The logarithmic Sobolev capacity in the Heisenberg group

Abstract

In this paper, we consider the logarithmic Sobolev capacity \(\text {Cap}^{\mathbb {H}^n}_{ \log ,\gamma ,p}\) in the Heisenberg group \(\mathbb {H}^n\), a capacity generated by the logarithmic Sobolev space \(W^{ \log ,\gamma }_{p}(\mathbb {H}^n)\), where \((\gamma , p)\in (0,\infty )\times [1,\infty )\). In particular, we investigate several properties of the space \(W^{ \log ,\gamma }_{p}(\mathbb {H}^n)\), including completeness, min-max estimation and density. In addition, we investigate the properties of the logarithmic Sobolev capacity and the logarithmic perimeter in the Heisenberg group. Furthermore, we deal with the relationship between \(\text {Cap}^{\mathbb {H}^{n}}_{ \log ,\gamma ,p}\) and the Hausdorff capacity in the Heisenberg group. As an application, we prove the corresponding capacity estimate and the tracing principle.