Sobolev空间与分数次光滑函数空间的迹及其应用

函数空间的迹及其迹空间的研究是函数空间实变理论中的重要课题。本项目拟在申请人及其合作者已有关于Sobolev空间、Besov空间等函数空间迹及其迹空间刻画的研究基础上，充分研究各种底空间的几何与分析性质，运用来自调和分析、泛函分析和几何分析等相关学科的一些基本理论与技巧，进一步研究Sobolev空间与分数次光滑函数空间的迹及其迹空间的刻画问题。具体研究内容如下：一，研究欧氏空间上临界权值的Sobolev空间、Besov空间以及其他分数次光滑函数空间的迹的存在性及其迹空间的刻画；二，在正则树上找到内蕴的仅依赖度量和测度的参数来刻画其上Sobolev函数的迹函数的存在性及可积性；三，将在欧氏空间以及正则树上的迹理论研究推广应用到一般度量空间上。这些研究以及迹理论在连接区域函数空间与对应边界函数空间时的桥梁作用，将为度量空间上的分析提供更多的研究方法，并在调和分析和偏微分方程等学科上有广泛应用。

Traces and trace spaces for function spaces are important topics in the research of function spaces. This project aims, based on the existing research works of the applicant and his collaborators about the characterization of traces and trace spaces for Sobolev spaces, Besov spaces and so on, via thoroughly revealing the geometric and analytic properties of the various underlying spaces, applying some theories and techniques from harmonic analysis, functional analysis, geometric analysis and other related analysis subjects, to further characterize the traces and the trace spaces for Sobolev spaces and other function spaces with fractional smoothness. More precisely, first, we consider the borderline cases of traces for weighted Sobolev spaces, Besov spaces and other function spaces with fractional order smoothness and characterize their trace spaces. Second, on the regular trees, we want to find an intrinsic parameter, which only depends on the metric and measure, to characterize the existence and the integrability of the traces for Sobolev functions. Third, we want to generalize the results on Euclidean setting and the regular trees to metric measure spaces. All of these researches on trace theory which can connect the function spaces on a domain and the function spaces on the corresponding boundary will provide more research methods on the analysis on metric spaces, and will have broad applications on subjects including harmonic analysis, partial differential equation and so on.