Sharp Bilinear Decoupling under Shell-Type Restrictions and Applications

Abstract

We establish a sharp bilinear decoupling inequality under shell-type Fourier support restrictions. The optimal coefficient exhibits a transition at $N_1=N_2^2$ and is strictly smaller than the corresponding thick-annulus coefficient of Fan--Staffilani--Wang--Wilson [FSWW18]. The proof combines a lossless $L^2$ cap--plate decomposition, a one-direction linear lifting argument, localized shell-type linear decoupling, and a complementary localization of the second factor. For applications, we also derive a corresponding bilinear Strichartz estimate, toral eigenfunction estimates, mixed additive-energy bounds on lattice spheres, and a separated nonlinear smoothing result for the periodic Zakharov system.

39 pages, 7 figures