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

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 $R^{-1}$-neighborhood of the paraboloid into $1\times R^{-1\mathrm{/}2}\times R^{-1}$ curved large-caps. In this article, we prove the sharp ${\mathrm{\ell }}^2(L^{10\mathrm{/}3})$ decoupling estimate for strips in ${\mathbb{R}}^3$. 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.