Dergiler / Turkish Journal of Mathematics / 2007 / Cilt: 31 - Sayı: 1
Estimates for Fourier Transform of Measures Supported on Singular Hypersurfaces
- Sayfa
- 1–21
- DOI
- —
Özet
We consider hypersurfaces S \subset \RR3 with zero Gaussian curvature at every ordinary point with surface measure dS and define the surface measure dm = y(x)dS(x) for smooth function y with compact support. We obtain uniform estimates for the Fourier transform of measures concentrated on such hypersurfaces. We show that due to the damping effect of the surface measure the Fourier transform decays faster than O(|x|-1/h), where h is the height of the phase function. In particular, Fourier transform of measures supported on the exceptional surfaces decays in the order O(|x|-1/2) (as |x| \to +\infty).
Abstract
We consider hypersurfaces S \subset \RR3 with zero Gaussian curvature at every ordinary point with surface measure dS and define the surface measure dm = y(x)dS(x) for smooth function y with compact support. We obtain uniform estimates for the Fourier transform of measures concentrated on such hypersurfaces. We show that due to the damping effect of the surface measure the Fourier transform decays faster than O(|x|-1/h), where h is the height of the phase function. In particular, Fourier transform of measures supported on the exceptional surfaces decays in the order O(|x|-1/2) (as |x| \to +\infty).