On the restriction of the Fourier transform to curves: endpoint results and the degenerate case

Abstract

For smooth curves Γ \Gamma in R n {{\mathbf {R}}^n} with certain curvature properties it is shown that the composition of the Fourier transform in R n {{\mathbf {R}}^n} followed by restriction to Γ \Gamma defines a bounded operator from L p ( R n ) {L^p}({{\mathbf {R}}^n}) to L q ( Γ ) {L^q}(\Gamma ) for certain p , q p,q . The curvature hypotheses are the weakest under which this could hold, and p p is optimal for a range of q q . In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.