Restriction estimates for hyperbolic cone in high dimensions

Abstract

In this paper, we obtain the L p L^p restriction estimates for the truncated conic surface Σ = { ( ξ ′ , ξ n , − ξ n − 1 ⟨ ξ ′ , N ξ ′ ⟩ ) : ( ξ ′ , ξ n ) ∈ B n − 1 ( 0 , 1 ) × [ 1 , 2 ] } \begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*} with N = I n − 1 − m ⊕ ( − I m ) N=I_{n-1-m}\oplus (-I_m) for m ≤ ⌊ n − 3 2 ⌋ m\leq \lfloor \tfrac {n-3}2\rfloor provided p > 2 ( n + 3 ) n + 1 p>\tfrac {2(n+3)}{n+1} . The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.