Boundedness of commutators of Marcinkiewicz integrals on nonhomogeneous metric measure spaces (Q901663)

Summary: Let \((\mathcal{X}, d, \mu)\) be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of \textit{T. Hytönen} [Publ. Mat., Barc. 54, No.~2, 485--504 (2010; Zbl 1246.30087)]. The aim of this paper is to establish the boundedness of commutator \(\mathcal{M}_b\) generated by the Marcinkiewicz integral \(\mathcal{M}\) and Lipschitz function \(b\). The authors prove that \(\mathcal{M}_b\) is bounded from the Lebesgue spaces \(L^p(\mu)\) to weak Lebesgue spaces \(L^q(\mu)\) for \(1 \leq p < n / \beta\), from the Lebesgue spaces \(L^p(\mu)\) to the spaces \(\operatorname{RBMO}(\mu)\) for \(p = n / \beta\), and from the Lebesgue spaces \(L^p(\mu)\) to the Lipschitz spaces \(\operatorname{Lip}_{(\beta - n / p)}(\mu)\) for \(n / \beta < p \leq \operatorname{\infty}\). Moreover, some results in Morrey spaces and Hardy spaces are also discussed.