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I'm reading Bourgain and Demeter's paper https://arxiv.org/abs/1403.5335 "The proof of the $l^2$ decoupling conjecture".

On page 1 the paper says, let $P^{n-1}=\{(\xi_1,...,\xi_{n-1},\xi_1^2+...+\xi_{n-1}^2)\in\mathbb{R}^n:|\xi_i|\leq 1/2\}$, and $N_\delta$ be the $\delta$ neighborhood of $P^{n-1}$ and let $P_\delta$ be a finitely overlapping cover of $N_\delta$ with curved regions $\theta$ of the form $$\theta=\{(\xi_1,...,\xi_{n-1},\eta+\xi_1^2+...+\xi_{n-1}^2):(\xi_1,...,\xi_{n-1})\in C_\theta,|\eta|\leq2\delta\}$$ where $C_\theta$ runs over all cubes $c+[-\frac{\delta^{1/2}}{2},\frac{\delta^{1/2}}{2}]^{n-1}$ with $c\in\frac{\delta^{1/2}}{2}\mathbb{Z}^{n-1}\cap [-1/2,1/2]^{n-1}$. We will denote by $f_\theta$ the Fourier restriction of $f$ to $\theta$.

My question is: how do we know the Fourier restriction on such $\theta$ is bounded on $L^p$?

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    $\begingroup$ The content of their decoupling theorem is that $||f||_{p}$ can be bounded by an an expression involving the functions $||f_\theta||_{p}$. If one of the $||f_\theta||_{p}$ is infinite the inequality has no content. That said, by Hausdorff-Young we have $||\hat{g} ||_{p} \leq ||g||_{p'}$ for $p \geq 2$ which shows $||f_\theta||_{p}$ is finite if $p\geq 2$ and its Fourier transform is, say, integrable and supported on in a finite ball. $\endgroup$
    – Mark Lewko
    Commented Feb 24, 2023 at 6:12
  • $\begingroup$ You might also find arxiv.org/pdf/1604.06032.pdf helpful. $\endgroup$
    – Mark Lewko
    Commented Feb 24, 2023 at 6:14

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