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 R-2d, d >= 3. These surfaces are defined by a complex curve gamma (z) of simple type, which is given by a mapping of the form z bar right arrow gamma (z) = (z, z(2), ..., z(d-1), phi(z)) where phi(z) is an analytic function on a domain Omega subset of C. This is regarded as a real mapping z = (x, y) bar right arrow gamma (x, y) from Omega subset of R-2 to R-2d. Our results cover the case phi(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 phi(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. (C) 2013 Elsevier Inc. All rights reserved.