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The main purpose of this paper is to prove that the boundedness of the commutatorM    κ  ,  b    ∗$\mathcal{M}_{\kappa,b}^{*} $generated by the Littlewood-Paley operatorM    κ    ∗$\mathcal{M}_{\kappa}^{*} $and RBMO ( μ ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel ofM    κ    ∗$\mathcal{M}_{\kappa}^{*} $satisfies a certain Hörmander-type condition, the authors prove thatM    κ  ,  b    ∗$\mathcal{M}_{\kappa,b}^{*} $is bounded on Lebesgue spaces  L p  ( μ ) for 1 &lt;  p  &lt; ∞, bounded from the space  L  log  L ( μ ) to the weak Lebesgue space  L  1,∞ ( μ ), and is bounded from the atomic Hardy spaces  H  1 ( μ ) to the weak Lebesgue spaces  L  1,∞ ( μ ).

Commutators of Littlewood-Paley g κ ∗ $g_{\kappa}^{*} $ -functions on non-homogeneous metric measure spaces