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

Abstract

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. Hytönen. 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 $\mathrm{RBLO}(\mu )$ (the regularized {\rm BLO}), which is a proper subset of $\mathrm{RBMO}(\mu )$ (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 $\mathrm{RBMO}(\mu )$, then it is also bounded from the finite atomic Hardy space $H_{\mathrm{fin}}^{1,\infty }(\mu )$ into $L^1(\mu )$. These results essentially improve the known results even for non-doubling measures.

arXiv commentSci. China Math. (to appear), 22 pages

References in corpus (2)

• The Hardy Space $H^1$ on Non-homogeneous Metric Spaces
• Spaces of Type BLO on Non-homogeneous Metric Measure Spaces

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• Multilinear fractional integral operators on non-homogeneous metric measure spaces