Quasiconformal and Sobolev mappings on metric measure spaces

在第一个项目中，我们考虑了加倍度量度量空间和Banach空间之间的q-绝对连续映射。研究了这些映射与超临界情形下的Sobolev映射之间的关系。特别地，我们证明了满足松弛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，我们研究了度量空间中BV和Soblev函数的非局部泛函的刻画，该度量空间具有加倍测度并支持Poincare不等式。与前人的工作相比，我们考虑了更一般的泛函。在p=1的情况下，我们还给出了一个反例，证明了在度量度量空间中，非局部函数的极限仅与变差度量是可比的。

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.