Decay Rates of Fourier Transforms of Curves
B. P. Marshall · Transactions of the American Mathematical Society · 1988
Let $d\mu$ be a smooth measure on a nondegenerate curve in ${{\mathbf {R}}^n}$. This paper examines the decay rate of spherical averages of its Fourier transform $\widehat {d\mu }$. Thus estimates of the following form are considered: \[ {\left ( {\int _{{\sum _r}} {|\widehat {d\mu }(\xi {|^p}d\xi } } \right )^{1/p}} \leqslant C{r^{ - \sigma }}||f||\] where ${\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\}$.