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arXiv · 2110.12210

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

Der-Chen Chang·Xuan Thinh Duong·Ji Li·Wei Wang·Qingyan Wu
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Abstract

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 non-zero 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 $ 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--Szegő 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.

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Der-Chen Chang, Xuan Thinh Duong, Ji Li, Wei Wang, Qingyan Wu. 2021-10-23. Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications. https://arxiv.org/abs/2110.12210

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