Restriction of Fourier transforms to some complex curves

TL;DR

This paper establishes Fourier restriction estimates for certain complex curves in higher-dimensional real spaces, extending previous results to more general polynomial and simple type curves in dimensions three and above.

Contribution

It proves new Fourier restriction estimates for complex curves of simple type, including polynomial cases, in all dimensions d ≥ 3, with uniform bounds when d=3.

Findings

Established restriction estimates for complex curves in R^{2d}

Extended results to polynomial curves of degree N in dimension 3

Provided uniform estimates for d=3 cases

Abstract

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 $\bbR^d$, $d\ge 3$.