Strip Decoupling Inequalities for the Paraboloid in $\mathbb{R}^3$

Abstract: Large-cap or large-ensemble decoupling inequalities were initially studied by Demeter. These are intimately connected to the reverse square-function conjecture and restriction conjecture in Fourier restriction theory. By strips, we mean a partition of the $R<sup>{-1}$-neighborhood of the paraboloid into $1\times R^{-1/2}\times R<sup>{-1}$ curved large-caps. In this article, we prove the sharp $\ell<sup>2(L<sup>{10/3})$ decoupling estimate for strips in $\mathbb{R}<sup>3$. We reduce the strip decoupling inequalities to a planar incidence problem, which is dealt with via two-ends Furstenberg incidence estimates. Our method is inspired by Wang--Wu on the restriction conjecture.