Decay rates of Fourier transforms of curves

Abstract

Let d μ d\mu be a smooth measure on a nondegenerate curve in R n {{\mathbf {R}}^n} . This paper examines the decay rate of spherical averages of its Fourier transform d μ ^ \widehat {d\mu } . Thus estimates of the following form are considered: \[ ( ∫ ∑ r | d μ ^ ( ξ | p d ξ ) 1 / p ⩽ C r − σ | | f | | {\left ( {\int _{{\sum _r}} {|\widehat {d\mu }(\xi {|^p}d\xi } } \right )^{1/p}} \leqslant C{r^{ - \sigma }}||f|| \] where ∑ r = { ξ ∈ R n : | ξ | = r } {\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\} .