Decay rates of Fourier transforms of curves

Marshall, B. P.

doi:10.1090/S0002-9947-1988-0948194-2

Decay rates of Fourier transforms of curves
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Trans. Amer. Math. Soc. 310 (1988), 115-126 Request permission

Abstract:

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 ${\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\}$.

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