Estimates for Fourier Transform of Measures Supported on Singular Hypersurfaces

Authors : İsroil A. IKROMOV

Pages : 1-21

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Publication Date : 0000-00-00

Article Type : Research Paper

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 = yinsert ignore into journalissuearticles values(x);dSinsert ignore into journalissuearticles values(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 Oinsert ignore into journalissuearticles values(|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 Oinsert ignore into journalissuearticles values(|x|-1/2); insert ignore into journalissuearticles values(as |x| \to +\infty);.
Keywords : Oscillatory Integrals, oscillation index, singular hypersurfaces, curvature