Commutators of fractional maximal functions on Orlicz spaces over non-homogeneous metric spaces

Abstract:

      Let $(\mathcal{X},d,\mu)$ be a non-homogeneous metric measure space satisfying the geometrically doubling and the upper doubling conditions in the sense of Hyt\"{o}nen. In this setting, the authors prove that the commutator $M^{(\alpha)}_{b}$ formed by $b\in\widetilde{\mathrm{RBMO}}(\mu)$ and the fractional maximal function $M^{(\alpha)}$ is bounded from Lebesgue spaces $L^{p}(\mu)$ into spaces $L^{q}(\mu)$, where $\frac{1}{q}=\frac{1}{p}-\alpha$ for $\alpha\in(0,1)$ and $p\in(1,\frac{1}{\alpha})$. Furthermore, the boundedness of the $M^{(\alpha)}_{b}$ on Orlicz spaces $L^{\Phi}(\mu)$ is established.