单边算子有界性和KdV型色散方程的适定性研究

Study of the boundedness of one-sided operators and the well-posedness of KdV type dispersion equations

数学和物理中许多重要问题均可归结为算子在某些函数空间中的有界性质.奇异积分算子有界性质的研究是调和分析理论的核心课题之一,由此发展起来的各种方法和技巧已广泛应用于偏微分方程的研究.借助奇异积分算子在Lebesgue空间或Morrey型空间中建立的时空估计和半群理论,可以得到非线性色散方程在低阶Sobolev空间中Cauchy问题的适定性.本文首次定义一类单边振荡奇异积分算子并研究该类算子的经典加权有界性质.受经典交换子刻画理论的启发,本文首次引入Morrey空间的交换子刻画理论.利用不同于常规极大函数的方法得到两类象征函数在Morrey空间中的交换子刻画.以上结果为偏微分方程的研究提供了新的工具.最后,结合能量方法和数论知识,本文解决几类KdV型色散方程的适定性问题.

Many important issues in Mathematics and Physics can often attribute to the boundedness of operators on some function spaces. The study for the boundedness of singular integral is one of the important components in Harmonic analysis. A variety of methods and techniques in Harmonic analysis have been widely used in the study of partial differential equations. By the space-time estimates for oscillatory type integral operators on Lebesgue spaces or Morrey type spaces and semigroup theory, one can obtain the well-posedness for the Cauchy problem of nonlinear dispersion equations in the low order Sobolev spaces. We give the definitions of one kind of one-sided oscillatory singular integral operators and study some classical boundedness of these operators. Inspired by the classical results for the characterizations via commutators, we introduce the idea of characterizations via commutators on Morrey spaces and obtain characterizations via commutators with symbol belonging to two kinds of function spaces on these spaces. The results obtained above offer some new tools to the study of partial differential equations. Finally, combining energy method and Number theory, we establish the well-posedness of the Cauchy problem for a class of KdV type dispersion equations.