We investigate the Cauchy--Szeg\H{ o} projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy--Szeg\H{ o} kernel and prove that the Cauchy--Szeg\H{ o} kernel is non-zero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy--Szeg\H{ o} 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 $2/3<p\leq1$. Moreover, we establish the characterisation of singular values of the commutator of Cauchy--Szeg\H{ o} projection based on the kernel estimates and on the recent new approach by Fan--Lacey--Li. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.
Categories: math.CV