Weighted restriction theorems for space curves ✩

Consider a nondegenerate Cn curve γ (t) in Rn, n 2, such as the curve γ0(t) = (t, t2, . . . , tn), t ∈ I , where I is an interval in R. We first prove a weighted Fourier restriction theorem for such curves, with a weight in aWiener 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. © 2007 Elsevier Inc. All rights reserved