Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces

Let (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal{X}, d, \mu )$\end{document} be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. In this paper, the authors prove the boundedness in Lp(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p} (\mu )$\end{document} of mth-order commutators Mb,mρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}^{\rho }_{b,m}$\end{document} generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal{X}, d, \mu )$\end{document}. In addition, the boundedness of the mth-order commutators Mb,mρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}^{\rho }_{b,m}$\end{document} on Morrey spaces Mpq(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M^{q}_{p}(\mu )$\end{document}, 1<p≤q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p \leq q< \infty $\end{document}, is also obtained for the parameter 0<ρ<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\rho <\infty $\end{document}.


Introduction
Marcinkiewicz integral operators and their commutators play a very important role in harmonic analysis. Therefore, many authors have focused on studying the operators and their commutators. In 1960, Hörmander [1] introduced the parametric Marcinkiewicz integral, defined by, 1 2 , (1.1) where ρ ∈ (0, ∞). Let be homogeneous of degree zero in R d for d ≥ 2, integrable and have mean value zero on the unit sphere S d-1 . Hörmander [1] proved that, if ∈ Lip α (S d-1 ) for some α ∈ (0, 1], then μ ρ is bounded on L p (R d ) for p ∈ (1, ∞). In 2001, Fan [2] obtained the boundedness of μ ρ from L 1 (R d ) to L 1,∞ (R d ) when ∈ L(logL)(S d-1 ). If ρ = 1 in (1.1), then it is the higher-dimensional Marcinkiewicz integral first introduced by Stein [3] in 1958, denoted μ . Stein [3] proved that μ is bounded on L p (R d ) for any 1 < p ≤ 2, and is also bounded from L 1 (R d ) to L 1,∞ (R d ). In 1990, Torchinsky and Wang [4] first introduced the commutator μ ,b generated by μ and BMO function b, defined as follows: and established its L p (R d ) boundedness for p ∈ (1, ∞). In 2002, Salman [5] reduced the condition of the kernel function to ∈ L(logL) 1 2 (S d-1 ), and proved that μ is bounded on L p (R d ) for p ∈ (1, ∞).
Recently, Gürbüz considered the boundedness of Marcinkiewicz integral operator with rough kernel associated with the Schrödinger operator and their commutators [6][7][8]. Gürbüz also proved some relevant conclusions about Marcinkiewicz operators, one may refer to [9][10][11][12]. In addition, Tao proved the boundedness of Marcinkiewicz integral operator with rough kernel [13][14][15] In this paper, we will discuss the boundedness of commutators of the parametric Marcinkiewicz integral on the non-homogeneous metric space. Let (X , d) be a metric space, and let μ be a positive Borel measure on X that satisfies the following growth condition: for all x ∈ X , r > 0, It is well known that the analysis on (X , d, μ) played key roles in many fields, for example, in solving Painlevé's problem [16]. In 2010, Hytönen [17] introduced a non-homogeneous metric measure space, of which the measure satisfies the geometrically doubling condition and the upper doubling condition. From then on, many researchers considered singular integral operators on (X , d, μ); see [18][19][20] for example. The purpose of this article is to consider the boundedness of the commutators generated by the Log-Dini-type parametric Marcinkiewicz integral with RBMO functions on (X , d, μ). Before stating our results, we recall some notions of geometrically doubling and upper doubling measure [17]. Definition 1.1 ([17]) Let (X , d) is a metric space; if there exists some N 0 ∈ N , and for any x ∈ X , r > 0, such that any ball B(x, r) ⊂ X can be covered by at most N 0 balls B(x i , r 2 ), we say (X , d) satisfies the geometrically doubling condition.
is a metric measure space, if μ is a Borel measure on X and there exist a dominating function λ(x, r) : X × R + → R + and a constant C λ > 0 such that r → λ(x, r) is increasing and for all x ∈ X , r > 0, then we say μ is an upper doubling measure.
We also need to recall other notions [17,21].
One can see from Lemma 3.2 of [17] that, if μ is upper doubling, for any α, β ∈ (1, ∞) and β > C log 2 a λ =: α ν , then for every ball B ⊂ X there exists j ∈ n, such that α j B is (α, β) doubling ball. Moreover, we see from Lemma 3.3 of [17] that, if (X , d) is geometrically doubling, there exists n 0 := log 2 N 0 , such that β > α n 0 , if μ is a Borel measure on X which is finite on bounded sets, then, for μ-a.e. x ∈ X , there exist arbitrarily small (α, β) doubling balls centred at x. Moreover, for any preassigned r > 0, their radius can be chosen to be of the form α j r, j ∈ n. Throughout this paper, fix τ ≥ 1, B is a (30τ , β 30τ ) doubling ball and For any τ ≥ 1, B ⊂ X ,B denotes the smallest (30τ , β 30τ ) doubling ball of the form (30τ ) j B.
As in [7], for any two balls B ⊂ S, r B and r S denote the radius of the ball B and S, respectively. And x B denotes the center of the ball B. We define K B,S andK B,S as follows: Let N B,S be the smallest integer satisfying 6 N B,S r B ≥ r S , we definẽ . (1.6) In the case that λ(x, ar) = a m λ(x, r) for all x ∈ X , a, r > 0, it is easy to show that K B,S K B,S . Nevertheless, in general, we only have K B,S ≤ CK B,S . Finally, we recall the definition of Morrey space [22] on (X , d, μ).
Next, we introduce the conditions of kernel discussed in this article. Let K(x, y) ∈ L 1 loc ((X ) 2 \ {(x, y) : x = y}), we say K(x, y) is the parametric Marcinkiewicz kernel of Log-Dini type, if there exists C > 0 such that the following size estimate and smoothness estimates hold: The parametric Marcinkiewicz integral M ρ with Log-Dini-type kernel K(x, y) satisfying (1.9), (1.10) and (1.11) is then defined, initially for f ∈ L ∞ with compact support, by (1.12) In case ρ = 1, M ρ , denoted by M, is just the Marcinkiewicz integral operator on (X , d, μ) with Log-Dini-type kernel.
In 2014, Lin and Yang [24] proved that M is bounded on L p (μ) if and only if M is bounded from L 1 (μ) to L 1,∞ (μ), if the kernel K(x, y) satisfies (1.9) and for all x, y, y ∈ X d(x,y)≥2d(y,y ) In 2016, Fu and Lin [25] proved that when the kernel K(x, y) satisfies (1.9) and (1. (1.14) In general, for all m ∈ N, the mth-order commutators M ρ b,m is defined by In 2015, Zhou [26] showed that the commutator M b is bounded on L p (μ), if M is bounded on L 2 (μ), and the kernel K(x, y) satisfies (1.9) and the following Hörmander type condition: (1. 16) In 2019, Tao [27] proved that, if the kernel satisfies (1.9) and (1.16), then M b is bounded on M q p (μ). In fact, we can see that (1.16) is stronger than (1.13). In case X = R d , the non-homogeneous Euclidean space, then for the kernel K(x, y) in the Marcinkiewicz integral it can be assumed that satisfies the following conditions with a constant C > 0: In 2007, Hu [28] obtained M is bounded on L p (μ), 1 < p < ∞, and is bounded from L 1 (μ) to L 1,∞ (μ). Later, Zhang [29] proved M is bounded on M q p (μ). For m ∈ N and b ∈ RBMO, the mth-order commutator for Marcinkiewicz integral is denoted by (1.20) In 2007, Hu [28] proved that M b,m is bounded on L p (μ) if the kernel K(x, y) satisfies (1.17) and the following condition: It is easy to see that (1.21) is stronger than (1.18). In 2010, Zhang [29] proved that M b is bounded on M q p (μ) under the same assumptions. Now we turn to stating the main results of this paper.  .12) and (1.14), then, for all f ∈ L p (μ), 1 < p < ∞, there exists a constant C > 0 such that In fact we will prove the L p (μ) boundedness for a more general mth-order commutator for the parametric Marcinkiewicz integral.  .15). If ω satisfies the following condition:

24)
then for all f ∈ L p (μ), 1 < p < ∞, there exists a constant C > 0 such that (1.25) Theorem 1.1 is the special case of Theorem 1.2 in which one can take m = 1. We will prove Theorem 1.2 in Sect. 2.
Moreover, we will establish the boundedness of M ρ b,m on the Morrey space.
By checking the proofs of Theorem 1.2 and Theorem 1.3, we can obtain the following two corollaries, which extend the results in [26] and [27].
Throughout this paper, d is the dimension of space; C denotes a positive constant that is independent of the parameters, furthermore, it value may differ from line to line; x B denotes the center of the ball B, r B denotes the radius of the ball B; for any p ∈ (1, ∞),

Proof of Theorem 1.2
We first recall the definition of a sharp maximal operator M f (x) [21] over (X , d, μ). For any f ∈ L 1 loc (μ), δ . We will use the following lemma about sharp maximal function on (X , d, μ) proved by Fu [18].
According to Theorem 4.2 in [21], we can easily get the following lemma.

Lemma 2.3
Let > 1, for b ∈ L 1 loc (μ). The following statements are equivalent: (ii) There exists a constant C > 0, such that, for all balls B,

5)
and for all (30τ , β 30τ ) doubling balls B ⊂ S, We need the following lemma about the boundedness of parametric Marcinkiewicz integral operators.
Proof In Theorem 2.1 of [25], the kernel function satisfies (1.9) and (1.13). It is easily to see that (1.13) is weaker than (1.10) and (1.11). So by similar argument as that in Theorem 2.1 of [25], we can prove the lemma. Hence, we omit the details.
To prove Theorem 1.2, we should first establish the following lemma. Lemma 2.6 Let K(x, y) satisfy (1.9), (1.10) and (1.11). Suppose M ρ be as in (1.12) is bounded on L 2 (μ), b ∈ RBMO(μ). If 0 < ρ < ∞, δ ∈ (0, 1) and ω satisfies (1.24), then there exists a constant C > 0, for all f ∈ L p (μ), such that (2.8) Proof Without loss of generality, we may assume that b RBMO(μ) = 1. In order to prove (2.8), it suffices to prove that, for all x ∈ X and balls B x, (2.9) and for all balls B ⊂ S, S is a (30τ , β 30τ ) doubling ball, Thus, we obtain Since 0 < δ < 1, To estimate E 1 , let γ , η > 1, be such that By Hölder's inequality and Lemma 2.4, we have For E 2 , by the Kolmogorov inequality, Lemma 2.5, Hölder's inequality and Lemma 2.4, we have As to the estimate E 3 , we observe that Hence In fact, for y, w ∈ B, we observe that In order to estimate E 3 , it suffices to estimate F 1 , F 2 , andF 3 . To estimate F 1 , for all y, w ∈ B, (c B , z). By the Minkowski inequality, (1.9), Hölder's inequality, and Lemma 2.4, we get We have used the following fact in the last inequality: Similarly, we get To estimate F 3 , for all y, w ∈ B, we have d(y, z) ∼ d(w, z) ∼ d(c B , z), using the Minkowski inequality, we get We have used the following fact in the last inequality: To estimate B 32 , for all y, w ∈ B, if ρ ∈ (0, ∞), by (1.9), Hölder's inequality and Lemma 2.4, we get Therefore, With the same argument as for E 1 , we get    d(y, z)) b(z) -mB(b) m f (z) 1 |d(y, z)| ρ dμ(z) Therefore, Similarly, We set