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A bilinear argument of the Fourier restriction (not bilinear Fourier restriction) can be described follows:

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This excerpt is from the book of C. Demeter, Fourier restriction, decoupling, applications. Using the coarea formula, we can reformulate this argument in other $\mathbb R^n$, not just in $\mathbb R^2$:

WLOG let $n=3$, using $E$ as the extension operator of the parabola, $$ \|Ef\|_3^2 = \| (Ef) ^2\|_{3/2}, $$ $T\colon \mathbb R^4 \to\mathbb R^3$ is $$ T(t,s) = (t+s, |t|^2+|s|^2) , $$ then by the coarea formula ($\mathrm J_T$ means the Jacobian of $T$) $$ \begin{aligned} (Ef)^2(x) & = \int_{(t,s)\in\mathbb R^4,t\neq s} f(t)f(s) e\bigl(\,\langle\bar x, (t+s)\rangle+ x_3(|t|^2+|s|^2)\,\bigr) \,d(t,s)\\ & = \int_{\mathbb R^3}\biggl[\int_{t\neq s, T(t,s)=\xi} \frac{f(t)f(s)}{\mathrm{J}_T(t,s)} \,d \mathcal H^1(t,s) \biggr]e(\langle x,\xi\rangle)\,d\xi. \end{aligned} $$ Hence by Hausdorff-Young, $\|(Ef)^2\|_{3/2}\leq \|F\|_3$, where $$ F(\xi) = \int_{t\neq s, T(t,s)=\xi} \frac{f(t)f(s)}{\mathrm{J}_T(t,s)} \,d \mathcal H^1(t,s). $$ By the rank theorem, the level set of $T$ is the union of some curves; since $|(t,s)|\leq1$, the $\mathcal H^1$-measure of the level set is uniformly bounded, hence $$ |F(\xi)| \lesssim \biggl( \int_{t\neq s, T(t,s)=\xi} \Bigl| \frac{f(t)f(s)}{\mathrm{J}_T(t,s)} \Bigr|^3 \,d \mathcal H^1(t,s) \biggr)^{1/3}, $$ and the $L^3$ norm of $F$ is $$ \|F\|_3^3\lesssim \int_{\mathbb R^3}\biggl[\int_{t\neq s, T(t,s)=\xi} \biggl|\frac{f(t)f(s)}{\mathrm{J}_T(t,s)}\biggr|^3 \,d \mathcal H^1(t,s) \biggr]\,d\xi = \int_{\mathbb R^4} \frac{|f(t)f(s)|^3}{|\mathrm{J}_T(t,s)|^2}\,d(t,s). $$ By Hardy-Littlewood-Sobolev estimate (for convenience, we assume that the Hardy-Littlewood-Sobolev inequality holds at the endpoints), and the fact that $$ \mathrm{J}_T(t,s) = c |t-s|, $$ we can deduce that $$ \|F\|_3^3\lesssim \| \,|f|^3\,\|_1 \Bigl\| |f|^3 *\frac1{|\cdot|^2}\Bigr\|_\infty \lesssim \|f\|_3^3 \|\,|f|^3\,\|_3 \lesssim \|f\|_\infty^6, $$ which implies the restriction conjecture. Therefore, there must be some mistakes in it, but I can't figure out where the error lies.

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  • $\begingroup$ @LVFToENlCx In 4-dim, the convolution kernel is locally integratable, hence if $\|f\|_\infty \leq 1$, then $\|F\|_3<\infty$, which can show that $\|F\|_3^3 \lesssim \|f\|_\infty^2$. $\endgroup$ Commented Sep 21 at 17:15
  • $\begingroup$ @LVFToENlCx The $DT(t,s)$ is $\left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 2 t_1 & 2 t_2 & 2 s_1 & 2 s_2 \\ \end{array} \right)$, hence $(DT)(DT)^\top$ is $\left( \begin{array}{ccc} 2 & 0 & 2 \left(s_1+t_1\right) \\ 0 & 2 & 2 \left(s_2+t_2\right) \\ 2 \left(s_1+t_1\right) & 2 \left(s_2+t_2\right) & 4 \left(s_1^2+s_2^2+t_1^2+t_2 ^2\right) \\ \end{array} \right)$, whose det is $8 \left(\left(s_1-t_1\right){ }^2+\left(s_2-t_2\right){}^ 2\right) = C|t-s|^2$. $\endgroup$ Commented Oct 7 at 14:49

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