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Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p<n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

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Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.