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

Abstract

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