On the restriction of the Fourier transform to a conical surface

Abstract

Let Γ \Gamma be the surface of a circular cone in R 3 {{\mathbf {R}}^3} . We show that if 1 ⩽ p > 4 / 3 1 \leqslant p > 4/3 , 1 / q = 3 ( 1 − 1 / p ) 1/q = 3(1 - 1/p) and f ∈ L p ( R 3 ) f \in {L^p}({{\mathbf {R}}^3}) , then the Fourier transform of f f belongs to L q ( Γ , d σ ) {L^q}(\Gamma ,d\sigma ) for a certain natural measure σ \sigma on Γ \Gamma . Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint p = 4 / 3 p = 4/3 , with logarithmic growth of the bound as the thickness of the annulus tends to zero.