$L^p$-weighted cone restriction estimates

We study Fourier extension estimates for the truncated cone in ${\mathbb{R}}^{n+1}$ with weights satisfying an $m$-dimensional ball condition only at radii between $1$ and $R^{1\mathrm{/}2}$, where $0<m\le n-1$. Within the discrete induction framework of Du–Li–Wang–Zhang, we obtain a family of bounds by combining multilinear refined Strichartz inequalities, narrow decoupling, and Lorentz rescaling. We also give lightplank, lattice, and hybrid lattice–Knapp examples.