dotsc + \xi_{n-1}^2) : 0 \leq \xi_i \leq 1\} $$ and the $\delta$-neighborhood of $\mathbb{P}^{n-1}$ above some set $U\subset [0,1]^{n-1}$ $$ N_\delta(U)=\{(\xi_1,\dotsc,\xi_{n-1},\xi_1^2+\dotsc+\xi_{n-1}^2+t):(\xi_1,\dotsc,\xi_{n-1})\in U,0\leq t \leq \delta\}. $$

Let us now restrict ourselves to $n=3$ dimensions. Let $L$ be the line in the $(\xi_1,\xi_2)$-plane given by $\eta=0$ (actually the authors choose $\eta = 1$ but this shouldn't change anything due to translation invariance) and let $S_L = \{(\xi_1,\xi_2)\in [0,1]^2:\text{dist}((\xi_1,\xi_2),L)<\frac{C}{K}\}$ be the fattened line for some constant $C>0$. Next, for a function $g:[0,1]^2\rightarrow \mathbb{C}$ we define the extension operator $$ E_{S_L}g(x,y,z)=\int_{S_L}g(\xi_1,\xi_2)e(\xi_1 x + \xi_2y+(\xi_1^2+\xi_2^2)z)d(\xi_1,\xi_2) $$ where $e(z)=e^{2\pi i z}$, which extends $g$ to $\mathbb{P}^2$. Now we fix some $y\in\mathbb{R}$ and define the restricted operator $$ E_{S_L,y}g(x,z):=E_{S_L}g(x,y,z). $$ So now we are in two dimensions.

The issue I have trouble with is the following: In order to obtain a decoupling inequality the authors claim that one can apply Thm. 5.1 to $E_{S_L,y}g$ which requires $E_{S_L,y}g$ being Fourier supported in $N_{C/K}([0,1])$ for some constant $C$. Their argument why this works is, because $E_{S_L,y}g$ is supported in the $O(K^{-1})$ neighborhood of the parabola $\eta=\xi^2$. But $N_{C/K}([0,1])$ is a strict subset of this neighborhood, so this argument is not sufficient to apply Thm. 5.1.

So to be precise, my question is the following:

How can one show, that $E_{S_L,y}g$ has Fourier support in $N_{C/K}([0,1])$?

By the uncertainty principle this is quite clear to me, but I don't know how to make this rigorous. My approach follows the standard ideas: Instead of $E_{S_L,y}g$ I consider $\phi E_{S_L,y}g$ for some Schwartz function $\phi$ which is Fourier supported in the cube $B(0,1/K)$ centered at the origin and with side length $1/K$. I deduce that $\phi E_{S_L,y}g$ is Fourier supported in the $O(K^{-1})$ neighborhood of the parabola $\eta=\xi^2$. And now I would like to apply Thm 5.1 to $\phi E_{S_L,y}g$, which I can't because of the reason discussed above.

Any help is appreciated. Thanks in advance.

Answer 1

The following should work. Depending on $C$, break $g$ into two pieces, $g_1$ which is supported on $\sim C$ of the $K^{-1} \times K^{-1/2}$ rectangles at the ends of the strip $S_L$ and $g_2$ supported on the rest of the $K^{-1} \times K^{-1/2}$ rectangles covering $S_L$. So for example if $C = 1$ take $g_1$ to be supported on the leftmost and rightmost $K^{-1} \times K^{-1/2}$ rectangles covering $S_L$.

With $\phi$ having Fourier support in $B(0, 1/K)$, the support of $g_2$ and the argument you gave shows that $\phi E_{S_L, y}g_2$ is Fourier supported in $N_{C/K}([0, 1])$ and you can apply Theorem 5.1. For $g_1$ use its support, Cauchy-Schwarz, and that the decoupling constant is $\gtrsim_{E, p} 1$ to obtain that \begin{align*} \|E_{S_L, y}g_1\|_{L^{p}_{x, z}([0, K]^2)} &\lesssim \sum_{U' \text{ at the ends}}\|E_{U'}g\|_{L^{p}_{x, z}([0, K])^{2})}\\ & \lesssim_{E, p} \operatorname{Dec}_{2}(K^{-1}, p, 10E)(\sum_{U}\|E_{U}g\|_{L^{p}_{x, z}([0, K])^{2})}^{2})^{1/2}. \end{align*}

Answer 2

For any function $f$, since $\hat{f}(x,y,z)=\int f(\xi_1,\xi_2,\xi_3)e(-\langle (x,y,z), \xi\rangle)\,d\xi$, for fixed $y$ we have that the function $(x,z)\mapsto \hat{f}(x,y,z)$ is the Fourier transform of $h_y(\xi_1,\xi_3)=\int f(\xi)e(-y\xi_2)\,d\xi_2$. Apply this to $f(\xi)=(\chi_Lg)(\xi_1,\xi_2)\delta(\xi_3-(\xi_1^2+\xi_2^2))$, to see that $h_y$ is supported in $N_{C/K^2}\subset N_{C/K}$. By the way, for $g$ bounded, $h_y$ blows-up right on the parabola, but it doesn't have any effect at the end.

I have been trying without success to get an alternative to Theorem 5.1. If I get it, I'd like to post it here.