函数空间与度量测度空间上的分析

In this project, we plan to study the function spaces，and analysis on metric measure spaces, and also apply them to several important geometric and analytic objects. Precisely, 1) we will establish the Hajlasz-type characterization of a large class of function spaces and use it to characterize the quasiconformal mappinps; 2) we consider the relations between the extension property of fractional and higher order Soboelv functions on domains and the geometric properties of domains, and answer the open question raised by Nezza-Palatucci -Valdinoci about the geometric characterization of fractional Sobolev extension domains;3) we will develop a theory about the L-infinity variational problem on metric measure spaces, and answer an open question of Juutinen-Shanmugalingam on arbitrary metric measure spaces that satisfy a doubling property and support a weak Poincare inequality;4) given a metric measure space endowed with a Dirichlet form (including Sierpinski gasket as an example), we will establish the (non-)coincidence of the intrinsic distance and differential structures, and use their coincidence to study the Ricci curvatures defined by Bakry-Emery and by Lott-Sturm-Villani and related questions.

申请人拟研究函数空间以及度量测度空间上的分析，并应用于几类重要的几何与分析对象。具体地，1）申请人将建立一大类函数空间的Haj?asz-型特征并用于刻画拟共形映照；2）考察区域上分数次和高阶Sobolev函数的扩张性质与区域几何性质之间的相互依赖关系，回答Nezza-Palatucci-Valdinoci的从几何上刻画分数次Sobolev扩张区域的公开问题；3）发展度量测度空间上L-无穷变分(无穷调和函数)理论，并在满足倍测度条件和Poincaré不等式的度量测度空间上回答Juutinen-Shanmugalingam的一个公开问题；4）在赋予Dirichlet形式的度量测度空间（包括Sierpinski gasket等分形）上，建立内蕴距离结构与微分结构的相容或不相容性，并用其相容性去研究Bakry-Emery及Lott-Sturm -Villani意义下的Ricci曲率及相关问题。

申请人在函数空间与非光滑分析中取得了系统的创新性研究成果。 1. 发展了分数次Sobolev空间，Q-空间和Triebel-Lizorkin（型）空间的包括Hajlasz-型特征在内的多种特征刻画，并由此建立了拟共形复合算子在Q-空间的有界性准则，给出了分数次Sobolev延拓、嵌入区域的几何刻画。2. 系统研究了非光滑空间上可测微分结构与内在度量结构的（弱）相容性，并由此发展了其上的无穷变分理论，获得了Sierpinski垫上可测黎曼结构与内在度量结构相容性，证明了度量结构与可测微分结构相容性是Bakry-Emery的Ricci曲率下有界与Lott-Sturm-Villani的Ricci曲率下有界相容的必要条件。已在JEMS, Adv Math 和ARMA等著名期刊发表SCI论文6篇。研究成果被国际数学家大会报告人Ambrosio和Saloff-Coste等多次引用。