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Quasiconformal and Sobolev mappings on metric measure spaces

度量测度空间上的拟共形映射和 Sobolev 映射

基本信息

批准号：
22K13947
负责人：
ZHOU Xiaodan
金额：
$ 2.91万
依托单位：
Okinawa Institute of Science and Technology Graduate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Early-Career Scientists
财政年份：
2022
资助国家：
日本
起止时间：
2022-04-01 至 2025-03-31
项目状态：
未结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-22K13947/
关键词：
Absolute Continuity Sobolev mappings Metric measure spaces Quasiconformal mappings Lusin property Nonlocal functional Quasiconformal mapping Sobolev mapping Poincare inequality

项目摘要

In the first project, we consider Q-absolutely continuous mappings between a doubling metric measure space and a Banach space. The relation between these mappings and Sobolev mappings in supercritical cases is investigated. In particular, we show that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Q-absolutely continuous.In the second project, we study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincare inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that in metric measure spaces the limit of the nonlocal functions is only comparable to the variation measure.

在第一个项目中，我们考虑了加倍度量测度空间和Banach空间之间的Q-绝对连续映射。在超临界情形下，研究了这些映射与Sobolev映射之间的关系.特别地，我们证明了满足放松拟共形条件的伪单调映射也是Q-绝对连续的.在第二个项目中，我们研究了在度量空间中BV函数和Sobolev函数通过非局部泛函的刻画，该度量空间具有一个加倍测度和一个Poincare不等式.与以前的工作相比，我们考虑更一般的泛函。在p=1的情形下，我们给出了一个反例，证明了在度量测度空间中，非局部函数的极限只与变差测度相当.