Restriction of the Fourier transform to some complex curves

Abstract

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in R2d, 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,z2,…,zd−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⊂R2 to R2d.Our results cover the case ϕ(z)=zN 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 Rd, d≥3.