Restriction of Fourier transforms to some complex curves

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in $\bbR^{2d}$, $d\ge 3$. These surfaces are defined by a complex curve $\gamma(z)$ of simple type, which is given by a mapping of the form % \[ z\mapsto \gamma (z) = \big(z, \, z^2,..., \, z^{d-1}, \, \phi(z) \big) \] % where $\phi(z)$ is an analytic function on a domain $\Omega \subset \bbC$. This is regarded as a real mapping $z=(x,y) \mapsto \gamma(x,y)$ from $\Omega \subset \bbR^2$ to $\bbR^{2d}$. Our results cover the case $\phi(z) = z^N$ for any nonnegative integer $N$, in all dimensions $d\ge 3$. Furthermore, when $d=3$, we have a uniform estimate, where $\phi(z)$ 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 $\bbR^d$, $d\ge 3$.