振荡型积分的有界性质及其在色散方程中的应用

The boundedness theory of oscillatory integral operators was one of the important components in harmonic analysis, and was widely used in the fields of partial differential equations, mathematical physics and so on. 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 of nonlinear dispersion equations in the low order Sobolev spaces. The applicant and his collaborators had obtained a series results on the boundedness of oscillatory type integral operators and their applications to the well-posedness of KdV type dispersion equations. Particularly, they had obtained the boundedness of oscillatory integral operators and their commutators on the weighted Morrey spaces. Also, they had established the local well-posedness for the dispersion generalized periodic KdV equations. This project will be devoted to the further research on the boundedness of oscillatory type integral operators and their applications to the well-posedness of dispersion equations, including establishing the boundedness of oscillatory type integral operators in Hardy type spaces and Morrey type spaces and establishing a criteria to determine the boundedness of certain oscillatory type integral operators; developing the boundedness of operators on Morrey type spaces and obtaining some creative characterizations of these spaces via the boundedness of commutators; establishing the well-posedness of the Cauchy problem for a class of KdV type dispersion equations combining the results obtained above with energy method and number theory.

振荡型积分理论在偏微分方程、数学物理等领域有着广泛应用, 该理论是调和分析重要内容之一. 借助振荡型积分算子在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空间中的适定性. 申请人与合作者已获得关于振荡型积分算子的有界性质及其在KdV型色散方程适定性中应用的一些结果. 特别地，研究了振荡积分算子及其交换子在加权 Morrey 空间中的有界性质；得到广义色散周期KdV 方程低正则条件下的局部适定性. 本项目拟深入研究振荡型积分算子有界性质及其在色散方程适定性中的应用，其中包括建立振荡型积分算子在Hardy型空间和Morrey型空间中的有界性质以及有界性判别法则；发展Morrey型空间的算子有界性理论并得到该类空间的交换子刻画；以上得到的有界性理论与能量方法和数论知识相结合解决一类KdV型色散方程 Cauchy问题的适定性.

振荡型积分理论和Morrey型空间理论在偏微分方程、数学物理等领域有着广泛应用, 该理论是调和分析重要内容之一. 借助振荡型积分算子在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 空间中的适定性. 申请人与合作者已获得关于振荡型积分算子的有界性质及其在KdV 型色散方程适定性中应用的一些结果. 特别地，研究了振荡积分算子及其交换子在加权 Morrey 空间中的有界性质；得到广义色散周期KdV 方程低正则条件下的局部适定性. 本项目深入研究了振荡型积分算子(包括单边情形)有界性质、Morrey 型空间刻画和一类周期色散方程适定性，其中包括建立振荡型积分算子在Hardy 型空间和Morrey 型空间中的有界性质以及有界性判别法则；发展了Morrey 型空间的算子有界性理论并得到该类空间(C-Z奇异积分算子、分数次积分算子和Hardy型算子)的交换子刻画；能量方法和数论知识相结合得到 一类周期 KdV 型色散方程（周期 Benjamin方程） Cauchy 问题的整体适定性和整体光滑性.