On weighted compactness of commutator of semi-group maximal function and fractional integrals associated to Schrödinger operators

Abstract

Let $\mathcal {T}^*$ and $\mathcal {I}_\alpha$ be the semi-group maximal function and fractional integrals associated to the Schrödinger operator $-\Delta +V(x)$, respectively, with V satisfying an appropriate reverse Hölder inequality. In this paper, we show that the commutator of $\mathcal {T}^*$ is a compact operator on $L^p(w)$ for $1<p<\infty$ if $b\in \text {CMO}_\theta (\rho )(\mathbb {R}^n)$ and $w\in A_p^{\rho ,\theta }(\mathbb {R}^n)$. We also show that the commutator of $\mathcal {I}_\alpha$ is a compact operator from $L^p(w^p)$ to $L^q(w^q)$ for $0<\alpha<n,\ 1<p<\alpha /n$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ if $w\in A_{(p,q)}^{\rho }$ and $b\in \text {CMO}_\theta (\rho )({\mathbb {R}^n})$. Here $\text { CMO}_\theta (\rho )(\mathbb {R}^n)$ denotes the closure of $\mathcal {C}_c^\infty (\mathbb {R}^n)$ in the $\text {BMO}_\theta (\rho )(\mathbb {R}^n)$ (which is larger than the classical $\text {BMO}(\mathbb {R}^n)$ space) topology. The space where b belongs and the weighs class w belongs are more larger than the usual $\text {CMO}(\mathbb {R}^n)$ space and the Muckenhoupt $A_p$ weights class, respectively.

1 Introduction

Consider the Schrödinger operator

$$\begin{aligned} L=-\Delta +V(x) \end{aligned}$$

in $\mathbb {R}^n,\ n\ge 3$. Where $\Delta$ is the Laplacian operator on $\mathbb {R}^n$ and the function V is a nonnegative potential belonging to certain reverse Hölder class $RH_q$ with an exponent $q>n/2$, that is, there exists a constant $C>0$ such that

$$\begin{aligned} \Big (\frac{1}{|B(x,r)|}\int _{B(x,r)}V^q(y)dy\Big )^{\frac{1}{q}}\le C\Big (\frac{1}{|B(x,r)|}\int _{B(x,r)}V(y)dy\Big ), \end{aligned}$$

for every ball $B(x,r)\subset \mathbb {R}^n$. It is worth pointing out that if $V\in RH_q$ for some $q>1$, then there exists $\epsilon >0$, such that $V\in RH_{q+\epsilon }$ (see [13]). On the other hand, the Hölder’s inequality gives that $RH_{q_1}\subset RH_{q_2}$ if $q_1\ge q_2>1$. Therefore, the assumption $q>n/2$ is equivalent to the case $q\ge n/2$. Thoughout this paper, we always assume that $V \not \equiv 0$ and $V\in RH_q$ with $q\ge n/2$.

The Schrödinger operator L with nonnegative potential is very useful in the study of certain subelliptic operators. For instance, by taking the partial Fourier transform in the t variable, the operator $-\Delta _x-V(x)\partial ^2_t$ is reduced to $-\Delta _x+V(x)\xi ^2$ (See [20]). Some basic results on L, including certain estimates of the fundamental solutions of L and the boundedness on $L^p$ of Riesz transforms $\Delta L^{-1/2}$ were obtained by Fefferman [12], Shen [21] and Zhong [32].

Attentions have also been paid to the study of function spaces associated to L. It was Dziubański and Zienkiewicz [11] who characterized the Hardy space $H^1_L$ related to the Schrödinger operator. Later on, for $0<p\le 1$, the $H^p_L$ space with potentials from reverse Hölder classes were studied in [9, 10]. Subsequently, Yang et al. [31] characterize the localized Hardy spaces by establishing the boundedness of Riesz transforms, maximal operators and endpoint estimates of fractional integrals associated with L.

For the classical Schrödinger operators L , there are many interesting results of its associated Riesz transforms, which essentially and heavily depend on the properties of $e^{-tL}$. The properties of semi-group $e^{-tL}$ such as the positivity, Gaussian estimates and off-diagonal estimates play a fundamental role in the study of Riesz transform. The maximal function defined by the semigroup $e^{-tL}\ (t>0)$ or the Riesz transforms $\Delta L^{-1/2}$ were further generalized by Lin et al. [17] to the setting of Heisenberg groups.

In order to introduce more results, we need to give some definitions. The semi-group maximal function associated to the Schrödinger operator L is defined by

$$\begin{aligned} \mathcal {T}^*(f)(x)=\sup _{t>0}|e^{-tL}f(x)|=\sup _{t>0}\Big |\int _{\mathbb {R}^n}k_t(x,y)f(y)dy\Big |, \end{aligned}$$

(1.1)

The fractional integral operator associated to the Schrödinger operator L is defined by

$$\begin{aligned} \mathcal {I}_\alpha (f)(x)= & {} L^{-\alpha /2}f(x)=\int _{{\mathbb {R}^n}}\int _{0}^{\infty }k_t(x,y)t^{\alpha /2}\frac{dt}{t}f(y)dy\nonumber \\= & {} \int _{{\mathbb {R}^n}}K_\alpha (x,y)f(y)dy,\ \ 0<\alpha <n. \end{aligned}$$

(1.2)

where $K_\alpha (x,y)=\int _{0}^{\infty }k_t(x,y)t^{\alpha /2-1}dt$ and $k_t$ is the kernel function of the operator $e^{-tL}$.

As in [21], we will use the auxiliary function $\rho$ defined for $\mathbb {R}^n$ as

$$\begin{aligned} \rho (x)=\sup _{r>0}\Big \{r:\frac{1}{r^{n-2}}\int _{B(x,r)}V(y)dy\le 1\Big \}. \end{aligned}$$

(1.3)

Remark 1.1

Under the above assumptions on V, it is easy to get that $0<\rho (x)<\infty$. In particular, $\rho (x)=1$ with $V=1$. For more details concerning the function $\rho (x)$ and its applications in studying the Schrödinger operator L, we refer the reader to [12, 21, 22].

For $1<p<\infty$, the $A_p^{\rho ,\theta }$ weights class is defined as follows.

Definition 1.2

($A_p^{\rho ,\theta }$ weights class, [2]). Let w be a nonnegative, locally integrable function on $\mathbb {R}^n$. For $1<p<\infty$ and $0<\theta <\infty$, we say that a weight w belongs to the class $A_p^{\rho ,\theta }$ if there exists a positive constant C such that for all balls $B=B(x,r)$, it holds that

$$\begin{aligned} \Big (\frac{1}{|B|}\int _Bw(y)dy\Big )\Big (\frac{1}{|B|}\int _Bw(y)^{-1/(p-1)}dy\Big )^{p-1}\le C\Big (1+\frac{r}{\rho (x)}\Big )^{\theta p}. \end{aligned}$$

(1.4)

w is said to satisfy the $A_1^{\rho ,\theta }$ condition if there exists a constant C such that for all balls B

$$\begin{aligned} M^\theta _V(w)(x)\le Cw(x),\ a.e. \ x\in \mathbb {R}^n, \end{aligned}$$

where

$$\begin{aligned} M^\theta _V(f)(x)=\sup _{x\in B}\frac{1}{\Psi _\theta (B)|B|}\int _B|f(y)|dy. \end{aligned}$$

Here, $\Psi _\theta (B)=(1+r/\rho (x))^\theta$ with $B=B(x,r)$.

Remark 1.3

Clearly, the classes $A_p^{\rho ,\theta }$ are increasing with $\theta$, and $A_p\subset A_p^{\rho ,\theta }$ for $1\le p<\infty$. Moreover, from the Remark 1.7 below, it is easy to see that $A_p\subsetneq A_p^{\rho ,\theta }$.

We introduce the $A_{(p,q)}^{\rho }$ class, which also depends on the parameter $\theta$.

Definition 1.4

($A_{(p,q)}^{\rho }$ weights class, [23]). We say that a weight w belongs to the class $A_{(p,q)}^{\rho }$ for $1\le p<\infty$ and $1\le q<\infty$, if there is a constant $C>0$ such that for any cube $Q=Q(x,r)$,

$$\begin{aligned} \Big (\frac{1}{\Psi _\theta (Q)|Q|}\int _{Q}w(y)^qdy\Big )^{1/q}\Big (\frac{1}{\Psi _\theta (Q)|Q|}\int _{Q}w(y)^{-p'}dy\Big )^{1/p'}\le C. \end{aligned}$$

Remark 1.5

Obviously, $w^{1/p}\in A_{(p,p)}^{\rho }$ if and only if $w\in A^{\rho ,\theta }_p$ for $1\le p<\infty$; $w\in A_{(p,q)}^{\rho }$ if and only if $w^q\in A^{\rho ,\theta }_{1+q/p'}$.

Definition 1.6

($\text {BMO}_\theta (\rho )({\mathbb {R}^n})$ space, [1]). For $\theta >0$, we defined the class $\text {BMO}_\theta (\rho )({\mathbb {R}^n})$ of locally integrable functions f such that

$$\begin{aligned} \frac{1}{|B(x,r)|}\int _{B(x,r)}|f(y)-f_B|dy\le C\Big (1+\frac{r}{\rho (x)}\Big )^{\theta }, \end{aligned}$$

(1.5)

for all $x\in \mathbb {R}^n$ and $r>0$, where $f_B=\frac{1}{|B|}\int _Bb$. A norm for $f\in \text {BMO}_\theta (\rho )({\mathbb {R}^n})$, denoted by $\Vert f\Vert _{\text {BMO}_\theta (\rho )}$ , is given by the infimum of the constants satisfying (1.5), after identifying functions that differ upon a constant. Clearly $\text {BMO}({\mathbb {R}^n})\subset \text {BMO}_\theta (\rho )({\mathbb {R}^n})\subset \text {BMO}_{\theta '}(\rho )({\mathbb {R}^n})$ for $0<\theta <\theta '$.

The commutators of $\mathcal {T^*}$ and $\mathcal {I}_\alpha$ with $b\in \text {BMO}_\theta (\rho )({\mathbb {R}^n})$ are defined by

$$\begin{aligned} \mathcal {T}_b^*(f)(x)=\sup _{t>0}\Big |\int _{\mathbb {R}^n}k_t(x,y)(b(x)-b(y))f(y)dy\Big |; \end{aligned}$$

(1.6)

and

$$\begin{aligned} \mathcal {I}_\alpha ^b(f)(x)=\int _{{\mathbb {R}^n}}K_\alpha (x,y)(b(x)-b(y))f(y)dy,\ \ 0<\alpha <n. \end{aligned}$$

(1.7)

In 2011, Bongioanni, Harboure and Salinas [1] considered the $L^p(\mathbb {R}^n)(1<p<\infty )$ boundedness of the commutators of Riesz transforms related to L with $\text {BMO}_\theta (\rho )({\mathbb {R}^n})$ functions. In another paper, they [2] established the weighted boundedness for the semi-group maximal function, Riesz transforms, fractional integrals and Littlewood–Paley functions related to L with weights belong to $A_p^{\rho ,\theta }$ class which includes the Muckenhoupt weight class $A_p$.

Theorem A

([2]). For $1<p<\infty$, the operator $\mathcal {T}^*$ is bounded on $L^p(w)$ when $w\in A_p^{\rho ,\theta }$.

Theorem B

([2]). For $1<p<n/\alpha$, the operator $\mathcal {I}_\alpha$ is bounded from on $L^p(w)$ into $L^q(w^{q/p})$ for $w^{q/p}\in A_{1+\frac{q}{p'}}^{\rho ,\infty }$, with $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$.

Recently, Tang [22] considered the weighted norm inequalities for $\mathcal {T}_b^*$.

Theorem C

([22]). Let $1<p<\infty$, $w\in A_p^{\rho ,\theta }$ and $b\in \text {BMO}_\theta (\rho )({\mathbb {R}^n})$, then there exists a constant C such that

$$\begin{aligned} \Vert \mathcal {T}^*_b(f)\Vert _{L^p(w)}\le C\Vert b\Vert _{\text {BMO}_\theta (\rho )}\Vert f\Vert _{L^p(w)}. \end{aligned}$$

Tang [23] established the weighted boundedness for fractional integrals $\mathcal {I}_\alpha$ and corresponding commutator $\mathcal {I}_\alpha ^b$ related to L. The Anh Bui [5] also gave weighted estimate for the commutator of fractional integrals related to L.

Theorem D

([23]). Let $0<\alpha<n,\ 1<p<\alpha /n$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$. if $w\in A_{(p,q)}^{\rho }$, then

$$\begin{aligned} \Big (\int _{{\mathbb {R}^n}}|\mathcal {I}_\alpha f(x)|^qw(x)^qdx\Big )^{1/q}\le C\Big (\int _{{\mathbb {R}^n}}| f(x)|^pw(x)^pdx\Big )^{1/p}. \end{aligned}$$

Theorem E

([5, 23]). Let $b\in \text {BMO}_\theta (\rho ){{\mathbb {R}^n}}$, $0<\alpha<n,\ 1<p<\alpha /n$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$. if $w\in A_{(p,q)}^{\rho }$, then

$$\begin{aligned} \Big (\int _{{\mathbb {R}^n}}|\mathcal {I}_\alpha ^b f(x)|^qw(x)^qdx\Big )^{1/q}\le C\Vert b\Vert _{\text {BMO}_\theta (\rho )}\Big (\int _{{\mathbb {R}^n}}| f(x)|^pw(x)^pdx\Big )^{1/p}. \end{aligned}$$

This paper is devoted to studying the weighted compactness for commutators of semi-group maximal function and fractional integrals related to Schrödinger operators. Before stating our main results, we recall some background for the compactness of the commutators of some classical operators. Given a locally integrable function b, the commutator [b, T] is defined by

$$\begin{aligned}{}[b,T](f)(x)=bTf(x)-T(bf)(x). \end{aligned}$$

Uchiyama [26] first studied the compactness of commutators of singular integrals and showed that the commutator $[b,T_\Omega ]$ is compact on $L^p(\mathbb {R}^n)$ for $1<p<\infty$ if and only if $b\in \mathrm{CMO}(\mathbb {R}^n)$, where $T_\Omega$ is a singular integral operator with rough kernel $\Omega \in \mathrm{Lip}_1(\mathrm{S}^{n-1})$ and $\mathrm{CMO}(\mathbb {R}^n)$ is the closure of $\mathcal {C}_c^\infty (\mathbb {R}^n)$ in the $\mathrm{BMO}(\mathbb {R}^n)$ topology.

Since then, the study on the compactness of commutators of different operators has attracted much more attention. Krantz and Li applied the compactness characterization of the commutator $[b,T_\Omega ]$ to study Hankel type operators on Bergman space in [15] and [16]. Wang [27] showed the compactness of the commutator of fractional integral operator form $L^p({\mathbb {R}^n})$ to $L^q({\mathbb {R}^n})$. In 2009, Chen and Ding [6] proved the commutator of singular integrals with variable kernels is compact on $L^p({\mathbb {R}^n})$ if and only if $b\in \mathrm CMO(\mathbb {R}^n)$ and they also establised the compactness of Littlewood–Paley square functions in [7]. Later on, Chen, Ding and Wang [8] obtained the compactness of commutators for Marcinkiewicz Integral in Morrey Spaces. Recently, Liu, Wang and Xue [19] showed the compactness of the commutator of oscillatory singular integrals with Hölder class kernels of non-convolutional type. We refer the reader to [3, 4, 18, 24, 25, 28, 30] for the compactness of commutators of multilinear operators.

The above compactness results are all concerned with the space $\text{ CMO }(\mathbb {R}^n)$. However, Theorem A shows that the $L^p$ boundedness holds for more larger space $\text {BMO}_\theta (\rho )(\mathbb {R}^n)$, rather than $\text {BMO}(\mathbb {R}^n)$ and the weights class $A_p^{\rho ,\theta }$ is more larger than $A_p$ weights class. Let $\text { CMO}_\theta (\rho )(\mathbb {R}^n)$ be the closure of $\mathcal {C}_c^\infty (\mathbb {R}^n)$ in the $\text {BMO}_\theta (\rho )(\mathbb {R}^n)$ topology. Then, it is quite natural to ask the following question:

Question: Is the operator $\mathcal {T}_b^*$ compact from $L^p(w)$ to $L^p(w)$ when $w\in A_p^{\rho ,\theta }$ and b belongs to the space $\text { CMO}_\theta (\rho )(\mathbb {R}^n)$? Is the operator $\mathcal {I}_\alpha ^b$ compact from $L^p(w^p)$ to $L^q(w^q)$ when $0<\alpha<n,\ 1<p<\alpha /n$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, $w\in A_{(p,q)}^{\rho }$ and b belongs to the space $\text { CMO}_\theta (\rho )(\mathbb {R}^n)$?

The main purpose of this paper is to give a firm answer to the above question. Our results are as follows:

Theorem 1.1

Let $1<p<\infty$, $w\in A_p^{\rho ,\theta }$ and $b\in \text {CMO}_\theta (\rho )({\mathbb {R}^n})$. If w satisfies the following condition

$$\begin{aligned} \lim _{A\rightarrow +\infty }A^{-np+n}\int _{|x|>1}\frac{w(Ax)}{{|x|}^{np}}dx=0, \end{aligned}$$

(1.8)

then the operator $\mathcal {T}^*_b$ defined by (1.1) is a compact operator from $L^p(w)$ to $L^p(w)$.

Theorem 1.2

Let $0<\alpha<n,\ 1<p<\alpha /n$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ and $w\in A_{(p,q)}^{\rho }$. If $b\in \text {CMO}_\theta (\rho )({\mathbb {R}^n})$, then the operator $\mathcal {I}_\alpha ^b$ is a compact operator from $L^p(w^p)$ to $L^q(w^q)$.

Remark 1.7

We give some comments about Theorem 1.1:

• (1) The weights class in Theorem 1.1 is more larger than the classical Muckenhoupt weights class $A_p$. In fact, if $w\in A_p$, the classical Muckenhoupt weights class, then the condition (1.8) holds. Let $0<\gamma <\theta$ and $w(x)=(1+|x|)^{-(n+\gamma )}$, it is easy to see that w satisfies (1.8) and $w(x)\notin A_p$ ($1\le p<\infty$), but $w\in A_1^{\rho ,\theta }\subset A_p^{\rho ,\theta }$ ($1<p<\infty$) provided that $V=1$ (see [22]).

• (2) Obviously, the space $\text {CMO}_\theta (\rho )({\mathbb {R}^n})$ where b belongs is more larger than $\text {CMO}({\mathbb {R}^n})$ space.

The paper is organized as follows. In Sect. 2 we give some definitions and preliminary lemmas, which are the main ingredients of our proofs. In Sect. 3 we will give the proof of Theorem 1.1 via smooth truncated techniques. The domain of integration will be divided into several cases. In actuality some cases are combinable, but various subcases also arise, which increases the difficulty we need to deal with. In Sect. 4, we will give the proof of 1.2.

Throughout the paper, the letter C or c, sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables. $A\sim B$ means that there exists constants $C_1>0$ and $C_2>0$ such that $C_2B\le A\le C_1B$.

2 Preliminaries

We first recall some notation. Given a Lebesgue measurable set $E\subset \mathbb {R}^n$, |E| will denote the Lebesgue measure of E. If $B = B(x, r)$ is a ball in $\mathbb {R}^n$ and $\lambda$ is a real number, then $\lambda B$ shall stand for the ball with the same center as B and radiu $\lambda$ times that of B. A weight w is a non-negative measurable function on $\mathbb {R}^n$. The measure associated with w is the set function given by $w(E)=\int _E wdx$. For $0<p<\infty$ we denote by $L^p(w)$ the space of all Lebesgue measurable function f(x) such that

$$\begin{aligned} \Vert f\Vert _{L^p(w)}=\Big (\int _{\mathbb {R}^n}|f(x)|^pw(x)dx|\Big )^{1/p}. \end{aligned}$$

The auxiliary function $\rho$ enjoys the following property.

Lemma 2.1

([21]). There exist contants $k_0\ge 1$ and $C>0$ such that for all $x,\ y\in {\mathbb {R}^n}$,

$$\begin{aligned} C^{-1}\rho (x)\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-k_0}\le \rho (y)\le C\rho (x)(1+\frac{|x-y|}{\rho (x)}\Big )^{\frac{k_0}{k_0+1}}. \end{aligned}$$

In particular, $\rho (x)\sim \rho (y)$ if $|x-y|<C\rho (x)$.

$A_p^{\rho ,\theta }$ weights class has some properties analogy to $A_p$ weights class for $1\le p<\infty$.

Lemma 2.2

([2, 23]). Let $1<p<\infty$ and $w\in A_p^{\rho ,\infty }=\bigcup _{\theta \ge 0}A_p^{\rho ,\theta }$. Then

• (i) If $1\le p_1<p_2<\infty$, then $A_{p_1}^{\rho ,\theta }\subset A_{p_2}^{\rho ,\theta }.$

• (ii) $w\in A_p^{\rho ,\theta }$ if and only if $w^{-\frac{1}{p-1}}\in A_{p'}^{\rho ,\theta }$, where $1/p+1/p'=1$.

• (iii) If $w\in A_p^{\rho ,\infty }$, $1<p<\infty$, then there exists $\epsilon >0$ such that $w\in A_{p-\epsilon }^{\rho ,\infty }$.

• (iv) if $w\in A_p^{\rho ,\theta }$, $1\le p<\infty$, then for any cube Q, we have

  $$\begin{aligned} \frac{1}{\Psi _\theta (Q)|Q|}\int _Q|f(y)|dy\le C\Big (\frac{1}{w(5Q)}\int _Q|f|^pw(y)dy\Big )^{1/p}, \end{aligned}$$

  where $w(E)=\int _Ew(x)dx$. In particularly, for any set $E\subset Q$, let$f=\chi _E$, then

  $$\begin{aligned} \frac{|E|}{\Psi _\theta (Q)|Q|}\le C\Big (\frac{w(E)}{w(5Q)}\Big )^{1/p}. \end{aligned}$$

It should be pointed out that (iii) of Lemma 2.2 was proved by Bongioanni et al. [2] and (iv) of Lemma 2.2 was proved by Tang [23].

The following result holds:

Lemma 2.3

([23]). Let $1<p<\infty$ and suppose that $w\in A_p^{\rho ,\theta }$. If $p<p_1<\infty$, then

$$\begin{aligned} \int _{\mathbb {R}^n}|M_V^\theta f(x)|^{p_1}w(x)dx\le C_p\int _{\mathbb {R}^n}|f(x)|^{p_1}w(x)dx. \end{aligned}$$

By the Lemma 2.3, $M_V^\theta$ may not be bounded on $L^p(w)$ for all $w\in A_p^{\rho ,\theta }$ and $1<p<\infty$. So we need the variant maximal operator $M_{V,\eta }$ defined by

$$\begin{aligned} M_{V,\eta } f(x)=\sup _{x\in B}\frac{1}{(\Psi _\theta (B))^\eta |B|}\int _B|f(y)|dy,\ \ \ 0<\eta <\infty . \end{aligned}$$

We have the following Lemma.

Lemma 2.4

([23]). Let $1<p<\infty$, $p'=p/(p-1)$ and suppose that $w\in A_p^{\rho ,\theta }$. Then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert M_{V,p'}f\Vert _{L^p(w)}\le C\Vert f\Vert _{L^p(w)}. \end{aligned}$$

Next, we will give the weighted strong type (p, q) inequality for a fractioal variant maximal operator $M_{\alpha ,V,\eta }$ defined by

$$\begin{aligned} M_{\alpha ,V,\eta } f(x)=\sup _{x\in Q}\frac{1}{(\Psi _\theta (Q))^\eta (\Psi _\theta (Q)|Q|)^{1-\frac{\alpha }{n}}}\int _B|f(y)|dy,\ \ \ 0<\eta <\infty . \end{aligned}$$

Lemma 2.5

([23]). Let $0\le \alpha <n$, $1<p<n/\alpha$, $p'=p/(p-1)$ and $1/q=1/p-\alpha /n$. If $w\in A_{(p,p)}^{\rho }$ and $\eta \ge (1-\alpha /n)p'/q$. Then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert M_{\alpha ,V,\eta }f\Vert _{L^q(w^q)}\le C\Vert f\Vert _{L^p(w^p)}. \end{aligned}$$

We also need the following properties of the kernel $k_t$ and $K_\alpha$.

Lemma 2.6

([9, 14]). For every N, there is a constant $C_N$ such that

$$\begin{aligned} 0<k_t(x,y)\le C_N t^{-\frac{n}{2}}e^{-\frac{|x-y|^2}{5t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}. \end{aligned}$$

Lemma 2.7

([10]). There exist $0< \delta <1$ and a constant $c>0$ such that for every $N>0$ there is a constant $C_N>0$ so that, for all $|h|\le \sqrt{t}$

$$\begin{aligned} |k_t(x+h)-k_t(x,y)|\le C_N\Big (\frac{|h|}{\sqrt{t}}\Big )^\delta t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}. \end{aligned}$$

Lemma 2.8

([10]). If $0<\alpha <n$, then there exists $0< \delta _0 <1$ such that for any $N>0$ there is a constant $C_N>0$ so that

$$\begin{aligned} 0<K_\alpha (x,y)\le C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{1}{|x-y|^{n-\alpha }} \end{aligned}$$

(2.1)

and

$$\begin{aligned} |K_\alpha (x,y)-K_\alpha (x,z)|\le C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|y-z|^{\delta _0}}{|x-y|^{n-\alpha +\delta _0}}, \end{aligned}$$

(2.2)

whenever $x,\ y,\ z\in {\mathbb {R}^n}$ and $|y-z|<|x-y|/2$.

Let X and Y be Banach spaces. Suppose T and $\{T_n\}$ be operators from X to Y, then we have the following Lemma.

Lemma 2.9

([29, p. 278, Theorem(iii)]) Let a sequence $\{T_n\}$ of compact operators converge to an operator T in sense of the uniform operator toplogy, i.e., $\lim \limits _{n\rightarrow \infty }\Vert T-T_n\Vert =0$. Then T is also compact.

We end this section by introducing the general weighted version of Frechet-Kolmogorov theorems, which was proved by Xue et al. [30].

Lemma 2.10

([30]). Let w be a weight on $\mathbb {R}^n$. Assume that $w^{-1/(p_0-1)}$ is also a weight on $\mathbb {R}^n$ for some $p_0>1$. Let $0<p<\infty$ and $\mathcal {F}$ be a subset in $L^p(w)$, then $\mathcal {F}$ is conditionally sequentially compact in $L^p(w)$ if the following three conditions are satisfied:

• (i) $\mathcal {F}$ is bounded, i.e., $\sup \limits _{f\in \mathcal {F}}\Vert f\Vert _{L^p(w)}<\infty$;

• (ii) $\mathcal {F}$ uniformly vanishes at infinity, i.e.,

  $$\begin{aligned} \lim \limits _{N\rightarrow \infty }\sup \limits _{f\in \mathcal {F}}\int _{|x|>N}|f(x)|^pw(x)dx=0; \end{aligned}$$

• (iii) $\mathcal {F}$ is uniformly equicontinuous, i.e.,

  $$\begin{aligned} \lim \limits _{|h|\rightarrow 0}\sup \limits _{f\in \mathcal {F}}\int _{\mathbb {R}^n}|f(\cdot +h)-f(\cdot )|^pw(x)dx=0. \end{aligned}$$

3 Proof of Theorem 1.1

Proof of Theorem 1.1

We shall prove Theorem 1.1 via smooth truncated techniques. First, we introduce the following smooth truncated function. Let $\varphi \in C^{\infty }([0,\infty ))$ satisfy

$$\begin{aligned} 0\le \varphi \le 1\ \ \ and\ \ \ \varphi (x)= {\left\{ \begin{array}{ll} 1, &{}x\in [0,1],\\ 0, &{}x\in [2,\infty ). \end{array}\right. } \end{aligned}$$

(3.1)

For any $\gamma >0$, let

$$\begin{aligned} k_{t,\gamma }(x,y)=k_t(x,y)\Big (1-\varphi (\gamma ^{-1}|x-y|)\Big ). \end{aligned}$$

(3.2)

Define

$$\begin{aligned} \mathcal {T}^*_\gamma f(x)=\displaystyle \sup _{t>0}\Big |\int _{{\mathbb {R}^n}}k_{t,\gamma }(x,y)f(y)dy\Big |. \end{aligned}$$

(3.3)

and

$$\begin{aligned} \mathcal {T}^*_{b,\gamma } f(x)=\displaystyle \sup _{t>0}\Big |\int _{{\mathbb {R}^n}}k_{t,\gamma }(x,y)(b(x)-b(y))f(y)dy\Big |. \end{aligned}$$

(3.4)

For any $b\in \mathcal {C}_c^\infty ({\mathbb {R}^n})$ and $\gamma ,\theta ,\ \eta >0$, by (3.2), (3.4) and Lemma 2.6 with $N=\theta \eta$, one has

$$\begin{aligned} \begin{aligned} |\mathcal {T}^*_{b} f(x)-\mathcal {T}^*_{b,\gamma } f(x)|&\le C\gamma \displaystyle \sup _{t>0}\int _{|x-y|<2\gamma }t^{-n/2}e^{-\frac{|x-y|^2}{5t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }|f(y)|dy\\&\le C\gamma \displaystyle \Big \{\sup _{\sqrt{t}<\gamma }\int _{|x-y|<\sqrt{t}}t^{-n/2}e^{-\frac{|x-y|^2}{5t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }|f(y)|dy\\&\quad +\displaystyle \sup _{\sqrt{t}<\gamma }\int _{\sqrt{t}\le |x-y|<2\gamma }t^{-n/2}e^{-\frac{|x-y|^2}{5t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }|f(y)|dy\\&\quad +\displaystyle \sup _{\sqrt{t}\ge \gamma }\int _{|x-y|<2\gamma }t^{-n/2}e^{-\frac{|x-y|^2}{5t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }|f(y)|dy\Big \}\\&=:C\gamma \{J_1+J_2+J_3\}. \end{aligned} \end{aligned}$$

(3.5)

One may obtain

$$\begin{aligned} \begin{aligned} J_1&\le \displaystyle \sup _{\sqrt{t}<\gamma }t^{-n/2}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|<\sqrt{t}}|f(y)|dy\\&\le CM_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.6)

and

$$\begin{aligned} \begin{aligned} J_3&\le 2^{\theta \eta }\displaystyle \sup _{\sqrt{t}\ge \gamma }\gamma ^{-n}\Big (1+\frac{2\gamma }{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|<2\gamma }|f(y)|dy\\&\le CM_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.7)

It remains to estimate $J_2$. Using the estimate $e^{-s}\le \frac{C}{s^{M/2}}$with $M>n+\theta \eta$ and splitting to annuli, it follows that

$$\begin{aligned} \begin{aligned} J_2&\le \displaystyle \sup _{\sqrt{t}<\gamma }\sum _{k=1}^{\infty }t^{\frac{M-n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|\sim 2^k\sqrt{t}}\frac{|f(y)|}{|x-y|^M}dy\\&\le \displaystyle \sup _{\sqrt{t}<\gamma }\sum _{k=1}^{\infty }\frac{2^{-k(M-n-\theta \eta )}}{(2^k\sqrt{t})^n\Big (1+\frac{2^k\sqrt{t}}{\rho (x)}\Big )^{\theta \eta }}\int _{|x-y|<2^k\sqrt{t}}|f(y)|dy\\&\le CM_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.8)

Combing (3.8) with (3.5), (3.6) and (3.7) may lead to

$$\begin{aligned} |\mathcal {T}^*_{b} f(x)-\mathcal {T}^*_{b,\gamma } f(x)|\le C\gamma M_{V,\eta }f(x). \end{aligned}$$

Then Lemma 2.4 with $p'\le \eta <\infty$ gives that

$$\begin{aligned} \Vert \mathcal {T}^*_{b}f-\mathcal {T}^*_{b,\gamma } f\Vert _{L^p(w)}\le C\gamma \Vert f\Vert _{L^p(w)}, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{\gamma \rightarrow 0}\Vert \mathcal {T}^*_{b}-\mathcal {T}^*_{b,\gamma }\Vert _{L^p(w)\rightarrow L^p(w)}=0. \end{aligned}$$

(3.9)

On the other hand, if $b\in \text {CMO}_\theta (\rho )({\mathbb {R}^n})$, then for any $\epsilon >0$, there exists $b_\epsilon \in \mathcal {C}_c^\infty ({\mathbb {R}^n})$ such that $\Vert b-b_\epsilon \Vert _{\text {BMO}_\theta (\rho )}<\epsilon$, so that, by Theorem C, we have

$$\begin{aligned} \Vert \mathcal {T}^*_bf-\mathcal {T}^*_{b_\epsilon } f\Vert _{L^p(w)}\le \Vert \mathcal {T}^*_{b-b_\epsilon }f\Vert _{L^p(w)}\le C\Vert b-b_\epsilon \Vert _{\text {BMO}_\theta (\rho )}\Vert f\Vert _{L^p(w)}\le C\epsilon . \end{aligned}$$

Thus, to prove $\mathcal {T}^*_b$ is compact on $L^p(w)$ for any $b\in \text {CMO}_\theta (\rho )$, it suffices to prove that $\mathcal {T}^*_b$ is compact on $L^p(w)$ for any $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$. By (3.9) and Lemma 2.9, it suffices to show that $\mathcal {T}^*_{b,\gamma }$ is compact for any $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$ when $\gamma >0$ is small enough. To this end, for arbitrary bounded set F in $L^p(w)$, let

$$\begin{aligned} \mathcal {F}=\{\mathcal {T}^*_{b,\gamma }f:f\in F\}. \end{aligned}$$

Then, we need to show that for $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$, $\mathcal {F}$ satisfies the conditions$\mathrm (i)$-$\mathrm (iii)$ of Lemma 2.10.

From the definition of $k_{t,\gamma }$, we know that $0<k_{t,\gamma }(x,y)\le k_t(x,y)$, then $\mathcal {T}^*_{\gamma }f(x)\le \mathcal {T}^*(|f|)(x)$ and $\mathcal {T}^*_{b,\gamma }f(x)\le \mathcal {T}^*(|f|)(x)$. Hence, the boundedness of $\mathcal {T}^*_{\gamma }$ and $\mathcal {T}^*_{b,\gamma }$ also holds. Thus, we have

$$\begin{aligned} \sup \limits _{f\in F}\Vert \mathcal {T}^*_{b,\gamma }f\Vert _{L^p(w)}\le C\sup \limits _{f\in F}\Vert f\Vert _{L^p(w)}\le C, \end{aligned}$$

which yields the fact that the set $\mathcal {F}$ is bounded.

Assume $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$ and ${\text {supp}}(b)\subset B(0,R)$, where B(0, R) is the ball of radius R center at the origin in $\mathbb {R}^n$. For any $|x|> A > 2R$, $w \in A_p^{\rho ,\theta }$, $1<p<\infty$ and $f\in F$. By Lemma 2.6 and the estimate $e^{-\frac{|x-y|^2}{5t}}\le C\frac{t^\frac{n}{2}}{|x-y|^n}$, we have

$$\begin{aligned} \begin{aligned} |\mathcal {T}^*_{b,\gamma }f(x)|&\le \displaystyle \sup _{t>0}\int _{|y|<R}k_t(x,y)|b(y)f(y)|dy\\&\le C\displaystyle \sup _{t>0}t^{-n/2}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-N}\int _{|y|<R}e^{-\frac{|x-y|^2}{5t}}|f(y)|dy\\&\le C|x|^{-n}\displaystyle \int _{|y|<R}|f(y)|dy\\&\le C|x|^{-n}\Vert f\Vert _{L^p(w)}\displaystyle \Big (\int _{|y|<R}w^{-p'/p}(y)dy\Big )^{1/p'}. \end{aligned} \end{aligned}$$

Therefore

$$\begin{aligned} \begin{array}{ll} \displaystyle \int _{|x|>A}|\mathcal {T}^*_{b,\gamma }f(x)|^pw(x)dx&{}\le C\displaystyle \int _{|x|>A}\frac{w(x)}{|x|^{np}}dx\\ &{}\displaystyle =C\sum _{j=0}^{\infty }\int _{2^jA<|x|<2^{j+1}A}\frac{w(x)}{|x|^{np}}dx\\ &{}\displaystyle =CA^{-np+n}\int _{|x|>1}\frac{w(Ax)}{{|x|}^{np}}dx. \end{array} \end{aligned}$$

This together with the condition (1.8) yields that

$$\begin{aligned} \lim _{A\rightarrow \infty }\int _{|x|>A}|\mathcal {T}^*_{b,\gamma }f(x)|^pw(x)dx=0, \end{aligned}$$

whenever $f\in F$.

It remains to show that the set $\mathcal {F}$ is uniformly equicontinuous. It suffices to verify that

$$\begin{aligned} \lim \limits _{|h|\rightarrow 0}\Vert \mathcal {T}^*_{b,\gamma }f(h+\cdot )-\mathcal {T}^*_{b,\gamma }f(\cdot )\Vert _{L^p(w)}=0, \end{aligned}$$

(3.10)

holds uniformly for $f\in F$.

In what follows, we fix $\gamma \in (0,\frac{1}{4})$ and $|h|<\frac{\gamma }{4}$. Then

$$\begin{aligned} \begin{aligned}&|\mathcal {T}^*_{b,\gamma }f(x+h)-\mathcal {T}^*_{b,\gamma }f(x)|\\&\quad \le \displaystyle \sup _{t>0}\int _{\mathbb {R}^n}|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)||b(x+h)-b(y)||f(y)|dy\\&\qquad +\displaystyle \sup _{t>0}\int _{\mathbb {R}^n}k_{t,\gamma }(x,y)|b(x+h)-b(x)||f(y)|dy\\&\quad =:I(x)+II(x). \end{aligned} \end{aligned}$$

(3.11)

For II(x), it holds that

$$\begin{aligned} \begin{array}{ll} &{}II(x)=\displaystyle |b(x+h)-b(x)|\sup _{t>0}\int _{\mathbb {R}^n}k_{t,\gamma }(x,y)|f(y)|dy\\ &{}\qquad \le C\displaystyle |h|\mathcal {T}^*_\gamma (|f|)(x). \end{array} \end{aligned}$$

Then, by the $L^p(w)$-bounds of $\mathcal {T}^*_\gamma$, we have

$$\begin{aligned} \Vert II\Vert _{L^p(w)}\le C|h|\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(3.12)

For I(x), we decompose it into two parts

$$\begin{aligned} \begin{aligned} I(x)\le&\displaystyle \sup _{\sqrt{t}\ge |h|}\int _{\mathbb {R}^n}|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)||b(x+h)-b(y)||f(y)|dy\\&\quad +\displaystyle \sup _{\sqrt{t}<|h|}\int _{\mathbb {R}^n}|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)||b(x+h)-b(y)||f(y)|dy\\&=:I_1(x)+I_2(x). \end{aligned} \end{aligned}$$

(3.13)

Contribution of $I_1$. For $I_1(x)$, if $|h|\le \sqrt{t}$, then by Lemmas 2.6 and 2.7, we have

$$\begin{aligned} \begin{aligned}&|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)|\\&\le |k_t(x+h,y)-k_t(x,y)|+|k_t(x+h,y)-k_t(x,y)|\varphi (\gamma ^{-1}|x+h-y|)\\&\quad +k_t(x,y)|\varphi (\gamma ^{-1}|x+h-y|)-\varphi (\gamma ^{-1}|x-y|)|\\&\le C\Big (\frac{|h|}{\sqrt{t}}\Big )^\delta t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}\\&\quad +C\frac{|h|}{\gamma } t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}. \end{aligned} \end{aligned}$$

(3.14)

Therefore, we have

$$\begin{aligned} \begin{aligned} I_1(x)&\le C\displaystyle \sup _{\sqrt{t}\ge |h|}\int _{{\mathbb {R}^n}}\Big (\Big (\frac{|h|}{\sqrt{t}}\Big )^\delta +\frac{|h|}{\gamma }\Big ) t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-N}\\&\quad \times |b(x+h)-b(y)||f(y)|dy\\&\le C\displaystyle \sup _{\sqrt{t}\ge 1}\Big \{\int _{|x-y|<\sqrt{t}}+\int _{|x-y|\ge \sqrt{t}}\Big \} \Big (\Big (\frac{|h|}{\sqrt{t}}\Big )^\delta +\frac{|h|}{\gamma }\Big )t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\\&\quad \times \Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-N}|b(x+h)-b(y)||f(y)|dy\\&\quad +C\displaystyle \sup _{|h|\le \sqrt{t}<1}\Big \{\int _{|x-y|<\sqrt{t}}+\int _{|x-y|\ge \sqrt{t}}\Big \}\Big (\Big (\frac{|h|}{\sqrt{t}}\Big )^\delta +\frac{|h|}{\gamma }\Big ) t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}\\&\quad \times \Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-N}|b(x+h)-b(y)||f(y)|dy\\&=:I_{11}(x)+I_{12}(x)+I_{13}(x)+I_{14}(x). \end{aligned} \end{aligned}$$

(3.15)

Now, we are in the position to estimate the above four terms.

For $I_{11}(x)$, if $\sqrt{t}\ge 1$, then $t^{-\delta /2}\le 1$. Taking $N=\theta \eta$ for any $\theta ,\eta >0$, then we have

$$\begin{aligned} \begin{aligned} I_{11}(x)&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{\sqrt{t}\ge 1}\int _{|x-y|<\sqrt{t}}t^{-\frac{n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{\sqrt{t}\ge 1}\frac{1}{(\sqrt{t})^n(1+\frac{\sqrt{t}}{\rho (x)})^{\theta \eta }}\int _{|x-y|<\sqrt{t}}|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|) M_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.16)

In order to estimate $I_{12}(x)$, we need the following ineqality: for any $M>0$, there exists a constant $C>0$, such that

$$\begin{aligned} e^{-\frac{c|x-y|^2}{t}}\le C\frac{t^{\frac{M}{2}}}{|x-y|^{M}}. \end{aligned}$$

(3.17)

Using (3.17) with $M>n+\theta \eta$, splitting into annuli, we obtain

$$\begin{aligned} \begin{aligned} I_{12}(x)&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{\sqrt{t}\ge 1}t^{\frac{M-n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|\ge \sqrt{t}}\frac{|f(y)|}{|x-y|^M}dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{\sqrt{t}\ge 1}\sum _{k=1}^{\infty }\frac{2^{-k(M-n)}}{(2^k\sqrt{t})^n(1+\frac{\sqrt{t}}{\rho (x)})^{\theta \eta }}\int _{|x-y|<2^k\sqrt{t}}|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{\sqrt{t}\ge 1}\sum _{k=1}^{\infty }\frac{2^{-k(M-n-\theta \eta )}}{(2^k\sqrt{t})^n(1+\frac{2^k\sqrt{t}}{\rho (x)})^{\theta \eta }}\int _{|x-y|<2^k\sqrt{t}}|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|) M_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.18)

If $\sqrt{t}<1$, then $t^{-\delta /2}<t^{-1/2}$. For any $\theta ,\eta >0$, taking $N=\theta \eta$. For $I_{13}(x)$, if $|h|\le \sqrt{t}$, $|x-y|<\sqrt{t}$ and $b\in \mathcal {C}_c^\infty ({\mathbb {R}^n})$, then

$$\begin{aligned} |b(x+h)-b(y)|\le C|x+h-y|\le C(|x-y|+|h|)\le C\sqrt{t}. \end{aligned}$$

Then, it follows that

$$\begin{aligned} I_{13}(x)&\le C|h|^\delta \displaystyle \sup _{|h|\le \sqrt{t}<1}t^{-\frac{n+1}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|<\sqrt{t}}|b(x+h)-b(y)||f(y)|dy\nonumber \\&\quad +C\gamma ^{-1}|h|\displaystyle \sup _{|h|\le \sqrt{t}<1}t^{-\frac{n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|<\sqrt{t}}|f(y)|dy\nonumber \\&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{|h|\le \sqrt{t}<1}t^{-\frac{n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|<\sqrt{t}}|f(y)|dy\nonumber \\&\le C\gamma ^{-1}(|h|^\delta +|h|) M_{V,\eta }f(x). \end{aligned}$$

(3.19)

For $I_{14}(x)$, if $|x-y|<2^k\sqrt{t}$, $k=1,2,\cdots$, and $|h|\le \sqrt{t}$, $b\in \mathcal {C}_c^\infty ({\mathbb {R}^n})$, then

$$\begin{aligned} |b(x+h)-b(y)|\le C|x+h-y|\le C(|x-y|+|h|)\le C2^k\sqrt{t}, \end{aligned}$$

which combining with (3.17) for $M>n+1+\theta \eta$ yields that

$$\begin{aligned} \begin{aligned} I_{14}(x)&\le C|h|^\delta \displaystyle \sup _{|h|\le \sqrt{t}<1}t^{\frac{M-n-1}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|\ge \sqrt{t}}\frac{|f(x)||b(x+h)-b(y)|}{|x-y|^M}dy\\&\quad + C\gamma ^{-1}|h| \displaystyle \sup _{|h|\le \sqrt{t}<1}t^{\frac{M-n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\int _{|x-y|\ge \sqrt{t}}\frac{|f(x)|}{|x-y|^M}dy\\&\le C|h|^\delta \displaystyle \sup _{|h|\le \sqrt{t}<1}t^{\frac{M-n-1}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\sum _{k=1}^{\infty }\frac{2^k\sqrt{t}}{(2^k\sqrt{t})^M}\int _{|x-y|\sim 2^{k}\sqrt{t}}|f(y)|dy\\&\quad +C\gamma ^{-1}|h|\displaystyle \sup _{|h|\le \sqrt{t}<1}t^{\frac{M-n}{2}}\Big (1+\frac{\sqrt{t}}{\rho (x)}\Big )^{-\theta \eta }\sum _{k=1}^{\infty }\frac{1}{(2^k\sqrt{t})^M}\int _{|x-y|\sim 2^{k}\sqrt{t}}|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|)\displaystyle \sup _{|h|\le \sqrt{t}<1}\sum _{k=1}^{\infty }\frac{2^{-k(M-n-1-\theta \eta )}}{(2^k\sqrt{t})^n(1+\frac{2^k\sqrt{t}}{\rho (x)})^{\theta \eta }}\int _{|x-y|<2^{k}\sqrt{t}}|f(y)|dy\\&\le C\gamma ^{-1}(|h|^\delta +|h|) M_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.20)

Sum up (3.15), (3.16), (3.18), (3.19) and (3.20) in all, we get

$$\begin{aligned} I_1(x)\le C\gamma ^{-1}(|h|^\delta +|h|) M_{V,\eta }f(x). \end{aligned}$$

(3.21)

Contribution of $I_2$. Next we will estimate $I_2(x)$. When $|x-y|<\frac{\gamma }{2}$ and $|h|<\frac{\gamma }{4}$, then $|x+h-y|<\frac{3\gamma }{4}$. Hence $\varphi (\gamma ^{-1}|x+h-y|)=1=\varphi (\gamma ^{-1}|x-y|).$ This implies $k_{t,\gamma }(x+h,y)=0=k_{t,\gamma }(x,y).$ For $I_{2}(x)$, we decompose it as follows:

$$\begin{aligned} \begin{aligned} I_{2}(x)&\le \displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\int _{|x-y|\ge \frac{\gamma }{2}}|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)||b(x+h)-b(y)||f(y)|dy\\&\quad +\displaystyle \sup _{\begin{array}{c} {\sqrt{t}<|h|}\\ {\rho (x)\le |h|} \end{array}}\int _{|x-y|\ge \frac{\gamma }{2}}|k_{t,\gamma }(x+h,y)-k_{t,\gamma }(x,y)||b(x+h)-b(y)||f(y)|dy\\&:=I_{21}(x)+I_{22}(x). \end{aligned} \end{aligned}$$

(3.22)

For $I_{21}(x)$, since $|x-y|>\frac{\gamma }{2}>2|h|$ and $\sqrt{t}<|h|<\rho (x)$, then $|x+h-y|\sim |x-y|$ and $|h|/\rho (x)<1$. Choosing $M>n+1+\theta \eta$ and using Lemma 2.6 and (3.17), we get

$$\begin{aligned} \begin{aligned} I_{21}(x)&=\displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\int _{|x-y|\ge \frac{\gamma }{2}}|k_{t,\gamma }(x+h,y){-}k_{t,\gamma }(x,y)||b(x{+}h) {-}b(y)||f(y)|dy\\&\le C\displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\int _{|x-y|\ge 2|h|}t^{-\frac{n}{2}}e^{-\frac{c|x-y|^2}{t}}|x-y||f(y)|dy\\&\le C|h|^{M-n}\displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\sum _{k=2}^{\infty }\int _{|x-y|\ge 2|h|}\frac{|f(y)|}{|x-y|^{M-1}}dy\\&\le C|h|^{M-n}\displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\sum _{k=2}^{\infty }\frac{1}{(2^k|h|)^{M-1}}\int _{|x-y|\sim 2^k|h|}|f(y)|dy\\&\le C|h|\displaystyle \sup _{\sqrt{t}<|h|<\rho (x)}\sum _{k=2}^{\infty }\frac{2^{-k(M-n-1-\theta \eta )}}{(2^k|h|)^n2^{k\theta \eta }}\int _{|x-y|<2^k|h|}|f(y)|dy\\&\le C|h|M_{V,\eta }f(x). \end{aligned} \end{aligned}$$

(3.23)

Finally, it remains to consider $I_{22}(x)$. Since $b\in \mathcal {C}_c^\infty ({\mathbb {R}^n})$, $|x-y|\ge 2|h|$, $\sqrt{t}<|h|$, then $|h|/\sqrt{t}>1$ and $|b(x+h)-b(y)|\le C|x-y|$. In addition, if $|x-y|<2^l\rho (x),\ l=1,2,\ldots$, then by Lemma 2.1, we have

$$\begin{aligned} \rho (y)\le C2^{\frac{k_0}{k_0+1}l}\rho (x). \end{aligned}$$

Then, it follows that

$$\begin{aligned}&\Big (1+\frac{\sqrt{t}}{\rho (x)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}+\Big (1+\frac{\sqrt{t}}{\rho (x+h)}+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}\\&\quad \le 2\Big (1+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}\le C_N\Big (1+\frac{2^{-\frac{k_0}{k_0+1}l}\sqrt{t}}{\rho (x)}\Big )^{-N}. \end{aligned}$$

Choosing M, N such that $N>M>n+1+(k_0+1)\theta \eta$, and applying Lemma 2.6 and (3.17) again, we obtain

$$\begin{aligned} I_{22}(x)\le & {} C|h|\displaystyle \sup _{\begin{array}{c} {\sqrt{t}<|h|}\\ {\rho (x)\le |h|} \end{array}}\int _{|x-y|\ge 2\rho (x)}t^{-\frac{n+1}{2}}e^{-\frac{c|x-y|^2}{t}}\Big (1+\frac{\sqrt{t}}{\rho (y)}\Big )^{-N}|x-y||f(y)|dy \nonumber \\\le & {} C|h|\displaystyle \sup _{\begin{array}{c} {\sqrt{t}<|h|} \nonumber \\ {\rho (x)\le |h|} \end{array}}t^{\frac{M-n-1}{2}}\sum _{l=2}^{\infty }\Big (1+\frac{2^{-\frac{k_0}{k_0+1}l}\sqrt{t}}{\rho (x)}\Big )^{-N}\int _{|x-y|\sim 2^l\rho (x)}\frac{|f(y)|}{|x-y|^{M-1}}dy \nonumber \\\le & {} C|h|\displaystyle \sup _{\begin{array}{c} {\sqrt{t}<|h|}\\ {\rho (x)\le |h|} \end{array}} \sum _{l=2}^{\infty }\Big (\frac{\sqrt{t}}{\rho (x)}\Big )^{M-n-1} \Big (1+\frac{2^{-\frac{k_0}{k_0+1}l}\sqrt{t}}{\rho (x)}\Big )^{-N}\frac{2^{-l(M-n-1-\theta \eta )}}{(2^l\rho (x))^n2^{l\theta \eta }}\nonumber \\&\quad \times \int _{|x-y|<2^l\rho (x)}|f(y)|dy \nonumber \\\le & {} C|h|\displaystyle \sup _{\begin{array}{c} {\sqrt{t}<|h|}\nonumber \\ {\rho (x)\le |h|} \end{array}}\sum _{l=2}^{\infty }\frac{2^{-l(\frac{M-n-1}{k_0+1}-\theta \eta )}}{(2^l\rho (x))^n2^{l\theta \eta }}\int _{|x-y|<2^l\rho (x)}|f(y)|dy\\\le & {} C|h|M_{V,\eta }f(x). \end{aligned}$$

(3.24)

Inequality (3.24) together with (3.22) and (3.23) gives that

$$\begin{aligned} I_2(x)\le C|h|M_{V,\eta }f(x). \end{aligned}$$

(3.25)

Therefore, by (3.13) and (3.21) we have

$$\begin{aligned} I(x)\le C(|h|+|h|^\delta )M_{V,\eta }f(x). \end{aligned}$$

By Lemma 2.4 for any $p'\le \eta <\infty$, it holds that

$$\begin{aligned} \Vert I\Vert _{L^p(w)}\le C(|h|+|h|^\delta )\Vert M_{V,\eta }f\Vert _{L^p(w)}\le C(|h|+|h|^\delta )\Vert f\Vert _{L^p(w)}. \end{aligned}$$

(3.26)

From (3.11), (3.12) and (3.26), we get

$$\begin{aligned} \Vert \mathcal {T}^*_{b,\gamma }f(h+\cdot )-\mathcal {T}^*_{b,\gamma }f(\cdot )\Vert _{L^p(w)}\le C(|h|+|h|^\delta )\Vert f\Vert _{L^p(w)}, \end{aligned}$$

which yields (3.10) and finishes the proof of Theorem 1.1. $\square$

4 Proof of Theorem 1.2

Proof of Theorem 1.2

Similar to the proof of Theorem 1.1, we define the smooth truncated function $\varphi \in C^{\infty }([0,\infty ))$ satisfy (3.1). For any $\gamma >0$, let

$$\begin{aligned} K_{\alpha ,\gamma }(x,y)=K_\alpha (x,y)\Big (1-\varphi (\gamma ^{-1}|x-y|)\Big ). \end{aligned}$$

(4.1)

Define

$$\begin{aligned} \mathcal {I}_{\alpha ,\gamma } f(x)=\int _{{\mathbb {R}^n}}K_{\alpha ,\gamma }(x,y)f(y)dy. \end{aligned}$$

(4.2)

and

$$\begin{aligned} \mathcal {I}_{\alpha ,\gamma }^b f(x)=\int _{{\mathbb {R}^n}}K_{\alpha ,\gamma }(x,y)(b(x)-b(y))f(y)dy. \end{aligned}$$

(4.3)

For any $b\in \mathcal {C}_c^\infty ({\mathbb {R}^n})$ and $\gamma ,\theta ,\ \eta >0$, by (4.1), (4.3) and Lemma 2.8 with $N=\theta \eta$, one has

$$\begin{aligned} \begin{aligned}&|\mathcal {I}_\alpha ^{b} f(x)-\mathcal {I}_{\alpha ,\gamma }^bf(x)|\le C\int _{|x-y|<2\gamma }\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-\theta (\eta +1-\frac{\alpha }{n})}\frac{|f(y)|}{|x-y|^{n-\alpha -1}}dy\\&\qquad \le C\gamma \sum _{k=1}^{\infty }\Big (1+\frac{2^{-k}\gamma }{\rho (x)}\Big )^{-\theta (\eta +1-\frac{\alpha }{n})}\frac{2^{-k}}{(2^{-k}\gamma )^{n-\alpha }}\int _{|x-y|<2^{-k}\gamma }|f(y)|dy\\&\qquad \le CM_{\alpha ,V,\eta }f(x). \end{aligned} \end{aligned}$$

(4.4)

Then by Lemma 2.5 with $\eta \ge (1-\alpha /n)p'/q$ gives that

$$\begin{aligned} \Vert \mathcal {I}_\alpha ^{b}f-\mathcal {I}_{\alpha ,\gamma }^b f\Vert _{L^p(w)}\le C\gamma \Vert f\Vert _{L^p(w^p)}, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{\gamma \rightarrow 0}\Vert \mathcal {I}_\alpha ^{b}f-\mathcal {I}_{\alpha ,\gamma }^b f\Vert _{L^p(w^p)}=0. \end{aligned}$$

(4.5)

Similar to the Proof of Theorem 1.1, if $b\in \text {CMO}_\theta (\rho )({\mathbb {R}^n})$, $w\in A_{(p,q)}^{\rho }$, then for any $\epsilon >0$, there exists $b_\epsilon \in \mathcal {C}_c^\infty ({\mathbb {R}^n})$ such that $\Vert b-b_\epsilon \Vert _{\text {BMO}_\theta (\rho )}<\epsilon$, so that, by Theorem E, we have

$$\begin{aligned} \Vert \mathcal {I}_\alpha ^{b}f-\mathcal {I}_\alpha ^{b_\epsilon }f\Vert _{L^q(w^q)}\le \Vert \mathcal {I}_\alpha ^{b-b_\epsilon }f\Vert _{L^q(w^q)}\le C\Vert b-b_\epsilon \Vert _{\text {BMO}_\theta (\rho )}\Vert f\Vert _{L^p(w^p)}\le C\epsilon . \end{aligned}$$

Thus, to prove $\mathcal {I}_\alpha ^{b}$ is compact on $L^q(w^q)$ for any $b\in \text {CMO}_\theta (\rho )$, it suffices to show that $\mathcal {I}_{\alpha ,\gamma }^b$ is compact on $L^q(w^q)$ for any $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$ when $\gamma >0$ is small enough. For arbitrary bounded set G in $L^q(w^q)$, let

$$\begin{aligned} \mathcal {G}=\{\mathcal {I}_{\alpha ,\gamma }^b f:f\in G\}. \end{aligned}$$

Then, we need to show that for $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$, $\mathcal {G}$ satisfies the conditions $\mathrm (i)$-$\mathrm (iii)$ of Lemma 2.3.

First of all, we demonstrate that $K_{\alpha ,\gamma }(x,y)$ satisfies the conditions (2.1)-(2.2) with

$$\begin{aligned} \tilde{C_N}=\max \{2C_N,2^{1+\delta _0}C_N\Vert \nabla \varphi \Vert _\infty \}. \end{aligned}$$

For any $x,y\in \mathbb {R}^n$,

$$\begin{aligned} 0<K_{\alpha ,\gamma }(x,y)|\le |K_\alpha (x,y)|\le \tilde{C_N}\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{1}{|x-y|^{n-\alpha }}. \end{aligned}$$

(4.6)

Below we shall prove that when $|y-z|<\frac{1}{2}|x-y|$, it holds that

$$\begin{aligned} |K_{\alpha ,\gamma }(x,y)-K_{\alpha ,\gamma }(x,z)|\le \tilde{C_N}\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|y-z|^{\delta _0}}{|x-y|^{n-\alpha +\delta _0}}. \end{aligned}$$

(4.7)

We consider the following four cases:

(i) ($|x-y|\ge 2\gamma$ and $|x-z|\ge 2\gamma$). In this case we have $K_{\alpha ,\gamma }(x,y)=K_\alpha (x,y)$ and $K_{\alpha ,\gamma }(x,z)=K_\alpha (x,z)$. This together with (2.2) yields (4.7).

(ii) ($|x-y|\ge 2\gamma$ and $|x-z|<2\gamma$). Note that $|y-z|<\frac{1}{2}|x-y|$. In this case it holds that $K_{\alpha ,\gamma }(x,y)=K_\alpha (x,y)$, $|x-z|>\frac{1}{2}|x-y|$, so that $\gamma \ge \frac{1}{4}|x-y|$. These together with (2.2) imply that

$$\begin{aligned} \begin{aligned} |K_{\alpha ,\gamma }(x,y)-K_{\alpha ,\gamma }(x,z)|&\le |K_\alpha (x,y)-K_\alpha (x,z)|(1+\varphi (\gamma ^{-1}|x-z|))\\&\quad +K_\alpha (x,y)|\varphi (\gamma ^{-1}|x-z|)-\varphi (\gamma ^{-1}|x-y|)|\\&\le 2C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|y-z|^{\delta _0}}{|x-y|^{n-\alpha +\delta _0}}\\&\quad +C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{\Vert \nabla \varphi \Vert _\infty }{|x-y|^{n-\alpha }}\frac{|y-z|}{\gamma }\\&\le 2C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|y-z|^{\delta _0}}{|x-y|^{n-\alpha +\delta _0}}\\&\quad +C_N\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{4\Vert \nabla \varphi \Vert _\infty }{|x-y|^{n-\alpha +\delta _0}}\Big (\frac{|y-z|}{|x-y|}\Big )^{1-\delta _0}\\&\le \tilde{C_N}\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|y-z|^{\delta _0}}{|x-y|^{n-\alpha +\delta _0}}. \end{aligned} \end{aligned}$$

which proves (4.7).

(iii) ($|x-y|<2\gamma$ and $|x-z|\ge 2\gamma$) and (iv) ($|x-y|<2\gamma$ and $|x-z|<2\gamma$). These case is similar to the case (ii).

From the definition of $K_{\alpha ,\gamma }$, we know that $0<K_{\alpha ,\gamma }(x,y)\le K_\alpha (x,y)$, then $\mathcal {I}_{\alpha ,\gamma }f(x)\le \mathcal {I}_{\alpha }(|f|)(x)$ and $\mathcal {I}_{\alpha ,\gamma }^bf(x)\le \mathcal {I}_{\alpha }(|f|)(x)$. Hence, the boundedness of $\mathcal {I}_{\alpha ,\gamma }$ and $\mathcal {I}_{\alpha ,\gamma }^b$ also holds. Thus, we have

$$\begin{aligned} \sup \limits _{f\in G}\Vert \mathcal {I}_{\alpha ,\gamma }^bf\Vert _{L^q(w^q)}\le C\sup \limits _{f\in G}\Vert f\Vert _{L^p(w^p)}\le C, \end{aligned}$$

which yields the fact that the set $\mathcal {G}$ is bounded.

Assume $b\in \mathcal {C}_c^\infty (\mathbb {R}^n)$ and ${\text {supp}}(b)\subset B(0,R)$, where B(0, R) is the ball of radius R center at origin in $\mathbb {R}^n$. For any $|x|> A > 2R$, by Lemma 2.8, we have

$$\begin{aligned} \begin{aligned} |\mathcal {I}_{\alpha ,\gamma }^bf(x)|&\le \int _{|y|<R}K_\alpha (x,y)|b(y)f(y)|dy\\&\le C_N\Vert b\Vert _\infty \displaystyle \int _{|y|<R}\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|f(y)|}{|x-y|^{n-\alpha }}dy\\&\le \frac{C_N\Vert b\Vert _\infty }{|x|^{(n-\alpha )}}\frac{\Vert f\Vert _{L^p(w^p)}}{(1+1/2|x|m_V(x))^N}\displaystyle \Big (\int _{|y|<R}w^{-p'}(y)dy\Big )^{1/p'}. \end{aligned} \end{aligned}$$

By Lemma 2.1, there exist $k_0\ge 1$ and $C_0>0$ such that

$$\begin{aligned} m_V(x)\ge C_0m_V(0)(1+|x|m_V(0))^{-\frac{k_0}{k_0+1}}. \end{aligned}$$

so that, we have

$$\begin{aligned} \begin{aligned} \frac{1}{1+1/2|x|m_V(x)}&\le \frac{2}{1+|x|m_V(x)}\le \frac{C}{1+|x|m_V(0)(1+|x|m_V(0))^{-\frac{k_0}{k_0+1}}}\\&\le \frac{C}{(1+|x|m_V(0))(1+|x|m_V(0))^{-\frac{k_0}{k_0+1}}}\\&=\frac{C}{(1+|x|m_V(0))^{\frac{1}{k_0+1}}}=\frac{C}{(1+|x|/\rho (0))^{\frac{1}{k_0+1}}} \end{aligned} \end{aligned}$$

Which leads to

$$\begin{aligned} \begin{array}{ll} &{}\quad \displaystyle \Big (\int _{|x|>A}|\mathcal {I}_{\alpha ,\gamma }^bf(x)|^qw^q(x)dx\Big )^{1/q}\\ &{}\le C\displaystyle \sum _{j=0}^{\infty }\frac{1}{(2^jA)^{n-\alpha }}\frac{1}{(1+2^jA/\rho (0))^{\frac{N}{k_0+1}}}\Big (\int _{2^jA<|x|<2^{j+1}A}w(x)^qdx\Big )^{1/q}\\ &{}\quad \times \Big (\int _{|y|<R}w^{-p'}(y)dy\Big )^{1/p'}\\ &{}\le C\displaystyle \sum _{j=0}^{\infty }\frac{1}{(2^jA)^{n-\alpha }}\frac{1}{(1+2^jA/\rho (0))^{\frac{N}{k_0+1}}}\Big (\int _{B(0,2^{j+1})}w(x)^qdx\Big )^{1/q}\\ &{}\quad \times \Big (\int _{B(0,R)}w^{-p'}(y)dy\Big )^{1/p'}. \end{array} \end{aligned}$$

Taking $Q=B(0,2^{j+1})$, $E=B(0,R)$. Since $w\in A_{(p,q)}^\rho$, then $w^q\in A_{1+q/p'}^{\rho ,\theta }$. Let $r=1+q/p'$, by Lemma 2.2, we can get

$$\begin{aligned} \begin{array}{ll} w^q(5Q)&{}\le C\displaystyle \Big (\frac{\Psi _\theta (Q)|Q|}{|E|}\Big )^rw^q(E)\\ &{}\le C\displaystyle w^q(B(0,R))\Big (\frac{(1+2^{j+1}A/\rho (0))^\theta (2^{j+1}A)^n}{R^n}\Big )^r\\ &{}\le C\displaystyle w^q(B(0,R))(1+2^{j+1}A/\rho (0))^{\theta r}\frac{(2^{j+1}A)^{nr}}{R^{nr}} \end{array} \end{aligned}$$

Taking $N>\max \{(k_0+1)\theta r/q,(k_0+1)((n+\theta )r/q+\alpha -n)\}$, then

$$\begin{aligned}&\displaystyle \Big (\int _{|x|>A}|\mathcal {I}_{\alpha ,\gamma }^bf(x)|^qw^q(x)dx\Big )^{1/q}\\&\quad \le C\displaystyle \sum _{j=0}^{\infty }\frac{(2^{j+1}A)^{nr/q}}{(2^jA)^{n-\alpha }}\frac{(1+2^{j+1}A/\rho (0))^{\theta r/q}}{(1+2^jA/\rho (0))^{\frac{N}{k_0+1}}}\Big (\frac{1}{(1+\frac{R}{\rho (0)})^\theta R^n}\\&\qquad \displaystyle \times \int _{B(0,R)}w(x)^qdx\Big )^{1/q}\Big (\frac{1}{(1+\frac{R}{\rho (0)})^\theta R^n}\int _{B(0,R)}w^{-p'}(y)dy\Big )^{1/p'}\\&\quad \le C\displaystyle \sum _{j=0}^{\infty }(2^{j}A)^{nr/q-n+\alpha }\frac{2(1+2^{j}A/\rho (0))^{\theta r/q}}{(1+2^jA/\rho (0))^{\frac{N}{k_0+1}}}\\&\quad \le C\sum _{j=0}^{\infty }(2^{j}A)^{-(\frac{N}{k_0+1}-(n+\theta )r/q+n-\alpha )}\\&\quad \le C(A)^{-(\frac{N}{k_0+1}-(n+\theta )r/q+n-\alpha )}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \lim _{A\rightarrow \infty }\int _{|x|>A}|\mathcal {I}_{\alpha ,\gamma }^bf(x)|^qw^q(x)dx=0, \end{aligned}$$

holds uniformly for $f\in G$.

It remains to show that the set $\mathcal {G}$ is uniformly equicontinuous. It suffices to verify that

$$\begin{aligned} \lim \limits _{|h|\rightarrow 0}\Vert \mathcal {I}_{\alpha ,\gamma }^bf(h+\cdot )-\mathcal {I}_{\alpha ,\gamma }^bf(\cdot )\Vert _{L^q(w^q)}=0, \end{aligned}$$

(4.8)

holds uniformly for $f\in G$.

In what follows, we fix $\gamma \in (0,\frac{1}{4})$ and $|h|<\frac{\gamma }{4}$. Then

$$\begin{aligned} \begin{aligned}&\mathcal {I}_{\alpha ,\gamma }^bf(x+h)-\mathcal {I}_{\alpha ,\gamma }^bf(x)\\&\quad =\displaystyle \int _{\mathbb {R}^n}(K_{\alpha ,\gamma }(x+h,y)-K_{\alpha ,\gamma }(x,y))(b(x+h)-b(y))f(y)dy\\&\qquad +\displaystyle \int _{\mathbb {R}^n}K_{\alpha ,\gamma }(x,y)(b(x+h)-b(x))f(y)dy\\&\quad =:{\tilde{I}}(x)+{\tilde{II}}(x). \end{aligned} \end{aligned}$$

(4.9)

For ${\tilde{II}}(x)$, it holds that

$$\begin{aligned} \begin{array}{ll} &{}{\tilde{II}}(x)=\displaystyle |b(x+h)-b(x)|\Big |\int _{\mathbb {R}^n}K_{\alpha ,\gamma }(x,y)f(y)dy\Big |\\ &{}\qquad \le C\displaystyle |h|\int _{\mathbb {R}^n}K_{\alpha }(x,y)|f(y)|dy\\ &{}\qquad = C\displaystyle |h|\mathcal {I}_{\alpha ,\gamma }(|f|)(x). \end{array} \end{aligned}$$

Then, by the $L^p(w)$-bounds of $\mathcal {I}_{\alpha ,\gamma }$, we have

$$\begin{aligned} {\tilde{II}}\Vert _{L^q(w^q)}\le C|h|\Vert f\Vert _{L^p(w^p)}. \end{aligned}$$

(4.10)

When $|x-y|<\frac{\gamma }{2}$ and $|h|<\frac{\gamma }{4}$, then $|x+h-y|<\frac{3\gamma }{4}$. Hence

$$\begin{aligned} \varphi (\gamma ^{-1}|x+h-y|)=1=\varphi (\gamma ^{-1}|x-y|). \end{aligned}$$

This implies

$$\begin{aligned} K_{\alpha ,\gamma }(x+h,y)=0=K_{\alpha ,\gamma }(x,y). \end{aligned}$$

When $|x-y|\le \frac{\gamma }{2}$ and $|h|<\frac{\gamma }{4}$, then $|h|<\frac{1}{2}|x-y|$. Therefore, by (4.7) with $N=\theta (\eta +1-\frac{n}{\alpha })$, we have

$$\begin{aligned} \begin{aligned} |{\tilde{I}}(x)|&=\displaystyle \Big |\int _{|x-y|\ge \frac{\gamma }{2}}(K_{\alpha ,\gamma }(x+h,y)-K_{\alpha ,\gamma }(x,y))(b(x+h)-b(y))f(y)dy\Big |\\&\le C|h|^{\delta _0}\displaystyle \int _{|x-y|\ge \frac{\gamma }{2}}\Big (1+\frac{|x-y|}{\rho (x)}\Big )^{-N}\frac{|f(y)|}{|x-y|^{n-\alpha +\delta _0}}\\&\le C\gamma ^{-\delta _0}|h|^{\delta _0}\sum _{k=1}^{\infty }\Big (1+\frac{2^k\gamma }{\rho (x)}\Big )^{-N}\frac{2^{-k\delta _0}}{(2^k\gamma )^{n-\alpha }}\int _{|x-y|\sim 2^k\gamma }|f(y)|dy\\&\le C\gamma ^{-\delta _0}|h|^{\delta _0}M_{\alpha ,V,\eta }f(x). \end{aligned} \end{aligned}$$

By Lemma 2.5 for any $\eta \ge (1-\alpha /n)p'/q$, it holds that

$$\begin{aligned} \Vert {\tilde{I}}\Vert _{L^q(w^q)}\le C|h|^{\delta _0}\Vert M_{\alpha ,V,\eta }f\Vert _{L^q(w^q)}\le C|h|^{\delta _0}\Vert f\Vert _{L^p(w^p)}. \end{aligned}$$

(4.11)

From (4.9), (4.10) and (4.11), we get

$$\begin{aligned} \Vert \mathcal {I}_{\alpha ,\gamma }^bf(h+\cdot )-\mathcal {I}_{\alpha ,\gamma }^bf(\cdot )\Vert _{L^q(w^q)}\le C(|h|+|h|^\delta )\Vert f\Vert _{L^p(w)}, \end{aligned}$$

which yields (4.8) and finishes the proof of Theorem 1.2. $\square$