Time-Spcae Estimates Of Higher Order Schr(?)dinger Equation

Harmonic analysis which originated from the work of some famous mathematicianssuch as Euler and Fourier has developed into one of the most interesting subjects of modernmathematics in the following 200 years. These powerful methods have a wider range ofapplications in many other subjects of mathematics and are especially successful for partialdifferential equations.The applications in partial differential equations can be presented in the following twoaspects: The first one is its application to elliptic partial differential equation problems withboundary value. In Lip border region, Stein together with his partners achieved the L~pestimates and regularities of the solution of second order elliptic equations with boundaryvalue.And another aspect is that harmonic analysis plays an important role in the researchof evolution equations.The L~p - L~q estimates and the corresponding time-space estimates of the linear equa-tions provide us with new research tools in dealing with nonlinear equation problem. Forexample: with time-space estimates and some nonlinear estimates,we can establish manyuseful estimates of nonlinear Schro¨dinger equation.This thesis will be organized as follows.In the first chapter,the author will give someintroduction of time-space estimates.Then in the following chapter,the author provedthe time-space estimate of higher order Schro¨dinger equation on Besov spcae.In the lastchapter,the auther gives some application of the locally smoothing effects of higher orderSchro¨dinger equations.