Restriction of Fourier transforms to some complex curves

Abstract

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in ${\mathbb{R}}^{2d}$, $d\ge 3$. These surfaces are defined by a complex curve $\gamma (z)$ of simple type, which is given by a mapping of the form % \[ z\mapsto γ(z) = \big(z, \, z^2,..., \, z^{d-1}, \, ϕ(z) \big) \] % where $\varphi (z)$ is an analytic function on a domain $\Omega \subset \mathbb{C}$. This is regarded as a real mapping $z=(x,y)\mapsto \gamma (x,y)$ from $\Omega \subset {\mathbb{R}}^2$ to ${\mathbb{R}}^{2d}$. Our results cover the case $\varphi (z)=z^N$ for any nonnegative integer $N$, in all dimensions $d\ge 3$. Furthermore, when $d=3$, we have a uniform estimate, where $\varphi (z)$ may be taken to be an arbitrary polynomial of degree at most $N$. These results are analogues of the uniform restricted strong type estimate in \cite{BOS3}, valid for polynomial curves of simple type and some other classes of curves in ${\mathbb{R}}^d$, $d\ge 3$.