Estimates for Fourier transform of measures supported on singular hypersurfaces

We consider hypersurfaces $S\subset \Bbb{R}^3$ with zero Gaussian curvature at every ordinary point with surface measure dS and define the surface measure $d_\mu = \psi(x)dS_(x)$ for smooth function ψ with compact support. We obtain uniform estimates for the Fourier transform of measures co...

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Bibliographic Details
Published in:Turkish journal of mathematics 2007-01, Vol.31 (1), p.1-21
Main Author: IKROMOV, Isroil A
Format: Article
Language:English
Subjects:
Değer biçme
estimate
Fourier dönüşümü
Fourier transformation
Gauss eğriliği
Gaussian curvature
Hiperyüzey
hypersurface
Matematik
Mathematics
Regle yüzey
Ruled surface
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Summary:We consider hypersurfaces $S\subset \Bbb{R}^3$ with zero Gaussian curvature at every ordinary point with surface measure dS and define the surface measure $d_\mu = \psi(x)dS_(x)$ for smooth function ψ 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 $O(|\xi|^{-1/h})$, where h is the height of the phase function. In particular, Fourier transform of measures supported on the exceptional surfaces decays in the order $O(|\xi|^{-1/2})(as |\xi|\rightarrow+\infty)$.
ISSN:1300-0098
1303-6149