欧氏空间中加倍测度的限制与延拓

项目名称： 欧氏空间中加倍测度的限制与延拓

项目编号： No.11201500

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王小华

作者单位： 中央财经大学

项目金额： 22万元

中文摘要： 本项目主要研究加倍测度理论中的几个热点问题，这些问题的深入研究对于丰富和发展几何测度论以及调和分析都具有十分重要的理论意义。本项目拟研究如下几个问题：(1)具有分片光滑边界的Jordan域上的加倍测度限制与延拓；(2)中间区间Cantor集上加倍测度的限制与延拓；(3)均匀Cantor集上Moran测度的加倍性质的刻画及其延拓；（4）加倍测度肥集、瘦集上加倍测度的限制与延拓。项目成员将综合运用分形几何、实分析、几何测度论、调和分析等方面的技巧和知识，坚持独立思考，努力发掘一些新思想和新方法，圆满解决这些问题。

中文关键词： 分形几何；加倍测度理论；几何测度论；；

英文摘要： This project mainly concerns with some hot issues in the doubling measure theory. A thorough study on these issues is of great theoretical importance to the understanding of the geometric measure theory and harmonic analysis as well. The following topics will be addressed in this project. The first is concerned with the restrictions and extensions of doubling measures on Jordan domains with piecewise smooth boundary. The second is concerned with the restrictions and extensions of doubling measures on the middle interval Cantor sets. The third is concerned with doubling properties of Moran measures on uniform Cantor sets and their extensions. The last topic is concerned with the restrictions and extensions of doubling measures on fat sets and thin sets for doubling measures. In the study of these issues we shall employ knowledge and techniques from various fields such as fractal geometry, real analysis, geometric measure theory, harmonic analysis，etc..

英文关键词： fractal geometry；doubling measure theory；geometric measure theory；；