Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures

Abstract

Under the assumption that μ is a nondoubling measure on ${\mathbb{R}}^d$ satisfying the growth condition, the author proves that the commutator ${\mathcal{M}}_b$ generated by the Marcinkiewicz integral operator and the Lipschitz function is bounded from the Hardy space $H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )$ into $L^q(\mu )$ for $1/q=1-\beta /n$ with the kernel satisfying a certain Hörmander-type condition. Moreover, the author shows that for $p=n/\beta$, ${\mathcal{M}}_b$ is bounded from the Morrey space ${\mathcal{M}}_q^p(\mu )$ into $RBMO(\mu )$, from $L^{n/\beta }(\mu )$ into $RBMO(\mu )$ and from ${\mathcal{M}}_q^p(\mu )$ into ${Lip}_{(\beta -\frac{n}{p})}(\mu )$, respectively.

MSC:42B25, 47B47, 42B20, 47A30.

1 Introduction

In recent years, harmonic analysis on spaces with nondoubling measures has become a very active research topic. There has been significant progress in the study of boundedness for singular integrals on these spaces; see [1–8]. Among a long list of research papers, some of them [9–11] are on the Marcinkiewicz integral operators. The motivation for developing the analysis with nondoubling measures and some important examples of nondoubling measures can be found in [12].

We recall that a nonnegative Radon measure μ on ${\mathbb{R}}^d$ is said to be a nondoubling measure if there is a positive constant $C_0$ such that for all $x\in {\mathbb{R}}^d$ and all $r>0$ it satisfies:

$$\mu (B(x,r))\le C_0{r}^n,$$

(1.1)

where n is a positive constant and $0<n\le d$, $B(x,r)$ is the open ball centered at x and having radius r.

Let $K(x,y)$ be a locally integrable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \{(x,y):x=y\}$. Assume that there exists a constant $C>0$ such that for any $x,y\in {\mathbb{R}}^d$ with $x\ne y$,

$$|K(x,y)|\le C{|x-y|}^{-(n-1)},$$

(1.2)

and for any $x,y,y^{\prime }\in {\mathbb{R}}^d$,

$${\int }_{|x-y|\ge 2|y-y^{\prime }|}[|K(x,y)-K(x,y^{\prime })|+|K(y,x)-K(y^{\prime },x)|]\frac{1}{|x-y|}d\mu (x)\le C.$$

(1.3)

The Marcinkiewicz integral ℳ associated to the kernel $K(x,y)$ and the measure μ as in (1.1) is defined by

$$\mathcal{M}(f)(x)={\left({\int }_0^{\mathrm{\infty }}{|{\int }_{|x-y|\le t}K(x,y)f(y)d\mu (y)|}^2\frac{dt}{t^3}\right)}^{1/2},x\in {\mathbb{R}}^d.$$

(1.4)

Let $b\in L_{\mathrm{loc}}(\mu )$, the Marcinkiewicz commutator ${\mathcal{M}}_b$ is formally defined by

$${\mathcal{M}}_b(f)(x)={\left({\int }_0^{\mathrm{\infty }}{|{\int }_{|x-y|\le t}[b(x)-b(y)]K(x,y)f(y)d\mu (y)|}^2\frac{dt}{t^3}\right)}^{1/2},x\in {\mathbb{R}}^d.$$

(1.5)

If μ is the d-dimensional Lebesgue measure in ${\mathbb{R}}^d$, and

$$K(x,y)=\frac{\mathrm{\Omega }(x-y)}{{|x-y|}^{d-1}}$$

with Ω homogeneous of degree zero and $\mathrm{\Omega }\in {Lip}_{\alpha }(S^{d-1})$ for some $\alpha \in (0,1]$, then it is easy to verify that $K(x,y)$ satisfies (1.2) and (1.3), and ℳ in (1.4) is just the higher dimensional Marcinkiewicz integral ${\mathcal{M}}_{\mathrm{\Omega }}$ defined by Stein in [13], which is important in classical harmonic analysis and is a focus of active research; see [14–20]. Particularly, we should mention the work of Torchinsky and Wang [21], where they established the $L^p({\mathbb{R}}^d)$ boundedness for the commutator generated by the Marcinkiewicz integral ${\mathcal{M}}_{\mathrm{\Omega }}$ and $BMO({\mathbb{R}}^d)$ function with $p\in (1,\mathrm{\infty })$. However, it is also worth to study the different behavior of another type commutator generated by the Marcinkiewicz integral ${\mathcal{M}}_{\mathrm{\Omega }}$ and ${Lip}_{\beta }({\mathbb{R}}^d)$ function, which was recently studied by Mo and Lu in [22] when Ω is homogeneous of degree zero and satisfies the cancellation condition. They obtained its boundedness from $L^p({\mathbb{R}}^d)$ into $L^q({\mathbb{R}}^d)$ for $1<p<n/\beta$ and $1/q=1/p-\beta /n$.

When μ satisfies growth condition (1.1), ℳ as in (1.4) was first introduced by Hu et al. in [9], where the boundedness of such an operator in $L^p(\mu )$ with $1<p<\mathrm{\infty }$ and the Hardy space $H^1(\mu )$ were established under the assumption that ℳ is bounded on $L^2(\mu )$ with the kernel $K(x,y)$ satisfying (1.2) and (1.3). Moreover, they got the same estimates for the commutator ${\mathcal{M}}_b$ defined as (1.5) with $b\in RBMO(\mu )$ when the kernel $K(x,y)$ satisfies (1.2) and (1.6), which is slightly stronger than (1.3) and is defined as follows:

$$\begin{array}{r}\underset{|y-y^{\prime }|\le l}{\underset{y,y^{\prime }\in {\mathbb{R}}^d,l>0,}{sup}}\sum _{k=1}^{\mathrm{\infty }}k{\int }_{2^{k}l<|x-y|\le 2^{k+1}l}[|K(x,y)-K(x,y^{\prime })|\\ +|K(y,x)-K(y^{\prime },x)|]\frac{1}{|x-y|}d\mu (x)\le C.\end{array}$$

(1.6)

However, in our problem, we discover that the kernels should satisfy some other kind of smoothness to replace condition (1.6).

Definition 1.1 Let $1\le s<\mathrm{\infty }$, $0<\epsilon <1$. We say that the kernel K satisfies a Hörmander-type condition if there exist $c_s>1$ and $C_s>0$ such that for any $x\in {\mathbb{R}}^d$ and $l>c_s|x|$,

$$\begin{array}{r}\underset{|y-y^{\prime }|\le l}{\underset{l>0,y,y^{\prime }\in {\mathbb{R}}^d}{sup}}\sum _{k=1}^{\mathrm{\infty }}{2}^{k\epsilon }{\left(2^{k}l\right)}^n(\frac{1}{{(2^{k}l)}^n}{\int }_{2^{k}l<|x-y|\le 2^{k+1}l}\left[\right(|K(x,y)-K(x,y^{\prime })|\\ {{+|K(y,x)-K(y^{\prime },x)|\left)\frac{1}{|x-y|}\right]}^{s}d\mu (x))}^{1/s}\le C_s.\end{array}$$

(1.7)

Directly, one can see that condition (1.7) can be rewritten as

We note that this kind of smoothness was not new. Condition (1.7′) is similar to the Hörmander-type condition which allows that the integral operator can be controlled by a maximal operator in doubling measure spaces, and also useful in the research of Schrödinger operators; see [23–25] for details. We denote by ${\mathcal{H}}^s$ the class of kernels satisfying this condition. It is clear that these classes are nested,

$${\mathcal{H}}^{s_2}\subset {\mathcal{H}}^{s_1}\subset {\mathcal{H}}^1,1<s_1<s_2<\mathrm{\infty }.$$

We should point out that ${\mathcal{H}}^1$ is not condition (1.6).

In [11], by supposing that the kernel K satisfies (1.2) and (1.3), the authors studied the commutator ${\mathcal{M}}_b$ in the case of $b\in {Lip}_{\beta }(\mu )$ and established that it is bounded from $L^p(\mu )$ into $L^q(\mu )$ for $1<p<n/\beta$ and $1/q=1/p-\beta /n$. Furthermore, when condition (1.3) is replaced by (1.7), ${\mathcal{M}}_b$ is bounded from $L^p(\mu )$ into ${Lip}_{\beta -n/p}(\mu )$ for some $0<\beta <1/2$ and $n/\beta <p<\mathrm{\infty }$, from $L^{n/\beta }(\mu )$ into $RBMO(\mu )$ for some $0<\beta <1$ and $n/\beta <p<\mathrm{\infty }$, respectively.

The purpose of this paper is to get some estimates for the commutator ${\mathcal{M}}_b$ with the kernel K satisfying (1.2) and (1.7) on the Hardy-type space and $RBMO(\mu )$ spaces. To be precise, we establish the boundedness of ${\mathcal{M}}_b$ in $H_{\mathrm{fin}}^{1,\mathrm{\infty }}(\mu )$ for $1/q=1-\beta /n$ in Section 2. In Section 3, we prove that ${\mathcal{M}}_b$ is bounded from $RBMO(\mu )$ to the Morrey space ${\mathcal{M}}_q^p(\mu )$, from $RBMO(\mu )$ to $L^{n/\beta }(\mu )$ for $p=n/\beta$.

Before stating our result, we need to recall some necessary notation and definitions. For a cube $Q\subset {\mathbb{R}}^d$, we mean a closed cube whose sides are parallel to the coordinate axes. We denote its center and its side length by $x_Q$ and $\ell (Q)$, respectively. Let $\alpha >1$, αQ denote the cube with the same center as Q and $\ell (\alpha Q)=\alpha \ell (Q)$. Given two cubes $Q\subset R$ in ${\mathbb{R}}^d$, set

$$S_{Q,R}=1+\sum _{k=1}^{N_{Q,R}}\frac{\mu (2^{k}Q)}{{[\ell (2^{k}Q)]}^n},$$

where $N_{Q,R}$ is the smallest positive integer k such that $\ell (2^{k}Q)\ge \ell (R)$. The concept $S_{Q,R}$ was introduced in [1], where some useful properties of $S_{Q,R}$ can be found.

The following characterization of the Lipschitz space ${Lip}_{\beta }(\mu )$ for $0<\beta \le 1$ in [26] plays a key role in the proof of theorems.

Lemma 1.1 For a function $b\in L_{\mathrm{loc}}^1(\mu )$, conditions I, II and III below are equivalent.

• (I) There is a constant $C_1\ge 0$ such that

  $$|b(x)-b(y)|\le C_1{|x-y|}^{\beta }$$

for μ-almost every x and y in the support of μ.

• (II) There exist some constant $C_2\ge 0$ and a collection of numbers $b_Q$ such that these two properties hold: for any cube Q,

  $$\frac{1}{\mu (2Q)}{\int }_Q|b(x)-b_Q|d\mu (x)\le C_2\ell {(Q)}^{\beta },$$

  (1.8)

and for any cube R such that $Q\subset R$ and $\ell (R)\le 2\ell (Q)$,

$$|m_Q(b)-m_R(b)|\le C_2\ell {(Q)}^{\beta }.$$

(1.9)

• (III) For any given p, $1\le p\le \mathrm{\infty }$, there is a constant $C(p)\ge 0$ such that for every cube Q, we have

  $${[\frac{1}{\mu (Q)}{\int }_Q{|b(x)-m_Q(b)|}^{p}d\mu (x)]}^{1/p}\le C(p)\ell {(Q)}^{\beta },$$

  (1.10)

where, and in the sequel,

$$m_Q(b)=\frac{1}{\mu (Q)}{\int }_{Q}b(y)d\mu (y),$$

and also for any cube R such that $Q\subset R$ and $\ell (R)\le 2\ell (Q)$,

$$|m_Q(b)-m_R(b)|\le C(p)\ell {(Q)}^{\beta }.$$

In addition, the quantities $inf\{C_1\}$, $inf\{C_2\}$ and $inf\{C(p)\}$ with a fixed p are equivalent and denoted by ${\parallel b\parallel }_{{Lip}_{\beta }}$.

Remark 1.1 Lemma 1.1 is a slight variant of Theorem 2.3 in [26]. To be precise, if we replace all balls in Theorem 2.3 of [26] by cubes, we then obtain Lemma 1.1.

Remark 1.2 For $0<\beta \le 1$, (1.9) is equivalent to

$$|b_Q-b_R|\le C_2^{\prime }{S}_{Q,R}\ell {(R)}^{\beta }$$

(1.11)

for any two cubes $Q\subset R$ with $\ell (R)\le 2\ell (Q)$; see Remark 2.7 in [26]. Note that for $\beta =0$ (1.9) and (1.10) is just the space $RBMO(\mu )$ of Tolsa; see [27]. Therefore, the space ${Lip}_{\beta }(\mu )$ for $0\le \beta \le 1$ can be seen as a member of a family containing $RBMO(\mu )$.

We also need the following lemma for the $L^p(\mu )$-boundedness of ${\mathcal{M}}_b$, which was proved in [11].

Lemma 1.2 Let $b\in {Lip}_{\beta }(\mu )$, $0<\beta \le 1$. Suppose that $K(x,y)$ satisfies (1.2) and (1.3) and that ${\mathcal{M}}_b$ is as in (1.5). If ℳ is bounded on $L^2(\mu )$, then there exists a positive constant $C>0$ such that for all bounded functions f with compact support,

$${\parallel {\mathcal{M}}_b(f)\parallel }_{L^q(\mu )}\le C{\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathrm{L}}^p(\mu )},$$

(1.12)

where $1<p<n/\beta$ and $1/q=1/p-\beta /n$.

Throughout this paper, we use the constant C with subscripts to indicate its dependence on the parameters. We denote simply by $A\lesssim B$ if there exists a constant $C>0$ such that $A\le CB$; and $A\sim B$ means that $A\lesssim B$ and $B\lesssim A$. For a μ-measurable set E, ${\chi }_E$ denotes its characteristic function. For any $p\in [1,\mathrm{\infty }]$, we denote by $p^{\prime }$ its conjugate index, namely, $1/p+1/p^{\prime }=1$.

2 Boundedness of ${\mathcal{M}}_b$ in Hardy spaces

This section is devoted to the behavior of the commutator ${\mathcal{M}}_b$ in Hardy spaces. In order to define the Hardy space $H^1(\mu )$, Tolsa introduced the ‘grand’ maximal operator $M_{\mathrm{\Phi }}$ in [27].

Definition 2.1 Given $f\in L_{\mathrm{loc}}^1(\mu )$, we define

$$M_{\mathrm{\Phi }}f(x)=\underset{\phi \sim x}{sup}\left|{\int }_{{\mathbb{R}}^d}f\phi d\mu \right|,$$

where the notation $\phi \sim x$ means that $\phi \in L^1(\mu )\cap C^1({\mathbb{R}}^d)$ and satisfies

• (i) ${\parallel \phi \parallel }_{L^1(\mu )}\le 1$,

• (ii) $0\le \phi (y)\le \frac{1}{{|y-x|}^n}$ for all $y\in {\mathbb{R}}^d$,

• (iii) $|{\phi }^{\prime }(y)|\le \frac{1}{{|y-x|}^{n+1}}$ for all $y\in {\mathbb{R}}^d$.

Based on Theorem 1.2 in [27], we can define the Hardy space $H^1(\mu )$ as follows; see also [1].

Definition 2.2 The Hardy space $H^1(\mu )$ is the set of all functions $f\in L^1(\mu )$ satisfying that ${\int }_{{\mathbb{R}}^d}fd\mu =0$ and $M_{\mathrm{\Phi }}f\in L^1(\mu )$. Moreover, we define the norm of $f\in H^1(\mu )$ by

$${\parallel f\parallel }_{H^1(\mu )}={\parallel f\parallel }_{L^1(\mu )}+{\parallel M_{\mathrm{\Phi }}f\parallel }_{L^1(\mu )}.$$

We recall the atomic Hardy space $H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )$ as follows.

Definition 2.3 Let $\rho >1$. A function $h\in L_{\mathrm{loc}}^1(\mu )$ is called an atomic block if

• (1) there exists some cube R such that $supph\subset R$,

• (2) ${\int }_{{\mathbb{R}}^d}h(x)d\mu (x)=0$,

• (3) for $i=1,2$, there are functions $a_i$ supported on cubes $Q_i\subset R$ and numbers ${\lambda }_i\in \mathbb{R}$ such that $h={\lambda }_1{a}_1+{\lambda }_2{a}_2$, and

  $${\parallel a_i\parallel }_{L^{\mathrm{\infty }}(\mu )}\le {[\mu (\rho Q_i)S_{Q_i,R}]}^{-1}.$$

Then we define

$${|h|}_{H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )}=|{\lambda }_1|+|{\lambda }_2|.$$

Define $H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )$ and $H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )$ as follows:

$${\parallel f\parallel }_{H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )}=inf\{\sum _j^{\mathrm{\infty }}{|h_j|}_{H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )}:f=\sum _{j=1}^{\mathrm{\infty }}{h}_j,{\{h_j\}}_{j\in \mathbb{N}}\text{ are }(1,\mathrm{\infty },0)\text{-atoms}\}$$

and

$${\parallel f\parallel }_{H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )}=inf\{\sum _{j=1}^k{|h_j|}_{H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )}:f=\sum _{j=1}^k{h}_j,{\{h_j\}}_{j=1}^k\text{ are }(1,\mathrm{\infty },0)\text{-atoms}\},$$

where the infimum is taken over all possible decompositions of f in atomic blocks, $H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )$ is the set of all finite linear combinations of $(1,\mathrm{\infty },0)$-atoms.

Remark 2.1 It was proved in [1] that for each $\rho >1$, the atomic Hardy space $H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )$ is independent of the choice of ρ.

To establish the boundedness of operators in Hardy-type spaces on ${\mathbb{R}}^n$, one usually appeals to the atomic decomposition characterization (see [28, 29]) of these spaces, which means that a function or distribution in Hardy-type spaces can be represented as a linear combination of atoms. Then the boundedness of linear operators in Hardy-type spaces can be deduced from their behavior on atoms in principle. However, Meyer [30] (see also [31]) gave an example of $f\in H^1({\mathbb{R}}^n)$ whose norm cannot be achieved by its finite atomic decompositions via $(1,\mathrm{\infty },0)$-atoms. Based on this fact, Bownik [31] (Theorem 2) constructed a surprising example of a linear functional defined on a dense subspace of $H^1({\mathbb{R}}^n)$, which maps all $(1,\mathrm{\infty },0)$-atoms into bounded scalars, but yet cannot extend to a bounded linear functional on the whole $H^1({\mathbb{R}}^n)$.

Recently, in [32], a boundedness criterion was established via Lusin function characterizations of Hardy spaces on ${\mathbb{R}}^n$ as follows: a sublinear operator T extends to a bounded sublinear operator from Hardy spaces $H^p({\mathbb{R}}^n)$ with $p\in (0,1]$ to some quasi-Banach space B if and only if T maps all $(p,2,s)$-atoms into uniformly bounded elements of B for some $s\ge [n(1/p-1)]$. Here and in what follows $[t]$ means the integer part of real t. This result shows the structural difference between atomic characterization of $H^p({\mathbb{R}}^n)$ via $(p,2,s)$-atoms and $(p,\mathrm{\infty },s)$-atoms. On the other hand, Meda et al. [33] independently obtained some similar results by grand maximal function characterizations of Hardy spaces on ${\mathbb{R}}^n$. In fact, let $p\in (0,1]$, $p<q\in [1,\mathrm{\infty }]$ and integer $s\ge [n(1/p-1)]$, and let $H_{\mathrm{fin}}^{p,q,s}({\mathbb{R}}^n)$ be the set of all finite linear combinations of $(p,q,s)$-atoms. Denote by $C({\mathbb{R}}^n)$ the set of all continuous functions. For any $f\in H_{\mathrm{fin}}^{p,q,s}({\mathbb{R}}^n)$, when $q<\mathrm{\infty }$ or $f\in H_{\mathrm{fin}}^{p,q,s}({\mathbb{R}}^n)\cap C({\mathbb{R}}^n)$ when $q=\mathrm{\infty }$, Meda et al. in [33] proved that $f\in H^p({\mathbb{R}}^n)$ can be achieved by a finite atomic decomposition via $(p,q,s)$-atom when $q<\mathrm{\infty }$ or continuous $(p,q,s)$-atom when $q=\mathrm{\infty }$; from this, they further deduced that if T is a linear operator and maps all $(1,q,0)$-atoms with $q\in (1,\mathrm{\infty })$ or all continuous $(1,q,0)$-atoms with $q=\mathrm{\infty }$ into uniformly bounded elements of some Banach space B, then T uniquely extends to a bounded linear operator from $H^1({\mathbb{R}}^n)$ to B which coincides with T on these $(1,q,0)$-atoms.

According to the theory of Meda et al. [33], we get the result as follows.

Theorem 2.1 Let $0<\beta \le 1$, $b\in {Lip}_{\beta }(\mu )$ and $1/q=1-\beta /n$. Suppose that K satisfies (1.2) and ${\mathcal{H}}^q$ condition. If $f\in H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )$, then ${\mathcal{M}}_b$ is bounded from the Hardy space into the Lebesgue space, namely, there exists a positive constant C such that

$${\parallel {\mathcal{M}}_b(f)\parallel }_{L^q(\mu )}\le C{\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{H_{\mathrm{fin}}^{1,\mathrm{\infty },0}(\mu )}.$$

Proof of Theorem 2.1 Via Remark 2.1, without loss of generality, we may assume that $\rho =4$ and $f=\sum h$ as a finite sum of atomic blocks defined in Definition 2.3. It is easy to see that we only need to prove the theorem for one atomic block h. Let R be a cube such that $supph\subset R$, ${\int }_{{\mathbb{R}}^d}h(x)d\mu (x)=0$, and

$$h(x)={\lambda }_1{a}_1(x)+{\lambda }_2{a}_2(x),$$

(2.1)

where ${\lambda }_i$ for $i=1,2$, is a real number, ${|h|}_{H_{\mathrm{atb}}^{1,\mathrm{\infty }}(\mu )}=|{\lambda }_1|+|{\lambda }_2|$, $a_i$ for $i=1,2$, is a bounded function supported on some cube $Q_i\subset R$ and it satisfies

$${\parallel a_i\parallel }_{L^{\mathrm{\infty }}(\mu )}\le {[\mu (4Q_i)S_{Q_i,R}]}^{-1}.$$

(2.2)

Write

$$\begin{array}{r}{\parallel {\mathcal{M}}_b(h)\parallel }_{L^q(\mu )}\\ \lesssim {({\int }_{2R}{|{\mathcal{M}}_b(h)(x)|}^{q}d\mu (x))}^{1/q}+{({\int }_{{\mathbb{R}}^d\setminus (2R)}{|{\mathcal{M}}_b(h)(x)|}^{q}d\mu (x))}^{1/q}\\ \lesssim {({\int }_{2R}{|{\mathcal{M}}_b(h)(x)|}^{q}d\mu (x))}^{1/q}\\ +\{{\int }_{{\mathbb{R}}^d\setminus (2R)}({\int }_0^{|x-x_R|+2\ell (R)}|{\int }_{|x-y|\le t}K(x,y)\\ {{{\times [b(x)-b(y)]h(y)d\mu (y)|}^2\frac{dt}{t^3})}^{q/2}d\mu (x)\}}^{1/q}\\ +{\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{\left({\int }_{|x-x_R|+2\ell (R)}^{\mathrm{\infty }}{|{\int }_{|x-y|\le t}K(x,y)[b(x)-b(y)]h(y)d\mu (y)|}^2\frac{dt}{t^3}\right)}^{q/2}d\mu (x)\}}^{1/q}\\ =\mathrm{I}+\mathrm{II}+\mathrm{III}.\end{array}$$

By (2.1), we have

$$\begin{array}{rcl}\mathrm{I}& \le & |{\lambda }_1|{({\int }_{2R}{|{\mathcal{M}}_b(a_1)(x)|}^{q}d\mu (x))}^{1/q}+|{\lambda }_2|{({\int }_{2R}{|{\mathcal{M}}_b(a_2)(x)|}^{q}d\mu (x))}^{1/q}\\ =& {\mathrm{I}}_1+{\mathrm{I}}_2.\end{array}$$

To estimate ${\mathrm{I}}_1$, we write

$$\begin{array}{rl}{\mathrm{I}}_1& \le |{\lambda }_1|{({\int }_{2Q_1}{|{\mathcal{M}}_b(a_1)(x)|}^{q}d\mu (x))}^{1/q}+|{\lambda }_1|{({\int }_{2R\setminus 2Q_1}{|{\mathcal{M}}_b(a_1)(x)|}^{q}d\mu (x))}^{1/q}\\ ={\mathrm{I}}_{11}+{\mathrm{I}}_{12}.\end{array}$$

Choose $p_1$ and $q_1$ such that $1<p_1<n/\beta$, $1<q<q_1$ and $1/q_1=1/p_1-\beta /n$. By the Hölder inequality, the fact that $S_{Q_1,R}\ge 1$ and the $(L^{p_1}(\mu ),L^{q_1}(\mu ))$-boundedness of ${\mathcal{M}}_b$ (Lemma 1.2), we have that

$$\begin{array}{rl}{\mathrm{I}}_{11}& \le |{\lambda }_1|{\left[{\int }_{2Q_1}\right|{\mathcal{M}}_b(a_1)(x){|}^{q_1}d\mu (x)]}^{1/q_1}\mu {(2Q_1)}^{1/q-1/q_1}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\parallel a_1\parallel }_{L^{p_1}(\mu )}\mu {(2Q_1)}^{1/q-1/q_1}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\parallel a_1\parallel }_{L^{\mathrm{\infty }}(\mu )}\mu {(2Q_1)}^{1/p_1+1/q-1/q_1}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|.\end{array}$$

Denote $N_{2Q_1,2R}$ simply by $N_1$. Invoking the fact that ${\parallel a_1\parallel }_{L^{\mathrm{\infty }}(\mu )}\le {[\mu (4Q_1)S_{Q_1,R}]}^{-1}$, we thus get

$$\begin{array}{rcl}{\mathrm{I}}_{12}& \lesssim & |{\lambda }_1|{\{\sum _{k=1}^{N_1+1}{\int }_{2^{k+1}{Q}_1\setminus 2^k{Q}_1}{\left[{\int }_0^{\mathrm{\infty }}{|{\int }_{|x-y|\le t}\frac{[b(x)-b(y)]}{{|x-y|}^{n-1}}{a}_1(y)d\mu (y)|}^2\frac{dt}{t^3}\right]}^{q/2}d\mu (x)\}}^{1/q}\\ \lesssim & |{\lambda }_1|{\{\sum _{k=1}^{N_1+1}{\int }_{2^{k+1}{Q}_1\setminus 2^k{Q}_1}{[{\int }_{Q_1}\frac{|b(x)-b(y)|}{{|x-y|}^{n-1}}|a_1(y)|{\left({\int }_{|x-y|}^{\mathrm{\infty }}\frac{dt}{t^3}\right)}^{1/2}d\mu (y)]}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & |{\lambda }_1|{\{\sum _{k=1}^{N_1+1}{\int }_{2^{k+1}{Q}_1\setminus 2^k{Q}_1}{[{\int }_{Q_1}\frac{|b(x)-b(y)|}{{|x-y|}^n}|a_1(y)|d\mu (y)]}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\{\sum _{k=1}^{N_1+1}\ell {\left(2^k{Q}_1\right)}^{q(\beta -n)}{\int }_{2^{k+1}{Q}_1\setminus 2^k{Q}_1}{[{\int }_{Q_1}|a_1(y)|d\mu (y)]}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\left\{\sum _{k=1}^{N_1+1}\ell {\left(2^k{Q}_1\right)}^{q(\beta -n)}\mu \left(2^{(k+1)}{Q}_1\right){\parallel a_1\parallel }_{L^{\mathrm{\infty }}(\mu )}^q\mu {(Q_1)}^q\right\}}^{1/q}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\left\{\sum _{k=1}^{N_1+1}\ell {\left(2^k{Q}_1\right)}^{q(\beta -n)}\mu {(4Q_1)}^{-q}{S}_{Q_1,R}^{-q}\mu \left(2^{(k+1)}{Q}_1\right)\mu {(Q_1)}^q\right\}}^{1/q}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|{\left\{S_{Q_1,R}^{-q}\sum _{k=2}^{N_1+1}\frac{\mu (2^k{Q}_1)}{\ell {(2^k{Q}_1)}^n}\right\}}^{1/q}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_1|,\end{array}$$

here we have used the fact that

$$\sum _{k=2}^{N_1+1}\frac{\mu (2^{k}Q)}{l{(2^{k}Q)}^n}\le C{S}_{Q_1,R};$$

see [1, 27] for details.

The estimates for ${\mathrm{I}}_{11}$ and ${\mathrm{I}}_{12}$ give the desired estimate for ${\mathrm{I}}_1$. A similar argument tells us that

$${\mathrm{I}}_2\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}|{\lambda }_2|.$$

Combining the estimates for ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$ yields the desired estimate for I.

For $i=1,2$, $y\in Q_i\subset R$, $x\in {\mathbb{R}}^d\setminus (2R)$, we have $|x-y|\sim |x-x_R|\sim |x-x_R|+2\ell (R)$. By the Minkowski inequality, we get

$$\begin{array}{rcl}\mathrm{II}& \lesssim & {\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{[{\int }_{{\mathbb{R}}^d}{\left({\int }_{|x-y|}^{|x-x_R|+2\ell (R)}\frac{dt}{t^3}\right)}^{1/2}\frac{|h(y)|}{{|x-y|}^{n-1}}|b(x)-b(y)|d\mu (y)]}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\int }_R\{{\int }_{{\mathbb{R}}^d\setminus (2R)}[{(\frac{1}{{(|x-x_R|+2\ell (R))}^2}-\frac{1}{{|x-y|}^2})}^{1/2}\\ {{\times \frac{|h(y)|}{{|x-y|}^{n-1}}|b(x)-b(y)|]}^{q}d\mu (x)\}}^{1/q}d\mu (y)\\ \lesssim & {\int }_R{\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{\left(\frac{\ell {(R)}^{1/2}}{{|x-y|}^{3/2}}\frac{|h(y)|}{{|x-y|}^{n-1}}|b(x)-b(y)|\right)}^{q}d\mu (x)\}}^{1/q}d\mu (y)\\ \lesssim & {\int }_R{\{\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{(k+1)}R\setminus 2^{k}R}{\left(\frac{\ell {(R)}^{1/2}}{{|x-y|}^{n-\beta +1/2}}{\parallel b\parallel }_{{Lip}_{\beta }}\right)}^{q}d\mu (x)\}}^{1/q}|h(y)|d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}(\sum _{j=1}^2|{\lambda }_j|{\parallel a_j\parallel }_{L^1(\mu )})\left\{\sum _{k=1}^{\mathrm{\infty }}\ell {(R)}^{1/2}\ell {\left(2^{k}R\right)}^{-n+\beta -1/2}\mu {\left(2^{k+1}R\right)}^{1/q}\right\}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}(\sum _{j=1}^2|{\lambda }_j|).\end{array}$$

For any $y\in R$, we have $t\ge |x-x_R|+2\ell (R)\ge |x-x_R|+|y-x_R|\ge |x-y|$. It follows that

$$\begin{array}{rcl}\mathrm{III}& \le & {\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{|{\int }_{R}K(x,y)[b(x)-b(y)]h(y)d\mu (y){\left({\int }_{|x-x_R|+2\ell (R)}^{\mathrm{\infty }}\frac{dt}{t^3}\right)}^{1/2}|}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{|{\int }_{R}K(x,y)[b(x)-b(y)]h(y)d\mu (y)\frac{1}{|x-x_R|+2\ell (R)}|}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{|{\int }_R\frac{K(x,y)h(y)}{|x-x_R|+2\ell (R)}[b(x)-m_R(b)]d\mu (y)|}^{q}d\mu (x)\}}^{1/q}\\ +{\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{|{\int }_R\frac{K(x,y)h(y)}{|x-x_R|+2\ell (R)}[m_R(b)-b(y)]d\mu (y)|}^{q}d\mu (x)\}}^{1/q}\\ =& {\mathrm{III}}_1+{\mathrm{III}}_2.\end{array}$$

For ${\mathrm{III}}_1$, by the Minkowski inequality, we have

$$\begin{array}{rcl}{\mathrm{III}}_1& =& {\{{\int }_{{\mathbb{R}}^d\setminus (2R)}{|[b(x)-m_R(b)]{\int }_R\frac{K(x,y)-K(x,x_R)}{|x-x_R|+2\ell (R)}h(y)d\mu (y)|}^{q}d\mu (x)\}}^{1/q}\\ \lesssim & {\int }_R\sum _{k=1}^m{({\int }_{2^{k+1}R\setminus 2^{k}R}{\left[{\parallel b\parallel }_{{Lip}_{\beta }}{|x-x_R|}^{\beta }\frac{|K(x,y)-K(x,x_R)|}{|x-y|}\right]}^{q}d\mu (x))}^{1/q}|h(y)|d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\int }_R|h(y)|\sum _{k=1}^m{({\int }_{2^{k+1}R\setminus 2^{k}R}{\left[\ell {\left(2^{k}R\right)}^{\beta }\frac{|K(x,y)-K(x,x_R)|}{|x-y|}\right]}^{q}d\mu (x))}^{1/q}d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\int }_R|h(y)|\sum _{k=1}^m{2}^{k(\beta -\epsilon -n+n/q)}\ell {(R)}^{\beta -n+n/q}{2}^{k\epsilon }{[2^k\ell (R)]}^n\\ \times {(\frac{1}{{[2^k\ell (R)]}^n}{\int }_{2^k\ell (R)<|x-y|\le 2^{k+1}\ell (R)}{\left[\frac{|K(x,y)-K(x,x_R)|}{|x-y|}\right]}^{q}d\mu (x))}^{1/q}d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{j=1}^2|{\lambda }_j|{\parallel a_j\parallel }_{L^1(\mu )}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{j=1}^2|{\lambda }_j|,\end{array}$$

here we used the fact that $1/q=1-\beta /n$ and $0<\epsilon \le 1$.

We now turn to estimate ${\mathrm{III}}_2$. Note that for any $y\in R$, $x\in {\mathbb{R}}^d\setminus 2R$, we have $|x-y|\sim |x-x_R|+2\ell (R)$, so by the Minkowski inequality,

$$\begin{array}{rcl}{\mathrm{III}}_2& \le & {\int }_R\sum _{k=1}^{\mathrm{\infty }}{({\int }_{2^{k+1}R\setminus 2^{k}R}{\left[\frac{|K(x,y)|}{|x-y|}|m_R(b)-b(y)|\right]}^{q}d\mu (x))}^{1/q}|h(y)|d\mu (y)\\ \lesssim & {\int }_R\sum _{k=1}^{\mathrm{\infty }}{({\int }_{2^{k+1}R\setminus 2^{k}R}{\left[\frac{|m_R(b)-b(y)|}{{|x-y|}^n}\right]}^{q}d\mu (x))}^{1/q}|h(y)|d\mu (y)\\ \lesssim & {\int }_R\sum _{k=1}^{\mathrm{\infty }}{\parallel b\parallel }_{{Lip}_{\beta }}\ell {(R)}^{\beta }\ell {\left(2^{k}R\right)}^{-n}\mu {\left(2^{k+1}R\right)}^{1/q}\sum _{j=1}^2|{\lambda }_j||a_j(y)|d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{j=1}^2|{\lambda }_j|{\parallel a_j\parallel }_{L^1(\mu )}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{j=1}^2|{\lambda }_j|.\end{array}$$

Then

$$\mathrm{III}\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{j=1}^2|{\lambda }_j|.$$

Combining the estimates for I, II and III yields that

$${\parallel {\mathcal{M}}_b(h)\parallel }_{L^q(\mu )}\le C{|h|}_{H_{\mathrm{atb}}^{1,\mathrm{\infty },0}(\mu )},$$

and this is the result of Theorem 2.1. □

3 Boundedness of ${\mathcal{M}}_b$ in $RBMO(\mu )$ space

In this section, we investigate the boundedness for the commutator ${\mathcal{M}}_b$ as in (1.5) in the space $RBMO(\mu )$ for $f\in {\mathcal{M}}_q^p(\mu )$ and $f\in L^{n/\beta }(\mu )$, respectively.

Firstly, we recall the definition of the Morrey space with nondoubling measure denoted by ${\mathcal{M}}_q^p(\mu )$, which was introduced by Sawano and Tanaka in [34–36].

Definition 3.1 Let $k>1$ and $1\le q\le p<\mathrm{\infty }$. We define the Morrey space ${\mathcal{M}}_q^p(\mu )$ as

$${\mathcal{M}}_q^p(\mu ):=\{f\in L_{\mathrm{loc}}^q(\mu )|{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}<\mathrm{\infty }\},$$

where the norm ${\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}$ is given by

$${\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}:=\underset{Q}{sup}\mu {(kQ)}^{\frac{1}{p}-\frac{1}{q}}{\left({\int }_Q{|f|}^{q}d\mu \right)}^{\frac{1}{q}}.$$

We should note that the parameter $k>1$ appearing in the definition does not affect the definition of the space ${\mathcal{M}}_q^p(\mu )$, and the space ${\mathcal{M}}_q^p(\mu )$ is a Banach space with its norm; see [34]. By using the Hölder inequality to (1.5), it is easy to see that for all $1\le q_2\le q_1\le p$, we have

$$L^p(\mu )={\mathcal{M}}_p^p(\mu )\subset {\mathcal{M}}_{q_1}^p(\mu )\subset {\mathcal{M}}_{q_2}^p(\mu ).$$

Theorem 3.1 Let $b\in {Lip}_{\beta }(\mu )$, $0<\beta \le 1$, $1\le q<p=\frac{n}{\beta }$. Suppose that K satisfies (1.2) and ${\mathcal{H}}^{p^{\prime }}$ condition, ℳ is bounded on $L^2(\mu )$ and ${\mathcal{M}}_b$ is defined as in (1.5). Then there exists a positive constant C such that for all $f\in {\mathcal{M}}_q^p(\mu )$,

$${\parallel {\mathcal{M}}_b(f)\parallel }_{RBMO(\mu )}\le C{\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

Theorem 3.2 Let $b\in {Lip}_{\beta }(\mu )$, $0<\beta \le 1$ and $p=n/\beta$. Suppose that K satisfies (1.2) and ${\mathcal{H}}^{n/(n-\beta )}$ condition. If ℳ is bounded on $L^2(\mu )$ and ${\mathcal{M}}_b$ is defined as in (1.5), then there is a constant $C>0$ such that for all bounded functions f with compact support,

$${\parallel {\mathcal{M}}_b(f)\parallel }_{RBMO(\mu )}\le C{\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{L^{n/\beta }(\mu )}.$$

Theorem 3.3 Let $b\in {Lip}_{\beta }(\mu )$, $0<\beta \le 1$, $1\le q<p$ and $p>n/\beta$. Suppose that K satisfies (1.2) and ${\mathcal{H}}^{p^{\prime }}$ condition, ℳ is bounded on $L^2(\mu )$ and ${\mathcal{M}}_b$ is defined as in (1.5). Then there exists a positive constant C such that for all $f\in {\mathcal{M}}_q^p(\mu )$,

$${\parallel {\mathcal{M}}_b(f)\parallel }_{{Lip}_{(\beta -\frac{n}{p})}}\le C{\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

Remark 3.1 By the Minkowski inequality and the kernel condition, we get that

$$\begin{array}{rcl}{\mathcal{M}}_b(f)(x)& =& {\left({\int }_0^{\mathrm{\infty }}{|{\int }_{|x-y|\le t}[b(x)-b(y)]K(x,y)f(y)d\mu (y)|}^2\frac{dt}{t^3}\right)}^{1/2}\\ =& {\left({\int }_0^{\mathrm{\infty }}{|\frac{1}{t}{\int }_{|x-y|\le t}[b(x)-b(y)]K(x,y)f(y)d\mu (y)|}^2\frac{dt}{t}\right)}^{1/2}\\ =& {\int }_{{\mathbb{R}}^d}|K(x,y)[b(x)-b(y)]f(y)|{\left({\int }_{|x-y|\le t}^{\mathrm{\infty }}\frac{dt}{t^3}\right)}^{1/2}d\mu (y)\\ \lesssim & {\int }_{{\mathbb{R}}^d}\frac{|[b(x)-b(y)]f(y)|}{{|x-y|}^{n-1}}{|x-y|}^{-1}d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\int }_{{\mathbb{R}}^d}\frac{|f(y)|}{{|x-y|}^{n-\beta }}d\mu (y)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{I}_{\beta }(|f|)(x),\end{array}$$

where $I_{\beta }$ is the fractional integral operator. Then ${\mathcal{M}}_b(f)\in L_{\mathrm{loc}}^1(\mu )$.

Remark 3.2 Theorem 3.2 can be deduced as a conclusion of Theorem 3.1 in the case of $p=q=\frac{n}{\beta }$.

Remark 3.3 Applying Lemma 1.1, a slight change in the proof of Theorem 3.1 actually shows Theorem 3.3 and we leave the details to the reader.

Proof of Theorem 3.1 For any cubes Q and R in ${\mathbb{R}}^d$ such that $Q\subset R$ satisfies $\ell (R)\le 2\ell (Q)$, let

$$a_Q=m_Q[{\mathcal{M}}_b(f{\chi }_{{\mathbb{R}}^d\setminus \frac{3}{2}Q})]$$

and

$$a_R=m_R[{\mathcal{M}}_b(f{\chi }_{{\mathbb{R}}^d\setminus \frac{3}{2}R})].$$

It is easy to see that $a_Q$ and $a_R$ are real numbers. By Lemma 1.1, we need to show that for some fixed $r>q$ there exists a constant $C>0$ such that

$${(\frac{1}{\mu (2Q)}{\int }_Q{|{\mathcal{M}}_b(f)(x)-a_Q|}^{r}d\mu (x))}^{1/r}\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}$$

(3.1)

and

$$|a_Q-a_R|\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

(3.2)

Let us first prove estimate (3.1). For a fixed cube Q and $x\in Q$, decompose $f=f_1+f_2$, where $f_1=f{\chi }_{\frac{3}{2}Q}$ and $f_2=f-f_1$. Write that

$$\begin{array}{r}\frac{1}{\mu (2Q)}{\int }_Q{|{\mathcal{M}}_b(f)(x)-a_Q|}^{r}d\mu (x)\\ \le \frac{1}{\mu (2Q)}{\int }_Q{|{\mathcal{M}}_b(f_1)(x)|}^{r}d\mu (x)+\frac{1}{\mu (2Q)}{\int }_Q{|{\mathcal{M}}_b(f_2)(x)-a_Q|}^{r}d\mu (x)\\ ={\mathrm{I}}_1+{\mathrm{I}}_2.\end{array}$$

For $1/r=1/q-\beta /n$ and $p=n/\beta$, it follows that

$$\begin{array}{rl}{\mathrm{I}}_1& \lesssim \frac{1}{\mu (2Q)}{[{\int }_Q{|{\mathcal{M}}_b(f_1)(x)|}^{r}d\mu (x)]}^{1/r}\\ \lesssim \frac{1}{\mu (2Q)}{\parallel b\parallel }_{{Lip}_{\beta }}^r{({\int }_{\frac{3}{2}Q}\left|f{(x)}^q\right|d\mu (x))}^{r/q}\\ \lesssim \frac{1}{\mu (2Q)}{\parallel b\parallel }_{{Lip}_{\beta }}^r{\left\{{(\mu {(2Q)}^{1/p-1/q}{\int }_{\frac{3}{2}Q}{|f(x)|}^{q}d\mu (x))}^{1/q}\right\}}^r\mu {(2Q)}^{(1/p-1/q)r}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}^r{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}^r\mu {(2Q)}^{(1/q-1/p)r-1}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}^r{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}^r\mu {(2Q)}^{(1/r+\beta /n-1/p)r-1}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}^r{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}^r.\end{array}$$

In order to estimate the term ${\mathrm{I}}_2$, set

$$\begin{array}{c}{\mathrm{D}}_1(x,y)={\left({\int }_0^{\mathrm{\infty }}{\left[{\int }_{|x-z|\le t<|y-z|}|K(x,z)|\right|b(z)-m_Q(b)||f_2(z)|d\mu (z)]}^2\frac{dt}{t^3}\right)}^{1/2},\hfill \\ {\mathrm{D}}_2(x,y)={\left({\int }_0^{\mathrm{\infty }}{\left[{\int }_{|y-z|\le t<|x-z|}|K(y,z)|\right|b(z)-m_Q(b)||f_2(z)|d\mu (z)]}^2\frac{dt}{t^3}\right)}^{1/2}\hfill \end{array}$$

and

$${\mathrm{D}}_3(x,y)={\left({\int }_0^{\mathrm{\infty }}{\left[{\int }_{\begin{array}{c}|y-z|\le t\\ |x-z|\le t\end{array}}|K(x,z)-K(y,z)|\right|b(z)-m_Q(b)||f_2(z)|d\mu (z)]}^2\frac{dt}{t^3}\right)}^{1/2}.$$

It is easy to get that for any $x,y\in Q$,

$$\begin{array}{r}|{\mathcal{M}}_b(f_2)(x)-{\mathcal{M}}_b(f_2)(y)|\\ =|{\left({\int }_0^{\mathrm{\infty }}{|{\int }_{|x-z|\le t}[b(x)-b(z)]K(x,z)f_2(z)d\mu (z)|}^2\frac{dt}{t^3}\right)}^{1/2}\\ -{\left({\int }_0^{\mathrm{\infty }}{|{\int }_{|y-z|\le t}[b(y)-b(z)]K(y,z)f_2(z)d\mu (z)|}^2\frac{dt}{t^3}\right)}^{1/2}|\\ \lesssim \sum _{j=1}^3{\mathrm{D}}_j(x,y).\end{array}$$

For ${\mathrm{D}}_1(x,y)$, since $x,y\in Q$, $z\in \frac{3}{2}Q$, we thus get

$$\begin{array}{rl}{\mathrm{D}}_1& \le {\left({\int }_0^{\mathrm{\infty }}{[{\int }_{|x-z|\le t<|y-z|}\frac{|b(z)-m_Q(b)|}{{|x-z|}^{n-1}}|f_2(z)|d\mu (z)]}^2\frac{dt}{t^3}\right)}^{1/2}\\ \lesssim {\int }_{|x-z|<|y-z|}\frac{|b(z)-m_Q(b)|}{{|x-z|}^{n-1}}|f_2(z)|{\left[{\int }_{|y-z|}^{|x-z|}\frac{dt}{t^3}\right]}^{1/2}d\mu (z)\\ \lesssim {\int }_{|x-z|<|y-z|}\frac{|b(z)-m_Q(b)|}{{|x-z|}^{n-1}}|f_2(z)|\frac{\ell {(Q)}^{1/2}}{{|x-z|}^{3/2}}d\mu (z)\\ \lesssim \ell {(Q)}^{1/2}{\int }_{{\mathbb{R}}^d\setminus \frac{3}{2}Q}\frac{|b(z)-m_Q(b)|}{{|x-z|}^{n+1/2}}|f(z)|d\mu (z)\\ \lesssim \ell {(Q)}^{1/2}\sum _{k=1}^{\mathrm{\infty }}{\int }_{\frac{3}{2}{2}^{k}Q\setminus \frac{3}{2}{2}^{k-1}Q}\frac{|b(z)-m_Q(b)|}{{|x-z|}^{n+1/2}}|f(z)|d\mu (z)\\ \lesssim \ell {(Q)}^{1/2}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{\ell {(\frac{3}{2}{2}^{k}Q)}^{n+1/2}}{\int }_{\frac{3}{2}{2}^{k}Q}|b(z)-m_Q(b)||f(z)|d\mu (z)\\ \lesssim \sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}\frac{1}{\ell {(2^{k}Q)}^n}{({\int }_{\frac{3}{2}{2}^{k}Q}{|b(z)-m_Q(b)|}^{q^{\prime }}d\mu (z))}^{1/q^{\prime }}{({\int }_{\frac{3}{2}{2}^{k}Q}{|f(z)|}^{q}d\mu (z))}^{1/q}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k/2}\ell {\left(2^{k}Q\right)}^{\beta -n}\mu {\left(\frac{3}{2}{2}^{k}Q\right)}^{1-1/q}\mu {\left(2^{k+2}Q\right)}^{1/q-1/p}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k/2}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )},\end{array}$$

here we used the Minkowski inequality, $\beta =n/p$, (1.10) of Lemma 1.1 and the fact that

$$|b(z)-m_Q(b)|\lesssim \ell {\left(2^{k}Q\right)}^{\beta }{\parallel b\parallel }_{{Lip}_{\beta }}\text{for }z\in {\mathbb{R}}^d\setminus \frac{3}{2}Q.$$

By a similar argument, it follows that

$${\mathrm{D}}_2\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

Finally, by the condition ${\mathcal{H}}^{p^{\prime }}$, which the kernel K satisfies, and the fact that $\beta =n/p$, applying the Minkowski inequality, we have

$$\begin{array}{rl}{\mathrm{D}}_3(x,y)=& {\left({\int }_0^{\mathrm{\infty }}{\left[{\int }_{\begin{array}{c}|y-z|\le t\\ |x-z|\le t\end{array}}|K(x,z)-K(y,z)|\right|b(z)-m_Q(b)||f_2(z)|d\mu (z)]}^2\frac{dt}{t^3}\right)}^{1/2}\\ \lesssim & {\int }_{{\mathbb{R}}^d}|K(x,z)-K(y,z)||b(z)-m_Q(b)||f_2(z)|{\left[{\int }_{\begin{array}{c}|y-z|\le t\\ |x-z|\le t\end{array}}\frac{dt}{t^3}\right]}^{1/2}d\mu (z)\\ \lesssim & \sum _{k=1}^{\mathrm{\infty }}{\int }_{\frac{3}{2}{2}^{k}Q\setminus \frac{3}{2}{2}^{k-1}Q}|K(x,z)-K(y,z)||b(z)-m_Q(b)|\frac{|f(z)|}{|y-z|}d\mu (z)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}\sum _{k=1}^{\mathrm{\infty }}\ell {\left(2^{k}Q\right)}^{\beta }{\int }_{\frac{3}{2}{2}^{k}Q\setminus \frac{3}{2}{2}^{k-1}Q}|K(x,z)-K(y,z)|\frac{|f(z)|}{|y-z|}d\mu (z)\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}\sum _{k=1}^{\mathrm{\infty }}\ell {\left(2^{k}Q\right)}^{\beta -n/p}\ell {\left(2^{k}Q\right)}^{n/q}\\ \times {({\int }_{\frac{3}{2}{2}^{k}Q\setminus \frac{3}{2}{2}^{k-1}Q}{\left[|K(x,z)-K(y,z)|\frac{1}{|y-z|}\right]}^{q^{\prime }}d\mu (z))}^{1/q^{\prime }}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}\sum _{k=1}^{\mathrm{\infty }}\ell {\left(2^{k}Q\right)}^n\\ \times {(\frac{1}{\ell {(2^{k}Q)}^n}{\int }_{\frac{3}{2}{2}^{k}Q\setminus \frac{3}{2}{2}^{k-1}Q}{\left[|K(x,z)-K(y,z)|\frac{1}{|y-z|}\right]}^{q^{\prime }}d\mu (z))}^{1/q^{\prime }}\\ \lesssim & {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.\end{array}$$

Combining these estimates, we conclude that

$${\mathrm{I}}_2\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}^r{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}^r,$$

and so estimate (3.1) is proved. □

We proceed to show (3.2). For any cubes $Q\subset R$ with $x\in Q$, where Q is arbitrary and R is a doubling cube with $\ell (R)\le \ell (Q)$, denote $N_{Q,R}+1$ simply by N. Write

$$\begin{array}{rl}|a_Q-a_R|\le & |m_R[{\mathcal{M}}_b(f{\chi }_{{\mathbb{R}}^d\setminus 2^{N}Q})]-m_Q[{\mathcal{M}}_b(f{\chi }_{{\mathbb{R}}^d\setminus 2^{N}Q})]|\\ +\left|m_Q[{\mathcal{M}}_b(f{\chi }_{2^{N}Q\setminus \frac{3}{2}Q})]\right|+\left|m_R[{\mathcal{M}}_b(f{\chi }_{2^{N}Q\setminus \frac{3}{2}R})]\right|\\ =& {\mathrm{E}}_1+{\mathrm{E}}_2+{\mathrm{E}}_3.\end{array}$$

As in the estimate for the term ${\mathrm{I}}_2$, we have

$${\mathrm{E}}_1\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

We conclude from $y\in R$, $z\in 2^{N}Q\setminus \frac{3}{2}Q$ that

$$\begin{array}{rl}{\mathcal{M}}_b(f{\chi }_{2^{N}Q\setminus \frac{3}{2}R})(y)& \lesssim {\int }_{2^{N}Q\setminus \frac{3}{2}R}|K(y,z)(b(y)-b(z))f(z)|{\left({\int }_{|y-z|}^{\mathrm{\infty }}\frac{dt}{t^3}\right)}^{1/2}d\mu (z)\\ \lesssim {\int }_{2^{N}Q\setminus \frac{3}{2}R}\frac{|b(y)-b(z)|}{{|y-z|}^n}|f(z)|d\mu (z)\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\int }_{2^{N}Q\setminus \frac{3}{2}R}\frac{|f(z)|}{{|y-z|}^{n-\beta }}d\mu (z)\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}\ell {(R)}^{\beta -n}{({\int }_{2^{N}Q}{|f(z)|}^{q}d\mu (z))}^{1/q}\mu {\left(2^{N}Q\right)}^{1/q^{\prime }}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}\ell {\left(2^{N}Q\right)}^{\beta -n+n-n/q+n/q-n/p}\\ \lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.\end{array}$$

Taking mean over $y\in R$, we obtain

$${\mathrm{E}}_3\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

Analysis similar to that in the estimate for ${\mathrm{E}}_3$ shows that

$${\mathrm{E}}_2\lesssim {\parallel b\parallel }_{{Lip}_{\beta }}{\parallel f\parallel }_{{\mathcal{M}}_q^p(\mu )}.$$

Finally, we get (3.2) and this is precisely the assertion of Theorem 3.1.