Restriction of the Fourier transform to some complex curves

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in R 2 d , d ≥ 3 . These surfaces are defined by a complex curve γ ( z ) of simple type, which is given by a mapping of the form z ↦ γ ( z ) = ( z , z 2 , … , z d − 1 , ϕ ( z ) ) where ϕ ( z ) is an analytic function on a domain Ω ⊂ C . This is regarded as a real mapping z = ( x , y ) ↦ γ ( x , y ) from Ω ⊂ R 2 to R 2 d . sults cover the case ϕ ( z ) = z N for any nonnegative integer N , in all dimensions d ≥ 3 . The main result is a uniform estimate, valid when d = 3 , where ϕ ( z ) may be taken to be an arbitrary polynomial of degree at most N . It is uniform in the sense that the operator norm is independent of the coefficients of the polynomial. These results are analogues of the uniform restricted strong type estimates in  [5], valid for polynomial curves of simple type and some other classes of curves in R d , d ≥ 3 .