Fundamental properties of Cauchy-Szego projection on quaternionic Siegel upper half space and applications

We investigate the Cauchy-Szego projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy-Szego kernel and prove that the Cauchy-Szego kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy-Szego projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space ( )H(p )on quaternionic Siegel upper half space for 2/3 < p <= 1. Moreover, we establish the characterisation of singular values of the commutator of Cauchy-Szego projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.