Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Abstract

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.

Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

“…Theorem 13. Let K satisfy (8) and ( 9), ω ∈ A ρ p ðμÞ, and ϕ : ð0, ∞Þ ⟶ ð0, ∞Þ be an increasing function satisfying (18) and (19). Suppose that M ρ θ defined in (10) is bounded on L 2 ðμÞ.…”

“…Since then, the research on the space has been widely focused, for example, some authors established the properties of function spaces on the nonhomogeneous metric measure space (see [10][11][12][13][14]). On the other hand, the boundedness of singular integral operators on various of spaces is also obtained; the readers can see [15][16][17][18][19][20] and so on.…”

Fractional Type Marcinkiewicz Integral and Its Commutator on Nonhomogeneous Spaces

“…In 2017, Lin et al [16] proved that the commutator , which is generated by Calderón–Zygmund operator T and , is bounded from space to space , and also bounded on space for all . For more progresses on the properties of function spaces and the boundedness of operators on , we can see [2], [4], [7], [15], [17]–[21] and so on.…”

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

“…So, it is interesting to generalize and improve the known results to the non-homogeneous metric measure spaces, see [16][17][18][19][20][21][22][23][24].…”