Cauchy–Szegö commutators on weighted Morrey spaces

Zunwei Fu, Ruming Gong, Elodie Pozzi and Qingyan Wu

Mathematische Nachrichten, 2023, vol. 296, issue 5, 1859-1885

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

Date: 2023
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