若干与模空间相关的问题及其应用

Modulation space is a class of very important function space, which not only shows its advantages in boundedness of operators, but also is applied significantly to the field of time-frequency analysis and wavelet analysis. Moreover, in recent years, with the appearance of new definition, the advantages of its applications in boundedness of operators have been further expanded. It also can perfect the theory of function spaces and further be discovered its wide application in partial differential equations. This project is dedicated to research on the properties of modulation space, boundedness of some operators on modulation space and their applications in partial differential equations. Firstly, this project aims to research the equivalent characterizations of modulation space and Wiener amalgam space, which helps to estimate non-integer power function and study of well posedness of some partial differential equations. Secondly, this project tries to study algebraic property of α-modulation space, which helps to characterize the domain of well posedness of Heat equation in α-modulation space. Thirdly, this project targets on boundedness of unimodular multipliers on modulation space, which helps to characterize the domain of well posedness of nonlinear Schrödinger equation and nonlinear Klein-Gordon equation in modulation space. Lastly, this project attempts to study the restrictions of Fourier transforms to curves by wave packet decomposition, and tries to improve the restrictions of Fourier transforms on surface.

模空间是函数空间理论中一类非常重要的空间，其不仅在研究算子有界性时有优势，还在时频分析和小波分析领域有着重要应用。近几年，随着新定义的出现，其在研究算子有界性时的优势被进一步扩大，也完善了函数空间理论，还被发现在偏微分方程领域有着广泛应用。本项目致力于研究模空间的性质、算子在模空间的有界性及其在偏微分方程中的应用。首先，本项目拟研究模空间以及Wiener amalgam空间的等价刻画，这些理论有助于研究非整数次幂函数在模空间的估计以及偏微分方程的适定性；其次，本项目将研究α-模空间的代数性质，该结论有助于刻画热方程在α-模空间的适定性范围；再次，本项目将研究幺模乘子在模空间的有界性问题，该理论有助于刻画非线性薛定谔方程和非线性Klein-Gordon方程在模空间适定性的范围；最后，本项目将尝试用波包分解研究曲线上的傅里叶限制性定理，试图对曲面上的限制性定理有所改进。

算子理论和函数空间理论是调和分析研究的两大重要领域。本项目主要研究模空间的性质和算子在模空间有界性的刻画，具体为：迭代球面平均算子在模空间、α-模空间、Besov空间、Tirebel空间等函数空间有界性的指标范围的充要条件、Sobolev空间和Morrey空间的范数刻画。本项目的主要研究成果如下：（1）得到了迭代球面平均算子在模空间有界性的全指标范围的充要条件；（2）分别得到了迭代球面平均算子在α-模空间、Besov空间、Tirebel空间有界的充分条件和必要条件，并利用Tirebel空间有界性的结论改进了该算子在Lebesgue空间有界性的结果；（3）用带广义测度的Littlewood-Paley函数刻画了Sobolev空间的范数，首次将该刻画推广到非紧支集的情况；（4）带粗糙核奇异积分算子的端点有界性的推广；（5）用带位势的Schrodinger方程的解刻画了Morrey空间的范数。本项目的研究不仅充实了调和分析理论框架，还丰富了算子理论和函数空间理论在逼近论、随机过程等学科的应用。