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 ${\text{bbR}}^{2d}$, $d\ge 3$. These surfaces are defined by a complex curve $\gamma (z)$ of simple type, which is given by a mapping of the form % $$z\mapsto \gamma (z)=(z,z^2,\mathrm{.}\mathrm{.}\mathrm{.},z^{d-1},\varphi (z))$$ % where $\varphi (z)$ is an analytic function on a domain $\mathrm{\Omega }\subset \text{bbC}$. 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 $\varphi (z)=z^N$ 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 ${\text{bbR}}^d$, $d\ge 3$.