【作者】 邓超；

【导师】 郑权；

【作者基本信息】 华中科技大学 ， 应用数学， 2008， 硕士

【摘要】 调和分析的起源可追溯到Euler和Fourier等著名数学家的研究.经过其后二百多年的发展,调和分析已成为当今数学的核心学科之一.调和分析方法源于分析的诸多领域,并且在数学的很多领域中都有着广泛的应用.调和分析方法在偏微分方程研究中的应用主要体现在以下两个方面:一是在研究椭圆型方程边值问题中的应用. Stein及其学派解决了具有Lip边界的区域中二阶椭圆型方程边值问题Lp估计以及解的正则性问题;二是它在研究发展型方程的解的相关问题中的有重要应用.以振荡积分估计为基础建立起来的线性发展方程解的Lp ? Lq估计以及相应的时空估计,为非线性发展方程的研究提供了崭新的研究手段.例如:借助于时空估计方法和非线性项的相关估计可以建立非线性Schro¨dinger方程的Cauchy问题在低阶Sobolev空间的适定性理论.本文将利用调和分析理论中的一般可微函数理论和线性发展方程解的时空估计研究高阶Schro¨dinger方程解在Besov空间中的时空估计,并利用象征为一类多项式时高阶Schro¨dinger方程解的局部光滑性估计研究Carleson猜想.这两个结果分别作为本文的第二、三章.更多还原

【Abstract】 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.更多还原