Wiener共合空间与模空间中的若干问题及其应用

Wiener共合空间是由Wiener在研究Tauberian定理时首先引入. 20世纪80年代,Feichtinger在此基础上引入了另一类函数空间——模空间. 由于这两类空间在偏微分方程的正则性、概率统计等研究领域中的重要作用，所以，研究各类算子在Wiener共合空间、模空间上的有界性是一项具有重大理论意义和应用价值的工作. 项目将致力于研究调和分析中三类重要的算子——拟微分算子、幺模Fourier乘子、C-Z奇异积分. 在这个项目中，我们主要研究以下几方面内容：其一，寻找象征函数的最小正则性条件，使得多线性拟微分算子在乘积Wiener共合空间、α模空间上有界；其二，考虑幺模Fourier乘子在这两类空间上的最佳渐近估计；其三，深入研究C-Z奇异积分在Wiener共合空间、模空间上的有界性. 最后，探究上述得到的结论在偏微分方程、概率统计等学科中的应用.

The Wiener amalgam space is constructed by Wiener in the study of Tauberian theorem. Based on it, in 1980s, Feichtinger introduced another space, the Modulation space. They play an important role in the regularity problem of partial differential equations, probability statistics, and so on. Thus,the study of operators on these spaces is one of the most active subjects in modern harmonic analysis. In this project, we will focus on three important operators in modern harmonic analysis. They are pseudodifferential operators, unimodular Fourier multipliers, Calderón-Zygmund singular integrals. Firstly, we will investigate the minimum regularity condition of the symbol function for the multilinear pseudodifferential operators to be bounded on product Wiener amalgam spaces or α modulation spaces. Secondly, we will consider the optimal asymptotic estimates for unimodular Fourier multipliers on these spaces. Thirdly, we will study the boundedness of Calderón-Zygmund singular integrals on the Wiener amalgam space and the modulation space. Finally, we will also focus on the applications of our theorems in other fields, such as partial differential equations, probability statistics, and so on.