The Properties And Applications Of Fourier Transform Of Lebesgue Measure Of Surface

Fourier transforms of surfaces play an important role in many fields of harmonic analysis,such as decaying and limitation.The geometric properties of surfaces themselves also have a significant impact on the properties of Fourier transforms of Lebesgue measures on surfaces.This thesis mainly explores the properties and applications of Fourier transformation of Lebesgue measures on curved surfaces.The research on threedimensional elliptic paraboloids and d-1 dimensional ellipsoids is mainly based on generalized spherical coordinate transformation and coordinate rotation transformation.Using Van der Korput lemma to analyze the decay of Bessel functions,using the properties of oscillatory integrals and the decay estimation of Bessel functions to derive the relevant properties of curved surfaces.The main work includes the following aspects: Firstly,we investigate the three properties of the first type oscillatory integral and obtain that under certain conditions,the first type oscillatory integral exhibits attenuation.Secondly,based on the three properties of oscillatory integration,the attenuation estimation and asymptotic expansion of Bessel functions are studied.In addition,other integral expressions of Bessel functions are derived through the Bessel equation.Then,by using generalized spherical coordinate transformation and coordinate rotation transformation,the Fourier transform of the Lebesgue measure of the three-dimensional elliptical paraboloid and d-1 dimensional ellipsoid is transformed into the integral form of the Bessel function,and the attenuation estimation of the Fourier transform of the Lebesgue measure of the surface is derived.Finally,the Fourier limit of hypersurface measure is studied by using Littlewood Paley theory,and the exact convergence estimate of the solvability coefficient is obtained.