Weighted restriction theorems for space curves

Abstract

Consider a nondegenerate C-n curve gamma(t) in R-n, n >= 2, such as the curve gamma(0)(t) = (t, t(2),..., t(n)), t epsilon I, where I is an interval in R. We first prove a weighted Fourier restriction theorem for such curves, with a weight in a Wiener amalgam space, for the full range of exponents p, q, when I is a finite interval. Next, we obtain a generalization of this result to some related oscillatory integral operators. In particular, our results suggest that this is a quite general phenomenon which occurs, for instance, when the associated oscillatory integral operator acts on functions f with a fixed compact support. Finally, we prove an analogue, for the Fourier extension operator (i.e. the adjoint of the Fourier restriction operator), of the two-weight norm inequality of B. Muckenhoupt for the Fourier transform. Here I may be either finite or infinite. These results extend two results of J. Lakey on the plane to higher dimensions. (c) 2007 Elsevier Inc. All rights reserved.