Cauchy-Szeg\"o operator, quaternionic Siegel upper half space, commutator, weighted Morrey space

TL;DR

This paper characterizes the boundedness and compactness of the commutator of the Cauchy--Szeg"o operator on weighted Morrey spaces over quaternionic Heisenberg groups, linking these properties to BMO and VMO function spaces.

Contribution

It provides necessary and sufficient conditions for the boundedness and compactness of the commutator on these spaces, extending operator theory in quaternionic analysis.

Findings

Boundedness of commutator characterized by BMO functions.

Compactness of commutator characterized by VMO functions.

Results apply to weighted Morrey spaces on quaternionic Heisenberg groups.

Abstract

In the setting of quaternionic Heisenberg group $\mathscr H^{n-1}$, we characterize the boundedness and compactness of commutator $[b,\mathcal C]$ for the Cauchy--Szeg\"o operator $\mathcal C$ on the weighted Morrey space $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ with $p\in(1, \infty)$, $\kappa\in(0, 1)$ and $w\in A_p(\mathscr H^{n-1}).$ More precisely, we prove that $[b,\mathcal C]$ is bounded on $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ if and only if $b\in {\rm BMO}(\mathscr H^{n-1})$. And $[b,\mathcal C]$ is compact on $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ if and only if $b\in {\rm VMO}(\mathscr H^{n-1})$.