Wiener 共合空间上奇异积分性质的研究

Wiener共合空间是时间频率分析和相空间分析中一类重要的函数空间.基于Wiener共合空间的良好性质，本项目主要讨论以下几方面的问题：第一，利用Fourier变换，振荡积分估计等现代调和分析的方法与技巧，讨论强奇异卷积算子在Wiener共合空间上的有界性. 另外，根据子流形所满足的几何特征，进一步讨论沿子流形的强奇异积分算子在Wiener共合空间上的有界性；第二，利用二进制分解和特征分解的方法，讨论当特征属于Sobolev空间时，相应的拟微分算子在Wiener共合空间上的各种性质；第三，根据Wiener共合空间与调幅空间之间的关系，进一步讨论上述具有非正则特征的拟微分算子在广义调幅空间上的有界性. 本课题的研究将经典调和分析、时间频率分析和偏微分方程紧密的结合在一起，对促进学科之间的交叉发展产生积极的意义.

Wiener amalgam spaces is a class of important function spaces in time-frequency analysis and phase-space analysis. Based on the good properties of wiener amalgam spaces, this project mainly includs the following detailed contents. Firstly, using some modern harmonic analysis methods and techniques,such as Fourier transformation、 oscillatory integral estimates, we will investigate the mapping properties of strongly singular convolution operators on wiener amalgam spaces. Furthermore, according to the geometric characteristics of the submanifolds, we will also discuss the boundedness of the strongly singular integral operator along submanifolds on wiener amalgam spaces. Secondly, utilizing binary decompositions and symbol decompositions, we will study the properties of the pseudodifferential operators with symbols on sobolev spaces on wiener amalgam spaces.Thirdly,in accordance with the relationships between wiener amalgam spaces and modulation spaces, we will establish the bounededness properties of pseudodifferential operators with irregular symbols on general modulation spaces. This project combines the classical harmonic analysis、time-frequency analysis with partial differential equations closely. It generates positive meanings to the promotion of cross-development of the subjects.