A single scale smooth Alpert trilinear characterization of the Fourier extension conjecture on P 2 $\mathbb {P}^{2}$ double struck upper P squared

Source abstract

Abstract We show that for every 0 &lt; δ &lt; 1 $0<\delta <1$ 0 less than delta less than 1 , the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality of the form: ‖ E Q s , U 1 η f 1 E Q s , U 2 η f 2 E Q s , U 3 η f 3 ‖ L q 3 ( B R 3 ( 0 , 2 s 1 − δ ) ) ≤ C δ , ε , ν 2 ε s ‖ f 1 ‖ L ∞ ( U 1 ) ‖ f 2 ‖ L ∞ ( U 2 ) ‖ f 3 ‖ L ∞ ( U 3 ) , $$\begin{align*}&\left\Vert \mathcal{E}\mathsf{Q}_{s,U_{1}}^{\eta}f_{1}\ \mathcal{E} \mathsf{Q}_{s,U_{2}}^{\eta}f_{2}\ \mathcal{E}\mathsf{Q}_{s,U_{3}}^{\eta} f_{3}\right\Vert _{L^{\frac{q}{3}}\left( B_{\mathbb{R}^{3}}\left( 0,2^{\frac{s}{1-\delta}}\right) \right) }\\ &\quad \leq C_{\delta,\varepsilon,\nu}2^{\varepsilon s}\left\Vert f_{1}\right\Vert _{L^{\infty}\left( U_{1}\right) }\left\Vert f_{2} \right\Vert _{L^{\infty}\left( U_{2}\right) }\left\Vert f_{3}\right\Vert _{L^{\infty}\left( U_{3}\right) }, \end{align*}$$ which is a variant of the analogous multiscale trilinear inequality in Rios and Sawyer (2025, Equivalence of linear and trilinear Kakeya conjectures in three dimensions ), where the smooth Alpert projections Q s , U k η $\mathsf {Q}_{s,U_{k}}^{\eta }$ sans serif upper Q Subscript s comma upper U Sub Subscript k Subscript Superscript eta were replaced more generally with Q s k , U k η $\mathsf {Q}_{s_{k},U_{k}}^{\eta }$ sans serif upper Q Subscript s Sub Subscript k Subscript comma upper U Sub Subscript k Subscript Superscript eta , where s k $s_{k}$ s Subscript k was close to s $s$ s .