Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces

Let $({\mathcal X},\,d,\,\mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hyt\"onen. In this paper, the authors prove that the $L^p(\mu)$ boundedness with $p\in(1,\,\infty)$ of the Marcinkiewicz integral is equivalent to either of its boundedness from $L^1(\mu)$ into $L^{1,\infty}(\mu)$ or from the atomic Hardy space $H^1(\mu)$ into $L^1(\mu)$. Moreover, the authors show that, if the Marcinkiewicz integral is bounded from $H^1(\mu)$ into $L^1(\mu)$, then it is also bounded from $L^\infty(\mu)$ into the space ${\mathop\mathrm{RBLO}}(\mu)$ (the regularized {\rm BLO}), which is a proper subset of ${\rm RBMO}(\mu)$ (the regularized {\rm BMO}) and, conversely, if the Marcinkiewicz integral is bounded from $L_b^\infty(\mu)$ (the set of all $L^\infty(\mu)$ functions with bounded support) into the space ${\rm RBMO}(\mu)$, then it is also bounded from the finite atomic Hardy space $H_{\rm fin}^{1,\,\infty}(\mu)$ into $L^1(\mu)$. These results essentially improve the known results even for non-doubling measures.