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

Abstract

Let \((\mathcal{X}, d, \mu )\) 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 \(L^{p} (\mu )\) of mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on \((\mathcal{X}, d, \mu )\). In addition, the boundedness of the mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) on Morrey spaces \(M^{q}_{p}(\mu )\), \(1< p \leq q< \infty \), is also obtained for the parameter \(0<\rho <\infty \).

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,

$$\begin{aligned} \mathcal{\mu }_{\Omega }^{\rho }(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{ \vert x-y \vert < t} \frac{\Omega (x-y)}{ \vert x-y \vert ^{d-\rho }} f(y) \,dy \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}, \end{aligned}$$
(1.1)

where \(\rho \in (0,\infty )\). Let Ω be homogeneous of degree zero in \(R^{d}\) for \(d \geq 2\), integrable and have mean value zero on the unit sphere \(S^{d-1}\). Hörmander [1] proved that, if \(\Omega \in Lip_{\alpha }(S^{d-1})\) for some \(\alpha \in (0,1]\), then \(\mathcal{\mu }_{\Omega }^{\rho }\) is bounded on \(L^{p}(R^{d})\) for \(p\in (1,\infty )\). In 2001, Fan [2] obtained the boundedness of \(\mathcal{\mu }_{ \Omega }^{\rho }\) from \(L^{1}(R^{d})\) to \(L^{1,\infty }(R^{d})\) when \(\Omega \in L(\mathrm{logL}) (S^{d-1})\).

If \(\rho =1\) in (1.1), then it is the higher-dimensional Marcinkiewicz integral first introduced by Stein [3] in 1958, denoted \(\mathcal{\mu }_{\Omega }\). Stein [3] proved that \(\mathcal{\mu }_{\Omega }\) is bounded on \(L^{p}(R^{d})\) for any \(1< p\leq 2\), and is also bounded from \(L^{1}(R^{d})\) to \(L^{1,\infty }(R^{d})\). In 1990, Torchinsky and Wang [4] first introduced the commutator \(\mathcal{\mu }_{\Omega,b}\) generated by \(\mathcal{\mu }_{\Omega }\) and BMO function b, defined as follows:

$$\begin{aligned} \mathcal{\mu }_{\Omega,b}(f) (x)=b(x)\mathcal{\mu }_{\Omega }(f) (x)- \mathcal{\mu }_{\Omega }(bf) (x),\quad x\in R^{d}, \end{aligned}$$

and established its \(L^{p}(R^{d})\) boundedness for \(p\in (1,\infty )\). In 2002, Salman [5] reduced the condition of the kernel function to \(\Omega \in L(\mathrm{logL})^{\frac{1}{2}}(S^{{d}-1})\), and proved that \(\mathcal{\mu }_{\Omega }\) is bounded on \(L^{p}(R^{d})\) for \(p\in (1,\infty )\).

Recently, Gürbüz considered the boundedness of Marcinkiewicz integral operator with rough kernel associated with the Schrödinger operator and their commutators [68]. Gürbüz also proved some relevant conclusions about Marcinkiewicz operators, one may refer to [912]. In addition, Tao proved the boundedness of Marcinkiewicz integral operator with rough kernel [1315]

In this paper, we will discuss the boundedness of commutators of the parametric Marcinkiewicz integral on the non-homogeneous metric space. Let \((\mathcal{X}, d)\) be a metric space, and let μ be a positive Borel measure on \(\mathcal{X}\) that satisfies the following growth condition: for all \(x\in \mathcal{X}, r>0\),

$$\begin{aligned} \mu \bigl(B(x,r)\bigr) \leq C_{0} r^{n}, \end{aligned}$$
(1.2)

where \(C_{0}>0\) and \(B(x,r):=\{y \in \mathcal{X}:d(x,y)< r \}\).

It is well known that the analysis on \((\mathcal{X}, d, \mu )\) 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 \((\mathcal{X}, d, \mu )\); see [1820] 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 \((\mathcal{X}, d, \mu )\). Before stating our results, we recall some notions of geometrically doubling and upper doubling measure [17].

Definition 1.1

([17])

Let \((\mathcal{X}, d)\) is a metric space; if there exists some \(N_{0}\in N\), and for any \(x\in \mathcal{X}, r>0\), such that any ball \(B(x,r)\subset \mathcal{X}\) can be covered by at most \(N_{0}\) balls \(B(x_{i},\frac{r}{2})\), we say \((\mathcal{X}, d)\) satisfies the geometrically doubling condition.

Definition 1.2

([17])

Let \((\mathcal{X}, d, \mu )\) is a metric measure space, if μ is a Borel measure on \(\mathcal{X}\) and there exist a dominating function \(\lambda (x,r):\mathcal{X} \times R_{+} \rightarrow R_{+}\) and a constant \(C_{\lambda }>0\) such that \(r \rightarrow \lambda (x,r)\) is increasing and

$$\begin{aligned} \mu \bigl(B(x,r)\bigr) \leq \lambda (x,r) \leq C_{\lambda }\lambda \biggl(x, \frac{r}{2}\biggr), \end{aligned}$$
(1.3)

for all \(x\in \mathcal{X}, r>0\), then we say μ is an upper doubling measure.

We also need to recall other notions [17, 21].

Definition 1.3

For \(\alpha, \beta \in (1,\infty )\), a ball \(B\subset \mathcal{X}\) is called \((\alpha, \beta )\) doubling if

$$\begin{aligned} \mu (\alpha B)\leq \beta \mu (B). \end{aligned}$$
(1.4)

One can see from Lemma 3.2 of [17] that, if μ is upper doubling, for any \(\alpha, \beta \in (1,\infty )\) and \(\beta > C_{\lambda }^{\mathrm{log}_{2}{a}}=:\alpha ^{\nu }\), then for every ball \(B\subset \mathcal{X}\) there exists \(j\in n\), such that \(\alpha ^{j} B\) is \((\alpha, \beta )\) doubling ball. Moreover, we see from Lemma 3.3 of [17] that, if \((\mathcal{X}, d)\) is geometrically doubling, there exists \(n_{0}:=\mathrm{log}_{2}{N_{0}}\), such that \(\beta >\alpha ^{n_{0}}\), if μ is a Borel measure on \(\mathcal{X}\) which is finite on bounded sets, then, for μ-a.e. \(x\in \mathcal{X}\), there exist arbitrarily small \((\alpha, \beta )\) doubling balls centred at x. Moreover, for any preassigned \(r>0\), their radius can be chosen to be of the form \(\alpha ^{j} r\), \(j\in n\). Throughout this paper, fix \(\tau \geq 1\), B is a \((30\tau, \beta _{30\tau })\) doubling ball and

$$\begin{aligned} \beta _{30\tau } > \max \bigl\{ (30\tau )^{3n}, c^{3\mathrm{log}_{2}(30\tau )}_{ \lambda } \bigr\} . \end{aligned}$$

For any \(\tau \geq 1\), \(B\subset \mathcal{X}\), denotes the smallest \((30\tau, \beta _{30\tau })\) doubling ball of the form \((30\tau )^{j}B\).

As in [7], for any two balls \(B\subset S\), \(r_{B}\) 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 \(K_{B,S}\) and \(\tilde{K}_{B,S}\) as follows:

$$\begin{aligned} K_{B,S}:= 1+ \int _{r_{B} \leq d(x,x_{B}) \leq r_{S}} \frac{1}{\lambda (x_{B},d(x,x_{B})} \,d\mu (x). \end{aligned}$$
(1.5)

Let \(N_{B,S}\) be the smallest integer satisfying \(6^{N_{B,S}}r_{B}\geq r_{S}\), we define

$$\begin{aligned} \tilde{K}_{B,S}:= 1+ \sum_{k=1}^{N_{B,S}} \frac{\mu (6^{k}B)}{\lambda (x_{B},6^{k} r_{B})}. \end{aligned}$$
(1.6)

In the case that \(\lambda (x,ar)=a^{m}\lambda (x,r)\) for all \(x\in \mathcal{X}, a,r>0\), it is easy to show that \(K_{B,S}\simeq \tilde{K}_{B,S}\). Nevertheless, in general, we only have \(K_{B,S}\leq C \tilde{K}_{B,S}\).

Finally, we recall the definition of Morrey space [22] on \((\mathcal{X}, d, \mu )\).

Definition 1.4

Let \(\kappa >1\) and \(1\leq p\leq q<\infty \), the definition of Morrey space are as follows:

$$\begin{aligned} M^{q}_{p}(\kappa,\mu )=\bigl\{ f\in L^{p}_{\mathrm{loc}}: \Vert f \Vert _{M^{q}_{p}(\kappa, \mu )}< \infty \bigr\} , \end{aligned}$$

where

$$\begin{aligned} \Vert f \Vert _{M^{q}_{p}(\kappa,\mu )}= \sup_{B}{ \mu (\kappa B)^{ \frac{1}{q}- \frac{1}{p}} \biggl( \int _{B} \vert f \vert ^{p} \,d\mu \biggr)^{\frac{1}{p}}}. \end{aligned}$$
(1.7)

We remark that, for any \(\kappa _{1}, \kappa _{2}>1\), \(M^{q}_{p}(\kappa _{1},\mu )=M^{q}_{p}(\kappa _{2},\mu )\) (see [23]). Particularly, if μ is a doubling measure, then \(M^{q}_{p}(\kappa,\mu ) = M^{q}_{p}(1,\mu )\) for any \(\kappa >0\), and denote it by \(M^{q}_{p}(\mu )\) for brevity. Moreover, it is easily to see that the space \(M^{q}_{p}(\mu )\) becomes the classical Morrey space whenever \(d\mu =dx\).

Next, we introduce the conditions of kernel discussed in this article.

Definition 1.5

Let \(\omega:[0,\infty )\rightarrow [0,\infty )\) be non-decreasing function that satisfies

$$\begin{aligned} \int _{0}^{1} \frac{\omega (t)}{t} \vert \mathrm{log}t \vert \,dt < \infty. \end{aligned}$$
(1.8)

Let \(K(x, y) \in L^{1}_{\mathrm{loc}} ( (\mathcal{X})^{2} \setminus \{(x,y):x=y\} )\), 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:

  1. (i)

    For \(x, y \in \mathcal{X}\) with \(x\neq y\),

    $$\begin{aligned} \bigl\vert K(x, y) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))}. \end{aligned}$$
    (1.9)
  2. (ii)

    For \(x, x',y \in \mathcal{X}\) and if \(2d(x,x')\leq d(x,y)\),

    $$\begin{aligned} \bigl\vert K(x, y)-K\bigl(x', y\bigr) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))} \omega \biggl( \frac{d(x,x')}{d(x,y)} \biggr). \end{aligned}$$
    (1.10)
  3. (iii)

    For \(x, y',y \in \mathcal{X}\) and if \(2d(y,y')\leq d(x,y)\),

    $$\begin{aligned} \bigl\vert K(x, y)-K\bigl(x, y'\bigr) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))} \omega \biggl( \frac{d(y,y')}{d(x,y)} \biggr). \end{aligned}$$
    (1.11)

The parametric Marcinkiewicz integral \(\mathcal{M}^{\rho }\) with Log-Dini-type kernel \(K(x, y)\) satisfying (1.9), (1.10) and (1.11) is then defined, initially for \(f \in L^{\infty }\) with compact support, by

$$\begin{aligned} \mathcal{M}^{\rho }(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{B(x,t)} \frac{K(x,y)}{ \vert d(x,y) \vert ^{1-\rho }} f(y) \,d \mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(1.12)

In case \(\rho =1\), \(\mathcal{M}^{\rho }\), denoted by \(\mathcal{M}\), is just the Marcinkiewicz integral operator on \((\mathcal{X}, d, \mu )\) with Log-Dini-type kernel.

In 2014, Lin and Yang [24] proved that \(\mathcal{M}\) is bounded on \(L^{p}(\mu )\) if and only if \(\mathcal{M}\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\), if the kernel \(K(x, y)\) satisfies (1.9) and for all \(x,y,y'\in \mathcal{X}\)

$$\begin{aligned} \int _{d(x,y) \geq 2d(y,y')} \bigl[ \bigl\vert K(x,y)-K\bigl(x,y' \bigr) \bigr\vert + \bigl\vert K(y,x)-K\bigl(y',x\bigr) \bigr\vert \bigr] \frac{d\mu (x)}{d(x,y)} \leq C. \end{aligned}$$
(1.13)

In 2016, Fu and Lin [25] proved that when the kernel \(K(x, y)\) satisfies (1.9) and (1.13), if \(\mathcal{M}^{\rho }\) is bounded on \(L^{p_{0}}(\mu )\) with some \(1< p_{0}<\infty \) then \(\mathcal{M}^{\rho }\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\).

Given \(b\in \operatorname{RBMO}(\mu )\), the commutators \(\mathcal{M}^{\rho }_{b}\) generated by \(\mathcal{M}^{\rho }\) with RBMO function b is defined by

$$\begin{aligned} \mathcal{M}^{\rho }_{b}(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{B(x,t)} \frac{K(x,y)}{ \vert d(x,y) \vert ^{1-\rho }} \bigl[b(x)-b(y) \bigr] f(y) \,d \mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{ \frac{1}{2}}. \end{aligned}$$
(1.14)

In general, for all \(m \in \mathrm{N}\), the mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) is defined by

$$\begin{aligned} \mathcal{M}^{\rho }_{b,m}(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{B(x,t)} \frac{K(x,y)}{ \vert d(x,y) \vert ^{1-\rho }} \bigl[b(x)-b(y) \bigr]^{m} f(y) \,d\mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(1.15)

In 2015, Zhou [26] showed that the commutator \(\mathcal{M}_{b}\) is bounded on \(L^{p}(\mu )\), if \(\mathcal{M}\) is bounded on \(L^{2}(\mu )\), and the kernel \(K(x, y)\) satisfies (1.9) and the following Hörmander type condition:

$$\begin{aligned} \begin{aligned} & \sup_{{r>0 \atop d(y,y')\leq r}} \sum _{i=1}^{\infty } i \int _{6^{i}r < d(x,y) \leq 6^{i+1}r} \bigl[ \bigl\vert K(x,y)-K\bigl(x,y' \bigr) \bigr\vert \\ &\quad{} + \bigl\vert K(y,x)-K\bigl(y',x\bigr) \bigr\vert \bigr] \frac{1}{d(x,y)} \,d\mu (x) \leq C. \end{aligned} \end{aligned}$$
(1.16)

In 2019, Tao [27] proved that, if the kernel satisfies (1.9) and (1.16), then \(\mathcal{M}_{b}\) is bounded on \(M^{q}_{p}(\mu )\). In fact, we can see that (1.16) is stronger than (1.13).

In case \(\mathcal{X}=R^{d}\), the non-homogeneous Euclidean space, then for the kernel \(K(x, y)\) in the Marcinkiewicz integral it can be assumed that \(K(x, y) \in L^{1}_{\mathrm{loc}} ( R^{d} \times R^{d} \setminus \{(x,y):x=y\} )\) satisfies the following conditions with a constant \(C>0\):

$$\begin{aligned} \bigl\vert K(x, y) \bigr\vert \leq C \vert x-y \vert ^{-(d-1)} \end{aligned}$$
(1.17)

and

$$\begin{aligned} \int _{ \vert x-y \vert \geq 2 \vert y-y' \vert } \bigl[ \bigl\vert K(x,y)-K\bigl(x,y' \bigr) \bigr\vert + \bigl\vert K(y,x)-K\bigl(y',x\bigr) \bigr\vert \bigr] \frac{1}{ \vert x-y \vert } \,d\mu (x) \leq C. \end{aligned}$$
(1.18)

for all \(x,y,y'\in R^{d}\) with \(x\neq y\). And the Marcinkiewicz integral \(\mathcal{M}\) is defined by

$$\begin{aligned} \mathcal{M}(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t} \int _{ \vert x-y \vert < t} K(x,y) f(y) \,d\mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(1.19)

In 2007, Hu [28] obtained \(\mathcal{M}\) is bounded on \(L^{p}(\mu ), 1< p<\infty \), and is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\). Later, Zhang [29] proved \(\mathcal{M}\) is bounded on \(M^{q}_{p}(\mu )\).

For \(m\in N\) and \(b\in\operatorname{RBMO}\), the mth-order commutator for Marcinkiewicz integral is denoted by

$$\begin{aligned} \mathcal{M}_{b,m}(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t} \int _{ \vert x-y \vert \leq t} K(x,y) \bigl[b(x)-b(y) \bigr]^{m} f(y) \,d \mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(1.20)

In 2007, Hu [28] proved that \(\mathcal{M}_{b,m}\) is bounded on \(L^{p}(\mu )\) if the kernel \(K(x, y)\) satisfies (1.17) and the following condition:

$$\begin{aligned} \begin{aligned} & \sup_{{r>0,y,y'\in R^{n} \atop \vert y-y' \vert \leq r}} \sum _{l=1}^{\infty } l^{m} \int _{2^{l}r< \vert x-y \vert \leq 2^{l+1}r} \bigl[ \bigl\vert K(x,y)-K\bigl(x,y' \bigr) \bigr\vert \\ &\quad{} + \bigl\vert K(y,x)-K\bigl(y',x\bigr) \bigr\vert \bigr] \frac{1}{ \vert x-y \vert } \,d\mu (x) \leq C. \end{aligned} \end{aligned}$$
(1.21)

It is easy to see that (1.21) is stronger than (1.18). In 2010, Zhang [29] proved that \(\mathcal{M}_{b}\) is bounded on \(M^{q}_{p}(\mu )\) under the same assumptions.

Now we turn to stating the main results of this paper.

Theorem 1.1

Let K satisfy (1.9), (1.10), and (1.11). \(\mathcal{M}^{\rho }\), \(\mathcal{M}^{\rho }_{b}\) be as in (1.12) and (1.14), respectively. Suppose that \(\mathcal{M}^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\), \(0<\rho <\infty \). If ω satisfies

$$\begin{aligned} \int _{0}^{1} \frac{\omega (t)}{t} \vert \mathrm{log}t \vert \,dt < \infty, \end{aligned}$$
(1.22)

then, for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), there exists a constant \(C>0\) such that

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b}(f) \bigr\Vert _{L^{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$
(1.23)

In fact we will prove the \(L^{p} (\mu )\) boundedness for a more general mth-order commutator for the parametric Marcinkiewicz integral.

Theorem 1.2

Under the same conditions of Theorem 1.1and \(\mathcal{M}^{\rho }_{b,m}\) be as in (1.15). If ω satisfies the following condition:

$$\begin{aligned} \int _{0}^{1} \frac{\omega (t)}{t} \vert \mathrm{log}t \vert ^{m} \,dt < \infty, \end{aligned}$$
(1.24)

then for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), there exists a constant \(C>0\) such that

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{L^{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}( \mu )}^{m} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$
(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 \(\mathcal{M}^{\rho }_{b,m}\) on the Morrey space.

Theorem 1.3

Assume the same conditions of Theorem 1.1and \(\mathcal{M}^{\rho }_{b,m}\) as in (1.15). If ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {M^{q}_{p} (\mu )}, 1< p\leq q<\infty \), such that

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{M^{q}_{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}( \mu )}^{m} \Vert f \Vert _{M^{q}_{p} (\mu )}. \end{aligned}$$
(1.26)

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].

Corollary 1.4

Let the kernel \(K(x, y)\) satisfy (1.9) and (1.16), \(\mathcal{M}^{\rho }\) and \(\mathcal{M}^{\rho }_{b,m}\) be as in (1.12) and (1.15), respectively. Suppose that \(\mathcal{M}^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu ), 0<\rho <\infty \). If ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), such that

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{L^{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}( \mu )}^{m} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$

Corollary 1.5

Under the same conditions of Corollary 1.4, there exists a constant \(C>0\), for all \(f \in {M^{q}_{p} (\mu )}, 1< p\leq q<\infty \), such that

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{M^{q}_{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}( \mu )}^{m} \Vert f \Vert _{M^{q}_{p} (\mu )}. \end{aligned}$$

Remark 1.6

If \(\rho =1,m=1\) on Corollary 1.4, which is Theorem 1.10 of [26]; if \(\rho =1,m=1\) on Corollary 1.5, which is Theorem 1.8 of [27], so our results contain their conclusions.

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 \(p\in (1,\infty )\), we denote by \(p'=\frac{p}{p-1}\) its conjugate index; \(m_{B}(b)\) is the mean value of B on B, namely \(m_{B}(b)=\frac{1}{\mu (B)} \int _{B} b(x)\,d\mu (x)\).

Proof of Theorem 1.2

We first recall the definition of a sharp maximal operator \(M^{\sharp }f(x)\) [21] over \((\mathcal{X}, d, \mu )\). For any \(f\in L^{1}_{\mathrm{loc}}(\mu )\),

$$\begin{aligned} \begin{aligned} M^{\sharp }f(x)= \sup_{x\in B}{ \frac{1}{\mu (6B)} \int _{B} \bigl\vert f-m_{ \tilde{B}}(f) \bigr\vert \,d \mu } +\sup_{(B,S)\in \Delta _{x}} { \frac{ \vert m_{B}(f)-m_{S}(f) \vert }{K_{B,S}} }, \end{aligned} \end{aligned}$$
(2.1)

here \(\Delta _{x}=\{(B,S):x\in B\subset S, B,S \text{are doubling balls}\}\). As usual, we let \(M^{\sharp }_{\delta }(f)(x) = [M^{\sharp }(|f(x)|^{\delta })]^{ \frac{1}{\delta }} \).

We will use the following lemma about sharp maximal function on \((\mathcal{X}, d, \mu )\) proved by Fu [18].

Lemma 2.1

\((\mathrm{i})\) Let \(p>1\), \(s\in [1,p), \zeta \in [5,\infty )\). For all \(f \in L^{1}_{\mathrm{loc}}(\mu )\) and \(x\in \mathcal{X}\),

$$\begin{aligned} \begin{aligned} M_{s,\zeta } f(x) =\sup _{x\in B}{ \biggl( \frac{1}{\mu (\zeta B)} \int _{B} \bigl\vert f(y) \bigr\vert ^{s} \,d \mu (y) \biggr)^{\frac{1}{s}}} \end{aligned} \end{aligned}$$
(2.2)

is bounded on \(L^{p}(\mu )\) and also bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\). If \(s=1\), then \(M_{s,\zeta } f =M_{(\zeta )} f\).

\((\mathrm{ii})\) For any \(\delta \in (0,1)\) and for \(f\in L^{1}_{\mathrm{loc}}(\mu )\), define

$$\begin{aligned} \begin{aligned} N_{\delta }f(x):=\sup_{x\in B:doubling}{ \biggl( \frac{1}{\mu (B)} \int _{B} \bigl\vert f(y) \bigr\vert ^{\delta } \,d \mu (y) \biggr)^{\frac{1}{\delta }}}, \end{aligned} \end{aligned}$$

then, for μ-almost every \(x\in \mathcal{X}\),

$$\begin{aligned} \bigl\vert f(x) \bigr\vert \leq N_{\delta }f(x). \end{aligned}$$
(2.3)

According to Theorem 4.2 in [21], we can easily get the following lemma.

Lemma 2.2

Let \(f\in L^{1}_{\mathrm{loc}}(\mu )\) satisfy \(\int _{\mathcal{X}} fd\mu =0\) when \(\| \mu \|:=\mu (\mathcal{X})<\infty \). Assume that \(\inf \{1,N_{\delta }f\}\in L^{p}(\mu )\), for any \(p\in (1,\infty )\), \(\delta \in (0,1)\), then there exists a constant \(C>0\),

$$\begin{aligned} \Vert N_{\delta }f \Vert _{L^{p} (\mu )} \leq C \bigl\Vert M^{\sharp }_{\delta }(f) \bigr\Vert _{L^{p} (\mu )}. \end{aligned}$$
(2.4)

The next two lemmas can be found in [30].

Lemma 2.3

Let \(\varrho >1\), for \(b\in L^{1}_{\mathrm{loc}}(\mu )\). The following statements are equivalent:

\((\mathrm{i})\) \(b\in \operatorname{RBMO}(\mu )\).

\((\mathrm{ii})\) There exists a constant \(C>0\), such that, for all balls B,

$$\begin{aligned} \frac{1}{\mu (\varrho B)} \int _{B} \bigl\vert b(x)-m_{\tilde{B}}(b) \bigr\vert \,d\mu (x) \leq C, \end{aligned}$$
(2.5)

and for all \((30\tau,\beta _{30\tau })\) doubling balls \(B\subset S\),

$$\begin{aligned} \bigl\vert m_{B}(b)-m_{S}(b) \bigr\vert \leq C K_{B,S}, \end{aligned}$$
(2.6)

where \(m_{B}(b)=\frac{1}{\mu (B)} \int _{B} b(x)\,d\mu (x)\). Furthermore, the infimum of all positive constants C satisfying (2.5) and (2.6) is an equivalent RBMO norm of b, denoted by \(\|b\|_{\operatorname{RBMO}(\mu )}\).

Lemma 2.4

Let \(\varrho >1, p\in [1,\infty )\), if \(b\in \operatorname{RBMO}\), for any ball B, then there exists a constant \(C>0\), we have

$$\begin{aligned} \biggl( {\frac{1}{\mu (\varrho B)} \int _{B} \bigl\vert b(x)-m_{ \tilde{B}}(b) \bigr\vert ^{p} \,d\mu (x)} \biggr)^{\frac{1}{p}} \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )}. \end{aligned}$$
(2.7)

We need the following lemma about the boundedness of parametric Marcinkiewicz integral operators.

Lemma 2.5

Let kernel \(K(x, y) \in L^{1}_{\mathrm{loc}} ( (\mathcal{X})^{2} \setminus \{(x,y):x=y\} )\) satisfy (1.9), (1.10) and (1.11), \(\mathcal{M}^{\rho }\) be as in (1.12), \(0<\rho <\infty \). If \(\mathcal{M}^{\rho }\) is bounded \(L^{p_{0}}(\mu ), 1< p_{0}<\infty \), then \(\mathcal{M}^{\rho }\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\).

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 \(\mathcal{M}^{\rho }\) be as in (1.12) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\). If \(0<\rho <\infty \), \(\delta \in (0,1)\) and ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}\), such that

$$\begin{aligned} \begin{aligned} M^{\sharp }_{\delta }\bigl[ \mathcal{M}^{\rho }_{b,m}(f) (x)\bigr] \leq{}& C \Biggl[ \sum _{k=0}^{m-1} \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m-k} M_{{\eta },30\tau }\bigl[ \mathcal{M}^{\rho }_{b,k}(f)\bigr](x) \\ &{} + \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} M_{{p},30\tau }(f) (x) \Biggr]. \end{aligned} \end{aligned}$$
(2.8)

Here \(\mathcal{M}^{\rho }_{b,1}=\mathcal{M}^{\rho }_{b}\) and \(\mathcal{M}^{\rho }_{b,0}=\mathcal{M}^{\rho }\).

Proof

Without loss of generality, we may assume that \(\|b\|_{\operatorname{RBMO}(\mu )}=1\). In order to prove (2.8), it suffices to prove that, for all \(x\in \mathcal{X}\) and balls \(B \ni x\),

$$\begin{aligned} & \biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert \mathcal{M}^{\rho }_{b,m}(f) (y)-h_{B} \bigr\vert ^{ \delta } \,d\mu (y) \biggr]^{\frac{1}{\delta }} \\ &\quad \leq C \Biggl[ \sum_{k=0}^{m-1} M_{{\eta },30 \tau } \bigl[\mathcal{M}^{\rho }_{b,k}(f)\bigr](x) + M_{{p},30\tau }(f) (x) \Biggr], \end{aligned}$$
(2.9)

and for all balls \(B\subset S\), S is a \((30\tau,\beta _{30\tau })\) doubling ball,

$$\begin{aligned} \begin{aligned} & \vert h_{B}-h_{S} \vert \\ & \quad\leq C [ K_{B,S} ]^{m} \Biggl[ \sum _{k=0}^{m-1} M_{{\eta },30\tau }\bigl[ \mathcal{M}^{\rho }_{b,k}(f)\bigr](x) + M_{{p},30\tau }(f) (x) \Biggr], \end{aligned} \end{aligned}$$
(2.10)

where

$$\begin{aligned} \begin{aligned} & h_{B}:=m_{B} \bigl[ \mathcal{M}^{\rho }\bigl(\bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6B}}}\bigr) \bigr], \\ & h_{S}:=m_{S} \bigl[ \mathcal{M}^{\rho }\bigl( \bigl[b-m_{\tilde{S}}(b)\bigr]^{m} f{ \chi _{\mathcal{X} \setminus {6S}}}\bigr) \bigr]. \end{aligned} \end{aligned}$$

Now we decompose the function f into two parts, i.e., \(f=f{\chi _{6B}}+f{\chi _{\mathcal{X} \setminus {6B}}}:=f_{1}+f_{2}\). We can write

$$\begin{aligned} \begin{aligned} & \bigl[b(y)-b(z)\bigr]^{m} \\ & \quad=\bigl[m_{\tilde{B}}(b)-b(z)\bigr]^{m} -\sum _{k=0}^{m-1} C_{m}^{k} \bigl[b(y)-b(z)\bigr]^{k} \bigl[m_{\tilde{B}}(b)-b(y) \bigr]^{m-k}. \end{aligned} \end{aligned}$$
(2.11)

Thus, we obtain

$$\begin{aligned} &\mathcal{M}^{\rho }_{b,m}(f) (y) \\ &\quad= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{B(y,t)} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \bigl[\bigl(b(y)-b(z)\bigr) \bigr]^{m}f(z) \,d\mu (z) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}} \\ &\quad\leq \sum_{k=0}^{m-1} C^{k}_{m} \bigl\vert m_{\tilde{B}}(b)-b(y) \bigr\vert ^{m-k} \mathcal{M}^{\rho }_{b,k}(f) (y) +\mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{ \tilde{B}}(b)\bigr]^{m} f \bigr) (y). \end{aligned}$$

Since \(0<\delta < 1\),

$$\begin{aligned} & \biggl[\frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert \mathcal{M}^{\rho }_{b,m}(f) (y)-h_{B} \bigr\vert ^{\delta } \,d\mu (y) \biggr]^{\frac{1}{\delta }} \\ &\quad\leq \Biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \Biggl\vert \sum_{k=0}^{m-1} C^{k}_{m} \bigl\vert m_{\tilde{B}}(b)-b(y) \bigr\vert ^{m-k} \mathcal{M}^{\rho }_{b,k}(f) (y) \Biggr\vert ^{\delta }\,d\mu (y) \Biggr]^{\frac{1}{\delta }} \\ &\qquad{} + \biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert \mathcal{M}^{ \rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{1} \bigr) (y) \bigr\vert ^{ \delta }\,d\mu (y) \biggr]^{\frac{1}{\delta }} \\ &\qquad{} + \biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert \mathcal{M}^{ \rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{2} \bigr) (y)-h_{B} \bigr\vert ^{\delta }\,d\mu (y) \biggr]^{\frac{1}{\delta }} \\ &\quad=:E_{1}+E_{2}+E_{3}. \end{aligned}$$

To estimate \(E_{1}\), let \(\gamma,\eta >1\), be such that

$$\begin{aligned} \frac{1}{\gamma } +\frac{1}{\eta } =\frac{1}{\delta }. \end{aligned}$$

By Hölder’s inequality and Lemma 2.4, we have

$$\begin{aligned} E_{1} \leq{}& \sum _{k=0}^{m-1} C^{k}_{m} \biggl[ \frac{1}{\mu (30\tau B)} \biggl( \int _{B} \bigl\vert \bigl[m_{\tilde{B}}(b)-b(y) \bigr]^{m-k} \bigr\vert ^{{\delta } \cdot {\frac{\gamma }{\delta }}}\,d\mu (y) \biggr)^{\frac{\delta }{\gamma }} \\ &{}\times \biggl( \int _{B} \bigl\vert \mathcal{M}^{\rho }_{b,k}(f) (y) \bigr\vert ^{{\delta }\cdot {\frac{\eta }{\delta }}}\,d\mu (y) \biggr)^{ \frac{\delta }{\eta }} \biggr]^{\frac{1}{\delta }} \\ \leq{}& C \sum_{k=0}^{m-1} \Vert b \Vert ^{m-k}_{\operatorname{RBMO}(\mu )} \biggl( \frac{1}{\mu (30\tau B)} \biggl( \int _{B} \bigl\vert \mathcal{M}^{\rho }_{b,k}(f) (y) \bigr\vert ^{\eta }\,d\mu (y) \biggr)^{\frac{1}{\eta }} \biggr) \\ \leq{}& C \sum_{k=0}^{m-1} M_{{\eta },30\tau } \bigl( \mathcal{M}^{\rho }_{b,k}(f) \bigr) (x). \end{aligned}$$

For \(E_{2}\), by the Kolmogorov inequality, Lemma 2.5, Hölder’s inequality and Lemma 2.4, we have

$$\begin{aligned} E_{2} &= \biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert \mathcal{M}^{\rho }\bigl(\bigl[b( \cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{1}\bigr) (y) \bigr\vert ^{ \delta }\,d\mu (y) \biggr]^{\frac{1}{\delta }} \\ &\leq \bigl\Vert \mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{1} \bigr) (y) \bigr\Vert _{L^{1,\infty }(6B,\frac{d\mu (y)}{\mu (30\tau B)})} \\ &\leq C \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert b(y)-m_{\tilde{B}}(b) \bigr\vert ^{m} \cdot \bigl\vert f_{1}(y) \bigr\vert \,d\mu (y) \\ &\leq C \biggl( \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert b(y)-m_{ \tilde{B}}(b) \bigr\vert ^{m\cdot p'} \,d\mu (y) \biggr)^{\frac{1}{p'}} \biggl( \frac{1}{\mu (30\tau B)} \int _{B} \bigl\vert f(y) \bigr\vert ^{p} \,d \mu (y) \biggr)^{\frac{1}{p}} \\ &\leq C \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} M_{{p},30\tau }(f) (x) \\ &\leq C M_{{p},30\tau }(f) (x). \end{aligned}$$

As to the estimate \(E_{3}\), we observe that

$$\begin{aligned} & \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{ \tilde{B}}(b)\bigr]^{m} f_{2} \bigr) (y) -h_{B} \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\mu (B)} \int _{B} \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(y,z)< t} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \\ &\qquad{}\times \bigl[b(z)-m_{\tilde{B}}(b)\bigr]^{m} f_{2}(z) \,d\mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (w) \\ &\qquad{} -\frac{1}{\mu (B)} \int _{B} \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(w,z)< t} \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \\ &\qquad{}\times \bigl[b(z)-m_{\tilde{B}}(b)\bigr]^{m} f_{2}(z) \,d\mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (w) \biggr\vert \\ &\quad= \biggl\vert \frac{1}{\mu (B)} \int _{B} \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b( \cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{2} \bigr) (y) -\mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b) \bigr]^{m} f_{2} \bigr) (w) \bigr\vert \,d\mu (w) \biggr\vert . \end{aligned}$$

Hence

$$\begin{aligned} E_{3} \leq{}& \biggl[ \frac{1}{\mu (30\tau B)} \int _{B} \biggl(\frac{1}{\mu (B)} \int _{B} \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b( \cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{2} \bigr) (y) \\ &{} -\mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b) \bigr]^{m} f_{2} \bigr) (\omega ) \bigr\vert \,d\mu (\omega ) \biggr)^{\delta }\,d\mu (y) \biggr]^{\frac{1}{\delta }}. \end{aligned}$$

In fact, for \(y, w\in B\), we observe that

$$\begin{aligned} & \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b)\bigr]^{m} f_{2} \bigr) (y) - \mathcal{M}^{\rho } \bigl( \bigl[b(\cdot )-m_{\tilde{B}}(b) \bigr]^{m} f_{2} \bigr) (\omega ) \bigr\vert \\ &\quad\leq \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(y,z)< t} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \\ &\qquad{} - \biggl\vert \int _{d(w,z)< t} \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \\ &\quad\leq \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(y,z)< t} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \\ &\qquad{} - \int _{d(w,z)< t} \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \\ &\quad\leq \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(y,z)\leq t\leq d(w,z)} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \\ &\qquad{}+ \biggl( \int _{0}^{\infty } \biggl\vert \int _{d(w,z)\leq t\leq d(y,z)} \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \\ &\qquad{}+ \biggl( \int _{0}^{\infty } \biggl\vert \int _{\max \{{d(y,z),d(w,z)}\}< t} \biggl( \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} - \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \biggr) \\ &\qquad{} \times \bigl[b(z)-m_{\tilde{B}}(b)\bigr]^{m} f_{2}(z) \,d \mu (z) \biggr\vert ^{2} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \\ &\quad:=F_{1}+F_{2}+F_{3}. \end{aligned}$$

In order to estimate \(E_{3}\), it suffices to estimate \(F_{1}, F_{2}, \text{and} F_{3}\). To estimate \(F_{1}\), for all \(y, w\in B\), we have \(d(y,z)\sim d(w,z)\sim d(c_{B},z)\). By the Minkowski inequality, (1.9), Hölder’s inequality, and Lemma 2.4, we get

$$\begin{aligned} F_{1}\leq{}& \int _{\mathcal{X} \setminus {6B}} \frac{ \vert K(y,z) \vert }{ \vert d(y,z) \vert ^{1-\rho }} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \biggl( \int _{d(y,z) \leq t \leq d(w,z)} \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (z) \\ \leq{}& C \int _{\mathcal{X} \setminus {6B}} \frac{ \vert d(y,z) \vert ^{\rho }}{\lambda (y,d(y,z))} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \frac{1}{ \vert d(y,z) \vert ^{\rho }} \biggl( \frac{d(w,y)}{d(w,z)} \biggr)^{\frac{1}{2}} \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } \int _{{6^{k+1}B} \setminus {6^{k}B}} \biggl( \frac{r_{B}}{6^{k}r_{B}} \biggr)^{\frac{1}{2}} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } \biggl( \frac{1}{6^{\frac{k}{2}}} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggr) \biggl[ \int _{6^{k+1}B} \bigl\vert b(z)-m_{\widetilde{6^{k+1}B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ & {} + \int _{6^{k+1}B} \bigl\vert m_{\widetilde{6^{k+1}B}}(b)-m_{ \tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \biggr] \\ \leq{}& C \sum_{k=1}^{\infty } \biggl( \frac{1}{6^{\frac{k}{2}}} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggr) \biggl( \int _{6^{k+1}{B}} \bigl\vert f(z) \bigr\vert ^{p} \,d \mu (z) \biggr)^{\frac{1}{p}} \\ &{}\times \biggl[ \biggl( \int _{6^{k+1}B} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}{B}}}(b) \bigr\vert ^{m\cdot p'}\,d\mu (z) \biggr)^{ \frac{1}{p'}} \\ &{} + k^{m} \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} \bigl[ \mu \bigl(30\tau \times 6 ^{k+1}B\bigr) \bigr]^{1-\frac{1}{p}} \biggr] \\ \leq{}& C \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} M_{p,30\tau }(f) (x) \sum_{k=1}^{\infty } \biggl( \frac{k^{m}+1}{6^{\frac{k}{2}}} \frac{\mu (30\tau \times 6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} \biggr) \\ \leq{}& C M_{p,30\tau }(f) (x). \end{aligned}$$

We have used the following fact in the last inequality:

$$\begin{aligned} \frac{\mu (30\tau \times 6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} &\leq \frac{\mu (6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} \\ &\leq \frac{\mu (6^{k+1}B)}{\mu (6^{k}B)} \cdot \frac{\mu (6^{k}B)}{\lambda (c_{B},6^{k}r_{B})} \\ &\leq C. \end{aligned}$$

Similarly, we get

$$\begin{aligned} F_{2}\leq C M_{p,30\tau }(f) (x). \end{aligned}$$

To estimate \(F_{3}\), for all \(y,w\in B\), we have \(d(y,z)\sim d(w,z)\sim d(c_{B},z)\), using the Minkowski inequality, we get

$$\begin{aligned} F_{3} \leq{}& \int _{\mathcal{X} \setminus {6B}} \biggl\vert \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} - \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \biggr\vert \\ & {}\times \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \biggl( \int _{d(y,z)}^{\infty } \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (z) \\ \leq{}& \int _{\mathcal{X} \setminus {6B}} \biggl\vert \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} - \frac{K(w,z)}{ \vert d(y,z) \vert ^{1-\rho }} \biggr\vert \\ &{}\times \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \biggl( \int _{d(y,z)}^{\infty } \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (z) \\ &{} + \int _{\mathcal{X} \setminus {6B}} \biggl\vert \frac{K(w,z)}{ \vert d(y,z) \vert ^{1-\rho }} - \frac{K(w,z)}{ \vert d(w,z) \vert ^{1-\rho }} \biggr\vert \\ &{}\times \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \biggl( \int _{d(y,z)}^{\infty } \frac{dt}{t^{2\rho +1}} \biggr)^{\frac{1}{2}} \,d\mu (z) \\ :={}&F_{31}+F_{32}. \end{aligned}$$

Next we estimate \(F_{31}\) and \(F_{32}\), respectively. For \(F_{31}\), by (1.10), Hölder’s inequality, Lemma 2.4 and (1.24), we have

$$\begin{aligned} F_{31} \leq{}& C \int _{\mathcal{X} \setminus {6B}} \biggl\vert \frac{d(y,z)}{\lambda (y,d(y,z))} \biggr\vert \omega \biggl( \frac{d(y,w)}{d(y,z)} \biggr) \frac{ \vert f(z) \vert }{ \vert d(y,z) \vert } \bigl\vert b(z)-m_{ \tilde{B}}(b) \bigr\vert ^{m} \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } \int _{{6^{k+1}B} \setminus {6^{k}B}} \omega \biggl( \frac{r_{B}}{6^{k}r_{B}} \biggr) \frac{1}{\lambda (c_{B},6^{k}r_{B})} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } \omega \bigl(6^{-k}\bigr) \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl[ \int _{6^{k+1}B} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ & {} + \int _{6^{k+1}B} \bigl\vert m_{\widetilde{6^{k+1}B}}(b)-m_{ \tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \biggr] \\ \leq {}&C \sum_{k=1}^{\infty } \omega \bigl(6^{-k}\bigr) \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl( \int _{6^{k+1}{B}} \bigl\vert f(z) \bigr\vert ^{p} \,d \mu (z) \biggr)^{\frac{1}{p}} \\ &{}\times \biggl[ \biggl( \int _{6^{k+1}B} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}{B}}}(b) \bigr\vert ^{m\cdot p'} \,d\mu (z) \biggr)^{ \frac{1}{p'}} \\ &{} + C \bigl\vert m_{\widetilde{6^{k+1}B}}(b)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl[\mu \bigl(6^{k+1}B\bigr) \bigr]^{1-\frac{1}{p}} \biggr] \\ \leq{}& C \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} M_{p,30\tau }(f) (x) \sum_{k=1}^{ \infty } \bigl(k^{m}+1 \bigr) \omega \bigl(6^{-k}\bigr) \frac{\mu (30\tau \times 6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} \\ \leq{}& C M_{p,30\tau }(f) (x). \end{aligned}$$

We have used the following fact in the last inequality:

$$\begin{aligned} \int _{0}^{1} \frac{\omega (t)}{t} \vert \mathrm{log}t \vert ^{m} \,dt \geq \sum_{k=1}^{\infty } \int _{6^{-k}}^{6^{1-k}} \frac{\omega (6^{-k})}{6^{1-k}} \bigl\vert \mathrm{log}6^{-k} \bigr\vert ^{m} \,dt \geq C \sum _{k=1}^{\infty } k^{m} \omega \bigl(6^{-k}\bigr). \end{aligned}$$

To estimate \(B_{32}\), for all \(y,w\in B\), if \(\rho \in (0,\infty )\), by (1.9), Hölder’s inequality and Lemma 2.4, we get

$$\begin{aligned} F_{32} \leq{}& C \int _{\mathcal{X} \setminus {6B}} \biggl\vert \frac{d(w,z)}{\lambda (w,d(w,z))} \biggr\vert \biggl\vert \frac{d(w,y)}{d(y,z)^{2-\rho }} \biggr\vert \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \frac{1}{ \vert d(y,z) \vert ^{\rho }} \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } 6^{-k} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \int _{{6^{k+1}B} \setminus {6^{k}B}} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ \leq{}& C \sum_{k=1}^{\infty } 6^{-k} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl[ \int _{6^{k+1}B} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ &{} + \int _{6^{k+1}B} \bigl\vert m_{\widetilde{6^{k+1}B}}(b)-m_{ \tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \biggr] \\ \leq{}& C \sum_{k=1}^{\infty } 6^{-k} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl( \int _{6^{k+1}{B}} \bigl\vert f(z) \bigr\vert ^{p} \,d \mu (z) \biggr)^{\frac{1}{p}} \\ & {}\times\biggl[ \biggl( \int _{6^{k+1}B} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}{B}}}(b) \bigr\vert ^{m\cdot p'} \,d\mu (z) \biggr)^{ \frac{1}{p'}} \\ &{} + k^{m} \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} \bigl[\mu \bigl(6^{k+1}B\bigr) \bigr]^{1- \frac{1}{p}} \biggr] \\ \leq{}& C \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} M_{p,30\tau }(f) (x) \sum_{k=1}^{\infty } \frac{k^{m}+1}{6^{k}} \frac{\mu (30\tau \times 6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} \\ \leq{}& C M_{p,30\tau }(f) (x). \end{aligned}$$

Then we have

$$\begin{aligned} F_{3}\leq C M_{p,(30\tau )}(f) (x). \end{aligned}$$

Moreover, combining the estimates of \(E_{1}, E_{2}, F_{1}, F_{2}\), and \(F_{3}\), we obtain the desired inequality (2.9).

Next we give the proof of (2.10). Write

$$\begin{aligned} & \vert h_{B}-h_{S} \vert \\ &\quad= \bigl\vert m_{B} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{ \chi _{\mathcal{X} \setminus {6B}}} \bigr) \bigr) -m_{S} \bigl( \mathcal{M}^{ \rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6S}}} \bigr) \bigr) \bigr\vert \\ &\quad\leq \bigl\vert m_{B} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) \bigr) -m_{S} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) \bigr) \bigr\vert \\ &\qquad{} + \bigl\vert m_{S} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) \bigr) -m_{S} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) \bigr) \bigr\vert \\ & \qquad{}+ \bigl\vert m_{B} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b)\bigr]^{m} f{\chi _{{6^{N_{1}}B} \setminus {6B}}} \bigr) \bigr) \bigr\vert + \bigl\vert m_{S} \bigl( \mathcal{M}^{\rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{m} f{\chi _{{6^{N_{1}}B} \setminus {6S}}} \bigr) \bigr) \bigr\vert \\ &\quad:=G_{1}+G_{2}+G_{3}+G_{4}. \end{aligned}$$

To estimate \(G_{1}\), it being similar to \(E_{3}\), we get

$$\begin{aligned} G_{1} \leq C M_{p,30\tau }(f) (x). \end{aligned}$$

For \(G_{2}\), we use

$$\begin{aligned} & \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m}-\bigl[b(z)-m_{S}(b)\bigr]^{m} \\ &\quad =\sum_{k=0}^{m-1} C_{m}^{k} \bigl[b(z)-m_{S}(b)\bigr]^{k} \cdot \bigl[m_{S}(b)-m_{ \tilde{B}}(b) \bigr]^{m-k}, \\ & \bigl[b(z)-m_{S}(b)\bigr]^{k} =\sum_{i=0}^{k} C_{k}^{i} \bigl[b(z)-b(y)\bigr]^{i} \cdot \bigl[b(y)-m_{S}(b) \bigr]^{k-i}. \end{aligned}$$

So we have

$$\begin{aligned} & \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b) \bigr]^{m} f{\chi _{ \mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) -\mathcal{M}^{\rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{m} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) \bigr\vert \\ &\quad\leq \bigl\vert \mathcal{M}^{\rho } \bigl[ \bigl( \bigl[b-m_{\tilde{B}}(b) \bigr]^{m} -\bigl[b-m_{S}(b)\bigr]^{m} \bigr) f{ \chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr] \bigr\vert \\ &\quad\leq \sum_{k=0}^{m-1} C^{k}_{m} \bigl\vert m_{\tilde{B}}(b)-m_{S}(b) \bigr\vert ^{m-k} \mathcal{M}^{\rho } \bigl( \bigl[b-m_{S}(b) \bigr]^{k} f{\chi _{ \mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) (y) \\ &\quad\leq C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \mathcal{M}^{\rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{k} f{\chi _{\mathcal{X} \setminus {6^{N_{1}}B}}} \bigr) (y) \\ &\quad\leq C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \Biggl[ \sum_{i=0}^{k} C^{i}_{k} \bigl\vert m_{S}(b)-b(y) \bigr\vert ^{k-i} \mathcal{M}^{\rho }_{b,i}( f{\chi _{ \mathcal{X} \setminus {6^{N_{1}}B}}} ) (y) \Biggr] \\ &\quad\leq C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \Biggl( \sum_{i=0}^{k} C^{i}_{k} \bigl\vert m_{S}(b)-b(y) \bigr\vert ^{k-i} \mathcal{M}^{\rho }_{b,i}(f) (y) \Biggr) \\ &\qquad{} +C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \mathcal{M}^{\rho } \bigl( \bigl\vert b-m_{S}(b) \bigr\vert ^{k} f{\chi _{{6^{N_{1}}B} \setminus 6B}} \bigr) (y) \\ &\qquad{} +C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \mathcal{M}^{\rho } \bigl( \bigl\vert b-m_{S}(b) \bigr\vert ^{k} f{\chi _{6B}} \bigr) (y). \end{aligned}$$

Therefore,

$$\begin{aligned} G_{2} ={}&m_{S} \Biggl[ C\sum _{k=0}^{m-1} [K_{B,S}]^{m-k} \Biggl( \sum_{i=0}^{k} \bigl\vert m_{S}(b)-b(y) \bigr\vert ^{k-i} M^{\rho }_{b,i}(f) \Biggr) \Biggr] \\ & {}+m_{S} \Biggl[ C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \mathcal{M}^{ \rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{k} f{\chi _{{6^{N_{1}}B} \setminus 6B}} \bigr) \Biggr] \\ &{} +m_{S} \Biggl[ C \sum_{k=0}^{m-1} [K_{B,S}]^{m-k} \mathcal{M}^{ \rho } \bigl( \bigl[b-m_{S}(b)\bigr]^{k} f{\chi _{6B}} \bigr) \Biggr] \\ :={}&H_{1}+H_{2}+H_{3}. \end{aligned}$$

With the same argument as for \(E_{1}\), we get

$$\begin{aligned} H_{1} \leq C [K_{B,S}]^{m} \sum _{k=0}^{m-1} M_{{\eta },30\tau }\bigl(M^{ \rho }_{b,k}(f) \bigr) (x). \end{aligned}$$

The estimates of \(H_{2}\) and \(H_{3}\) is very similar to \(G_{4}\) and \(E_{2}\), respectively, then we have

$$\begin{aligned} \begin{aligned}& H_{2} \leq C [K_{B,S}]^{m} M_{{p},30\tau }(f) (x), \\ &H_{3} \leq C [K_{B,S}]^{m} M_{{p},30\tau }(f) (x). \end{aligned} \end{aligned}$$

Therefore, combining \(H_{1},H_{2},H_{3}\), we have

$$\begin{aligned} \begin{aligned} G_{2} \leq C [K_{B,S}]^{m} \Biggl[ \sum_{k=0}^{m-1} M_{{\eta },30\tau } \bigl( \mathcal{M}^{\rho }_{b,k}(f)\bigr) (x) + M_{{p},30\tau }(f) (x) \Biggr]. \end{aligned} \end{aligned}$$

For \(G_{3}\), by (1.9), the Minkowski inequality, Hölder’s inequality, we obtain

$$\begin{aligned} & \bigl\vert \mathcal{M}^{\rho } \bigl( \bigl[b-m_{\tilde{B}}(b) \bigr]^{m} f_{{6^{N_{1}}B} \setminus {6B}} \bigr) \bigr\vert \\ &\quad= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{d(y,t)< t} \frac{K(y,z)}{ \vert d(y,z) \vert ^{1-\rho }} \bigl[b(z)-m_{\tilde{B}}(b) \bigr]^{m} f{\chi _{{6^{N_{1}}B} \setminus 6B}} \,d\mu (z) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{ \frac{1}{2}} \\ &\quad\leq \int _{{6^{N_{1}}B} \setminus {6B}} \frac{ \vert d(y,z) \vert ^{\rho }}{\lambda (y,d(y,z))} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \frac{1}{ \vert d(y,z) \vert ^{\rho }} \,d\mu (z) \\ &\quad\leq C \sum_{k=1}^{N_{1}-1} \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \bigl\vert b(z)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ &\quad\leq C \sum_{k=1}^{N_{1}-1} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl[ \int _{6^{k+1}B} \bigl\vert b(z)-m_{6^{k+1}B}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \\ &\qquad{} + \int _{6^{k+1}B} \bigl\vert m_{6^{k+1}B}(b)-m_{\tilde{B}}(b) \bigr\vert ^{m} \bigl\vert f(z) \bigr\vert \,d\mu (z) \biggr] \\ &\quad\leq C \sum_{k=1}^{N_{1}-1} \frac{1}{\lambda (c_{B},6^{k}r_{B})} \biggl( \int _{6^{k+1}{B}} \bigl\vert f(z) \bigr\vert ^{p} \,d \mu (z) \biggr)^{ \frac{1}{p}} \\ & \qquad{}\times\biggl[ \biggl( \int _{6^{k+1}B} \bigl\vert b(z)-m_{6^{k+1}{B}}(b) \bigr\vert ^{m\cdot p'} \,d\mu (z) \biggr)^{\frac{1}{p'}} \\ &\qquad{} +k^{m} \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} \bigl[ \mu \bigl(30\tau \times 6^{k+1}B\bigr) \bigr]^{1-\frac{1}{p}} \biggr] \\ &\quad \leq C \Vert b \Vert ^{m}_{\operatorname{RBMO}(\mu )} M_{p,30\tau }(f) (x) \sum_{k=1}^{N_{1}-1} \biggl[ \bigl(k^{m}+1\bigr) \frac{\mu (30\tau \times 6^{k+1}B)}{\lambda (c_{B},6^{k}r_{B})} \biggr] \\ &\quad\leq C M_{p,30\tau }(f) (x). \end{aligned}$$

Therefore,

$$\begin{aligned} G_{3} \leq C M_{p,30\tau }(f) (x). \end{aligned}$$

Similarly,

$$\begin{aligned} G_{4} \leq C M_{p,30\tau }(f) (x). \end{aligned}$$

Since (2.10) has been proved, Lemma 2.6 follows directly.

By Lemma 2.5 and the Marcinkiewicz interpolation theorem, we have

$$\begin{aligned} \begin{aligned} \bigl\Vert \mathcal{M}^{\rho }(f) \bigr\Vert _{L^{p} (\mu )} &\leq C \Vert f \Vert _{L^{p} (\mu )}. \end{aligned} \end{aligned}$$
(2.12)

Then using (2.3), (2.4), Lemma 2.6 and Lemma 2.1, we get

$$\begin{aligned} & \bigl\Vert \mathcal{M}^{\rho }_{b}(f) \bigr\Vert _{L^{p} (\mu )} \\ &\quad \leq \bigl\Vert N_{\delta }\bigl(\mathcal{M}^{\rho }_{b}(f) \bigr) \bigr\Vert _{L^{p} (\mu )} \\ &\quad \leq \bigl\Vert M^{\sharp }_{\delta }\bigl(\mathcal{M}^{\rho }_{b}(f) \bigr) \bigr\Vert _{L^{p} (\mu )} \\ &\quad \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )} \bigl\Vert \bigl[ M_{{\eta },30\tau }\bigl(\mathcal{M}^{ \rho }(f)\bigr) (x) + M_{{p},30\tau }(f) (x) \bigr] \bigr\Vert _{L^{p} (\mu )} \\ &\quad \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$

We set

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m-1}(f) \bigr\Vert _{L^{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}}^{m-1} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$

Finally, using mathematical induction, (2.3), (2.4), Lemma 2.6 and Lemma 2.1, we obtain

$$\begin{aligned} & \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{L^{p} (\mu )} \\ &\quad\leq \bigl\Vert N_{\delta }\bigl( \mathcal{M}^{\rho }_{b,m}(f) \bigr) \bigr\Vert _{L^{p} (\mu )} \leq \bigl\Vert M^{\sharp }_{ \delta } \bigl(\mathcal{M}^{\rho }_{b,m}(f)\bigr) \bigr\Vert _{L^{p} (\mu )} \\ &\quad\leq C \Biggl[ \sum_{k=0}^{m-1} \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m-k} \bigl\Vert M_{{\eta },30 \tau }\bigl( \mathcal{M}^{\rho }_{b,k}(f)\bigr) \bigr\Vert _{L^{p} (\mu )}+ \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} \bigl\Vert M_{{p},30\tau }(f) \bigr\Vert _{L^{p}( \mu )} \Biggr]. \\ &\quad\leq C \Vert b \Vert _{\operatorname{RBMO}}^{m} \Vert f \Vert _{L^{p}(\mu )}. \end{aligned}$$

Then Theorem 1.2 is proved. □

Proof of Theorem 1.3

In this section, we will prove Theorem 1.3. We recall the boundedness in Morrey space \(M^{q}_{p}(\mu )\) of the sharp maximal function on \((\mathcal{X},d,\mu )\) [22, 31].

Lemma 3.1

Let \(f\in L^{1}_{\mathrm{loc}}(\mu )\) satisfy \(\int _{\mathcal{X}} f(x) \,d\mu (x)=0\) when \(\| \mu \|:=\mu (\mathcal{X})<\infty \). Let \(1< p\leq q<\infty \), \(\delta \in (0,1)\). If \(\inf \{1,N_{\delta }f\} \in M^{q}_{p}(\mu )\), then there exists a constant \(C>0\), such that

$$\begin{aligned} \Vert N_{\delta }f \Vert _{M^{q}_{p}(\mu )} \leq C \bigl\Vert M^{\sharp }_{\delta }(f) \bigr\Vert _{M^{q}_{p}( \mu )}. \end{aligned}$$
(3.1)

Lemma 3.2

Let \(\zeta >1, 1< s< p\leq q<\infty \), \(M_{s,\zeta }f\) be as in (2.2) is bounded on Morrey space \(M^{q}_{p}(\mu )\), that is,

$$\begin{aligned} \Vert M_{s,\zeta } f \Vert _{M^{q}_{p}(\mu )} \leq C \Vert f \Vert _{M^{q}_{p}(\mu )}. \end{aligned}$$
(3.2)

Next we give the proof of Theorem 1.3.

In combination with Lemma 2.6, the differences between the proof of Theorem 1.2 and Theorem 1.3 are as follows:

By (1.7) and (2.12), it is easy to see

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }(f) \bigr\Vert _{M^{q}_{p}(\mu )} & =\sup_{B}{ \mu (\kappa B)^{\frac{1}{q}-\frac{1}{p}} \biggl( \int _{B} \bigl\vert \mathcal{M}^{\rho }(f) \bigr\vert ^{p} \,d\mu \biggr)^{\frac{1}{p}} } \\ & =\sup_{B}{ \mu (\kappa B)^{\frac{1}{q}-\frac{1}{p}} \bigl\Vert \mathcal{M}^{\rho }(f) \bigr\Vert _{L^{q} (\mu )} } \\ & \leq C \sup_{B}{ \mu (\kappa B)^{\frac{1}{q}-\frac{1}{p}} \Vert f \Vert _{L^{q} (\mu )} } \\ & \leq C \Vert f \Vert _{M^{q}_{p}(\mu )}. \end{aligned}$$

Then using (2.3), (3.1), Lemma 2.6 and (2.2), we have

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b}(f) \bigr\Vert _{M^{q}_{p}(\mu )}& \leq \bigl\Vert N_{\delta }\bigl( \mathcal{M}^{\rho }_{b}(f) \bigr) \bigr\Vert _{M^{q}_{p}(\mu )} \leq \bigl\Vert M^{\sharp }_{ \delta } \bigl(\mathcal{M}^{\rho }_{b}(f)\bigr) \bigr\Vert _{M^{q}_{p}(\mu )} \\ &\leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )} \bigl( \bigl\Vert M_{{\eta },30\tau } \bigl(\mathcal{M}^{ \rho }(f)\bigr) \bigr\Vert _{M^{q}_{p}(\mu )} + \bigl\Vert M_{{p},30\tau }(f) \bigr\Vert _{M^{q}_{p}(\mu )} \bigr) \\ &\leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )} \Vert f \Vert _{M^{q}_{p}(\mu )}. \end{aligned}$$

We set

$$\begin{aligned} \bigl\Vert \mathcal{M}^{\rho }_{b,m-1}(f) \bigr\Vert _{M^{q}_{p}(\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}}^{m-1} \Vert f \Vert _{M^{q}_{p}(\mu )}. \end{aligned}$$

Finally, by mathematical induction, (2.3), (3.1), Lemma 2.6 and (2.2), we get

$$\begin{aligned} & \bigl\Vert \mathcal{M}^{\rho }_{b,m}(f) \bigr\Vert _{M^{q}_{p}(\mu )}\\ &\quad \leq \bigl\Vert N_{\delta }\bigl( \mathcal{M}^{\rho }_{b,m}(f) \bigr) \bigr\Vert _{M^{q}_{p}(\mu )} \leq \bigl\Vert M^{\sharp }_{ \delta } \bigl(\mathcal{M}^{\rho }_{b,m}(f)\bigr) \bigr\Vert _{M^{q}_{p}(\mu )} \\ &\quad\leq C \Biggl[ \sum_{k=0}^{m-1} \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m-k} \bigl\Vert M_{{\eta },30 \tau }\bigl( \mathcal{M}^{\rho }_{b,k}(f)\bigr) \bigr\Vert _{M^{q}_{p}(\mu )} + \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} \bigl\Vert M_{{p},30\tau }(f) \bigr\Vert _{M^{q}_{p}( \mu )} \Biggr]. \\ &\quad\leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} \Vert f \Vert _{M^{q}_{p}(\mu )}. \end{aligned}$$

So, the proof of Theorem 1.3 is finished.

Proof of Corollary 1.4

If \(\rho =1,m=1\) on Corollary 1.4, which is Theorem 1.10 of [26]. The different between Corollary 1.4 and Theorem 1.10 of [26] is to estimate \(F_{31}\). So, in order to complete the proof of Corollary 1.4, it suffices to show that

$$\begin{aligned} F_{31} \leq{}& \Vert f \Vert _{L^{\infty }(\mu )} \sum _{k=1}^{\infty } \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \bigl\vert b(z)-m_{ \tilde{B}}(b) \bigr\vert ^{m} \,d\mu (z) \\ \leq{}& \Vert f \Vert _{L^{\infty }(\mu )} \sum_{k=1}^{\infty } \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \bigl\vert b(z)-m_{ \widetilde{6^{k+1}B}}(b) \bigr\vert ^{m} \,d\mu (z) \\ &{} + \Vert f \Vert _{L^{\infty }(\mu )} \sum_{k=1}^{\infty } \bigl\vert m_{ \widetilde{6^{k+1}B}}(b)-m_{\tilde{B}}(b) \bigr\vert ^{m} \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \,d\mu (z) \\ =:{}&F_{31}^{1}+F_{31}^{2}. \end{aligned}$$

Using Lemma 2.3, Lemma 2.4 in [26], (1.9) and (1.16), we have

$$\begin{aligned} F_{31}^{1} \leq{}& \Vert f \Vert _{L^{\infty }(\mu )} \sum _{k=1}^{\infty } \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \\ &{} \times \mathrm{log}^{m} \biggl[ 2+6^{k} \cdot \mu \bigl(30 \tau \times 6^{k}B\bigr) \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \biggr] \,d \mu (z) \\ &{} + \Vert f \Vert _{L^{\infty }(\mu )} \sum_{k=1}^{\infty } \frac{1}{6^{k} \cdot \mu (30\tau \times 6^{k}B)} \int _{{6^{k+1}B}} \mathrm{exp} \bigl( \bigl\vert b(z)-m_{\widetilde{6^{k+1}B}}(b) \bigr\vert \bigr)\,d\mu (z) \\ \leq{}& \Vert f \Vert _{L^{\infty }(\mu )}+ \Vert f \Vert _{L^{\infty }(\mu )} \sum_{k=1}^{ \infty } k^{m} \int _{{6^{k+1}B} \setminus {6^{k}B}} \frac{ \vert K(y,z)-K(w,z) \vert }{d(y,z)} \\ & {}\times \mathrm{log}^{m} \biggl( 2+ \frac{\mu (30\tau \times 6^{k}B)}{\lambda (c_{B},d(y,z))} \biggr)\,d\mu (z) \\ \leq{}& \Vert f \Vert _{L^{\infty }(\mu )} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} F_{31}^{2} &\leq \Vert f \Vert _{L^{\infty }(\mu )} \sum_{k=1}^{\infty } k^{m} \int _{{6^{k+1}B} \setminus {6^{k}B}} \bigl\vert K(y,z)-K(w,z) \bigr\vert \frac{1}{d(y,z)} \,d\mu (z) \\ &\leq \Vert f \Vert _{L^{\infty }(\mu )}. \end{aligned} \end{aligned}$$

This completes the proof of Corollary 1.4.

Using the similar to the argument in the proof of Corollary 1.4, we can get Corollary 1.5.

Some applications

Now we give the applications of Theorem 1.2 and Theorem 1.3 for the classical parametric Marcinkiewicz integral.

Let Ω be homogeneous of degree zero in \(R^{d}\) for \(d \geq 2\), integrable and have mean value zero on the unit sphere \(S^{d-1}\). In addition, Ω satisfies the following condition: with a constant \(C > 0\), for \(x, x^{\prime },y \in R^{d}\) and \(\vert x-x^{\prime } \vert \leq \frac{|x-y|}{2}\),

$$\begin{aligned} \bigl\vert \Omega (x-y)- \Omega \bigl(x'-y\bigr) \bigr\vert \leq C \omega \biggl( \frac{ \vert x-x' \vert }{ \vert x-y \vert } \biggr), \end{aligned}$$
(5.1)

where ω satisfies (1.24).

\({\mu }_{\Omega }^{\rho }\) be as in (1.1), where Ω satisfies the above condition (5.1). Moreover, \({\mu }^{\rho }_{\Omega,b,m}\) is generated by \({\mu }_{\Omega }^{\rho }\) with RBMO functions b, defined by

$$\begin{aligned} \mathcal{\mu }_{\Omega,b,m}^{\rho }(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{ \vert x-y \vert < t} \frac{\Omega (x-y)}{ \vert x-y \vert ^{d-\rho }} \bigl[b(x)-b(y) \bigr]^{m} f(y) \,dy \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}} \end{aligned}$$
(5.2)

where \(0<\rho <d\).

Theorem 5.1

Let \(0<\rho <d\) and Ω satisfies (5.1). \({\mu }_{\Omega,b,m}^{\rho }(f)\) be as in (5.2), and ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (R^{d})}, 1< p<\infty \) such that

$$\begin{aligned} \bigl\Vert {\mu }^{\rho }_{\Omega,b,m}(f) \bigr\Vert _{L^{p} (R^{d})} \leq C \Vert b \Vert _{\operatorname{RBMO}(R^{d})}^{m} \Vert f \Vert _{L^{p} (R^{d})}. \end{aligned}$$

For all \(f \in {M^{q}_{p} (R^{d})},1< p\leq q<\infty \), such that

$$\begin{aligned} \bigl\Vert {\mu }^{\rho }_{\Omega,b,m}(f) \bigr\Vert _{M^{q}_{p} (R^{d})} \leq C \Vert b \Vert _{\operatorname{RBMO}(R^{d})}^{m} \Vert f \Vert _{M^{q}_{p} (R^{d})}. \end{aligned}$$

Next, we give the applications of Theorem 1.2 and Theorem 1.3 for the parametric Marcinkiewicz integral operator in Euclidean space where μ satisfies the growth condition (1.2).

Let ω satisfy (1.24), K satisfy (1.17) and the following conditions hold with a constant \(C > 0\):

$$\begin{aligned} \bigl(\mathrm{b'}\bigr) \quad\bigl\vert K(x, y)-K \bigl(x^{\prime }, y \bigr) \bigr\vert \leq C \frac{1}{ \vert x-y \vert ^{d-1}} \omega \biggl( \frac{ \vert x-x^{\prime } \vert }{ \vert x-y \vert } \biggr), \end{aligned}$$

where \(x, x^{\prime },y \in R^{d}\) and \(\vert x-x^{\prime } \vert \leq \frac{|x-y|}{2}\).

$$\begin{aligned} \bigl(\mathrm{c'}\bigr)\quad \bigl\vert K(x, y)-K \bigl(x, y^{\prime } \bigr) \bigr\vert \leq C \frac{1}{ \vert x-y \vert ^{d-1}} \omega \biggl( \frac{ \vert y-y^{\prime } \vert }{ \vert x-y \vert } \biggr), \end{aligned}$$

where \(x, y^{\prime },y \in R^{d}\) and \(\vert y-y^{\prime } \vert \leq \frac{|x-y|}{2}\).

Define the parametric Marcinkiewicz integral operator \(M^{\rho }\) with respect to the kernel above as follows:

$$\begin{aligned} M^{\rho }(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{ \vert x-y \vert < t} \frac{K(x,y)}{ \vert x-y \vert ^{1-\rho }} f(y) \,d\mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}, \quad 0< \rho < \infty \end{aligned}$$
(5.3)

\(M_{b,m}^{\rho }\) is generated by \(M^{\rho }\) with RBMO functions b, defined by

$$\begin{aligned} M_{b,m}^{\rho }(f) (x)= \biggl( \int _{0}^{\infty } \biggl\vert \frac{1}{t^{\rho }} \int _{ \vert x-y \vert < t} \frac{K(x,y)}{ \vert x-y \vert ^{1-\rho }} \bigl[b(x)-b(y) \bigr]^{m} f(y) \,d\mu (y) \biggr\vert ^{2} \frac{dt}{t} \biggr)^{\frac{1}{2}}, \end{aligned}$$
(5.4)

Theorem 5.2

Let \(0<\rho <\infty \), and K satisfythe above conditions (1.17), \((\mathrm{b}')\) and \((\mathrm{c}')\). Let \(M^{\rho }, M_{b,m}^{\rho }\) be as in (5.3) and (5.4). Suppose that \(M^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\), ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}, 1< p<\infty \) such that

$$\begin{aligned} \bigl\Vert M_{b,m}^{\rho }(f) \bigr\Vert _{L^{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} \Vert f \Vert _{L^{p} (\mu )}. \end{aligned}$$

For all \(f \in {M^{q}_{p} (R^{d})},1< p\leq q<\infty \), we have

$$\begin{aligned} \bigl\Vert M_{b,m}^{\rho }(f) \bigr\Vert _{M^{q}_{p} (\mu )} \leq C \Vert b \Vert _{\operatorname{RBMO}(\mu )}^{m} \Vert f \Vert _{M^{q}_{p} (\mu )}. \end{aligned}$$

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Acknowledgements

The authors thank the anonymous referee for reading the paper carefully and giving some useful advices.

Funding

This work was supported partially by the NNSF of China (No. 11771399, 11961056) and Zhejiang University of Science and Technology graduate research innovation fund.

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Xiangxing, T., Qiange, Z. Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces. J Inequal Appl 2021, 115 (2021). https://doi.org/10.1186/s13660-021-02651-6

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MSC

  • 42B20
  • 42B25

Keywords

  • Log-Dini-type parametric Marcinkiewicz operator
  • \(\operatorname{RBMO}(\mu )\)
  • Commutator
  • Non-homogeneous metric measure space