Two Problems Of Oscillatory Integrals And Higher-order Schr(?)dinger Equations

Oscillatory integrals have been an essential part of harmonic analysis. Great impetus for the study of oscillatory integrals came with their applications to the as-ymptotics of Fourier transforms of special functions, Fourier integral operators and pseudo-differential operators. In last twenty years, oscillatory integrals play a more and more important role in the study of linear and nonlinear partial differential equations. Some topics of partial differential equations have been new impetus for the study of oscillatory integrals.Since the 1920s the theory of Schr(o|¨)dinger operators has constantly been a central studied subject of modern mathematical physics. Moreover, space-time estimates, Kato local smoothing estimates and maximal operator estimates have been important topics in the recent twenty years because the investigation of those topics has abundant applied backgrounds as well as very strong theoretic meaning.The main purpose of this thesis is to study the estimate of an oscillatory integral with a parameter and the local smoothing estimate for a higher-order Schr(o|¨)dinger equation.This thesis consists of two parts: the first part studies the estimates of oscillatory integrals involving phase functions with a parameter and degenerate critical points; The second part studies the local smoothing estimate for the solution of a higher-order Schr(o|¨)dinger equation. In the first part, backgrounds and recent results of the problem are presented firstly. Then two estimates of phase functions are proved; Finally the main results are given by virtue of the conclusions already acquried. In the second part, backgrounds and recent results of the problem are presented firstly. Then the local smoothing estimate is obtained by using the spherical harmonic decomposition of L~2 and Hankel transform.