On the restriction of the Fourier transform to a conical surface

Abstract:

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