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 $γ(z)$ 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 $ϕ(z)$ is an analytic function on a domain $Ω\subset \bbC$. This is regarded as a real mapping $z=(x,y) \mapsto γ(x,y)$ from $Ω\subset \bbR^2$ to $\bbR^{2d}$. Our results cover the case $ϕ(z) = z^N$ for any nonnegative integer $N$, in all dimensions $d\ge 3$. Furthermore, when $d=3$, we have a uniform estimate, where $ϕ(z)$ 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 $\bbR^d$, $d\ge 3$.

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