Quasiconformal and Sobolev mappings on metric measure spaces

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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.