Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {Hᵖ} on quaternionic Siegel upper half space for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mfrac> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:mfrac> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {2/3<p≤ 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő 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.