本项目将围绕 Carnot-Caratheodory 空间上的几何测度论和微分不等式中几个重要的问题进行研究。首先我们将在一类特殊的 Carnot-Caratheodory 空间－海森堡群上研究等周不等式的最佳常数问题，并利用容量对等周不等式进行改进从而得到等容量不等式；其次在海森堡群上研究 BV 容量以及该容量跟 Sobolev 容量和 Hausdorff 容量的关系, 引入方向容量并建立仿射 Sobolev 不等式；再次将在 Grushin 平面和海森堡群上引入 ∞-容量，研究该容量的性质和 ∞-容量跟 ∞-拉普拉斯方程及粘性解的关系；另外，研究 Carnot-Caratheodory 空间上的极小梯度问题的极小值的存在性、唯一性和正则性；最后研究 Grushin 平面上的分数阶拉普拉斯微分不等式和海森堡群上的抛物型分数阶微分不等式。

This project will investigate some important problems of geometric measure theory and differential inequalities on the Carnot-Caratheodory space. Firstly, we study the optimal constant problems of the isoperimetric inequality on the Heisenberg group, which is a special Carnot-Caratheodory space; we improve the isoperimetric inequality and obtain the isocapacity inequality via the capacity. Secondly, we investigate the BV capacity on the Heisenberg group and its relation with Sobolev capacity and Hausdorff capacity. We introduce the directional capacity on the Heisenberg group and establish the affine Sobolev inequaltiy；Thirdly, we introduce ∞-capacity on the Heisenberg group and Grushin plane and study its properties and the relation between ∞-capacity and ∞-laplace equation on the Heisenberg group and the Grushin plane. Moreover, we investigate the existence, uniqueness and regularity of the minimizers of the least gradient on the Carnot-Caratheodory space. Finally, we investigate the fractional Laplacian differential inequalities on the Grushin plane and the parabolic fractional differential inequalities on the Heisenberg group.