\(\Theta \)-type fractional Marcinkiewicz integral operators and their commutators on some spaces over RD-spaces

The aim of this paper is to establish the boundedness of an \(\theta \) $\theta$ -type fractional Marcinkiewicz integral \({\mathcal {M}}_{q,\rho ,\alpha ,\theta }\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta }$ and its commutator \({\mathcal {M}}_{q,\rho ,\alpha ,\theta ,b}\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta ,b}$ on weighted Lebesgue spaces \(L^{p}(\omega )\) $L^p(\omega )$ , weighted Morrey spaces \({\mathcal {M}}^{p,\kappa }(\mu )\) ${\mathcal{M}}^{p,\kappa }(\mu )$ and generalized weighted Morrey spaces \({\mathcal {L}}^{p,\Phi }(\omega )\) ${\mathcal{L}}^{p,\mathrm{\Phi }}(\omega )$ over RD-spaces satisfying the doubling and reverse doubling conditions. Under assumption that the functions \(\omega \) $\omega$ and \(\Phi \) $\mathrm{\Phi }$ satisfy some certain conditions, the authors prove that the \({\mathcal {M}}_{q,\rho ,\alpha ,\theta }\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta }$ is bounded from spaces \(L^{p}(\omega )\) $L^p(\omega )$ into spaces \(L^{p}(\omega )\) $L^p(\omega )$ , bounded from spaces \({\mathcal {M}}^{p,\kappa }(\omega )\) ${\mathcal{M}}^{p,\kappa }(\omega )$ into spaces \({\mathcal {M}}^{p,\kappa }(\omega )\) ${\mathcal{M}}^{p,\kappa }(\omega )$ , and it is also bounded from spaces \({\mathcal {L}}^{p,\Phi }(\omega )\) ${\mathcal{L}}^{p,\mathrm{\Phi }}(\omega )$ into spaces \({\mathcal {L}}^{p,\Phi }(\omega )\) ${\mathcal{L}}^{p,\mathrm{\Phi }}(\omega )$ , where \(p\in (1,\infty )\) $p\in (1,\infty )$ , \(\kappa \in (0,1)\) $\kappa \in (0,1)$ , \(\omega \in A_{p}(\mu )\) $\omega \in A_p(\mu )$ and \(\Phi (\cdot )\) $\mathrm{\Phi }(·)$ is a non-decreasing and non-negative function defined on \((0,\infty )\) $(0,\infty )$ . Furthermore, by establishing the sharp maximal estimate for the commutator \({\mathcal {M}}_{q,\rho ,\alpha ,\theta ,b}\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta ,b}$ which is formed by \(b\in \textrm{BMO}(\mu )\) $b\in \text{BMO}(\mu )$ and the \({\mathcal {M}}_{q,\rho ,\alpha ,\theta }\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta }$ , the boundedness of the \({\mathcal {M}}_{q,\rho ,\alpha ,\theta ,b}\) ${\mathcal{M}}_{q,\rho ,\alpha ,\theta ,b}$ on spaces \(L^{p}(\omega )\) $L^p(\omega )$ , spaces \({\mathcal {M}}^{p,\kappa }(\omega )\) ${\mathcal{M}}^{p,\kappa }(\omega )$ and spaces \({\mathcal {L}}^{p,\Phi }(\omega )\) ${\mathcal{L}}^{p,\mathrm{\Phi }}(\omega )$ is obtained, respectively.