Fundamental properties of Cauchy--Szeg\H{o} projection on quaternionic Siegel upper half space and applications

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 $2\mathrm{/}3<p\le 1$. 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.