解析Morrey空间中的几个基本问题

Morrey spaces were firstly introduced by mathematician Ch. Morrey in 1938. Since Morrey spaces have a finer structure , they were applied to the study of harmonic analysis, potential theory and partial differential equations. Several important results of real Morrey spaces have obtained by investigators, but only a few results on analytic Morrey spaces have been obtained. Like the real case of Morrey spaces, complex analytic Morrey spaces also have its own interesting. In this project, we devoted to study several basic problems on analytic Morrey space, by using the theory of analytic functions, techniques in harmonic analysis and theory of Banach spaces. Firstly, we study the boundedness of the Volterra type operators on analytic Morrey spaces. Secondly, we investigate the interpolation problem on analytic Morrey spaces. Finally, Analogous to Hardy spaces and BMOA space, we establish the relationship between Brownian Motion and analytic Morrey space. The study of analytic Morrey spaces is of great value in the applications in complex differential equation and signal analysis.

Morrey空间是由数学家Ch. Morrey 于1938年首次引入. 由于Morrey空间中函数具有较好的正则性, 使得Morrey空间成为调和分析, 位势理论和偏微分方程中的重要研究对象. 国内外学者已在Morrey空间上取得了一系列重要成果, 复解析Morrey空间的研究较少. 与实变量Morrey空间一样, 解析Morrey空间的理论研究也非常有意义. 本项目主要是利用复分析中解析函数理论, 调和分析的技巧和Banach空间理论, 来研究解析Morrey空间的几个基本问题. 首先研究解析Morrey空间的Volterra型积分算子的有界性. 其次研究解析Morrey 空间作为Banach空间的插值定理. 最后, 类似于Hardy空间和BMOA空间, 建立Morrey空间与复布朗运动的联系. 解析Morrey空间的理论研究在复微分方程以及信号分析方面有一定的应用价值.

本项目主要研究解析Morrey空间中的一些基本问题。 在项目执行期间，项目组首先给出了Dirichlet 型空间的新型嵌入定理. 作为应用可得出Morrey型空间F(p,p-1,1)空间的嵌入定理. 其次, 本项目完成了Bergman-Morrey空间的嵌入定理和积分算子的有界性问题. 最后, 本项目组研究了相关解析函数空间上的积分算子刚性和奇异性问题.