An explicit formula of Cauchy-Szego kernel for quaternionic Siegel upper half space and applications

In this paper we obtain an explicit formula of Cauchy--Szego kernel for quaternionic Siegel upper half space, and then based on this, prove that the projection Heisenberg group is a Calderon--Zygmund operator via verifying size regularity conditions kernel. Next, also suitable version pointwise lower bound kernel, which further implies characterisations boundedness compactness commutator BMO VMO spaces group, respectively.