# Estimates for the commutators of operator $$V^{\alpha }\nabla (-\Delta +V)^{-\beta }$$

## Abstract

Let a function b belong to the space $$\operatorname{BMO}_{\theta }(\rho )$$, which is larger than the space $$\operatorname{BMO}(\mathbb{R}^{n})$$, and let a nonnegative potential V belong to the reverse Hölder class $$\mathit{RH}_{s}$$ with $$n/2< s< n$$, $$n\geq 3$$. Define the commutator $$[b,T_{\beta }]f=bT_{ \beta }f-T_{\beta }(bf)$$, where the operator $$T_{\beta }=V^{\alpha } \nabla \mathcal{L}^{-\beta }$$, $$\beta -\alpha =\frac{1}{2}$$, $$\frac{1}{2}< \beta \leq 1$$, and $$\mathcal{L}=-\Delta +V$$ is the Schrödinger operator. We have obtained the $$L^{p}$$-boundedness of the commutator $$[b,T_{\beta }]f$$ and we have proved that the commutator is bounded from the Hardy space $$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$$ into weak $$L^{1}(\mathbb{R}^{n})$$.

## Introduction and results

Let $$\mathcal{L}=-\Delta +V$$ be the Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class $$\mathit{RH}_{s}$$ with $$s> n/2$$, $$n\geq 3$$. Many papers related to Schrödinger operator have appeared (see [1,2,3,4,5]). In recent years, some researchers have studied the boundedness of the commutators generated by the operators associated with $$\mathcal{L}$$ and the BMO type space (see [6,7,8,9]). In this paper, we investigated the boundedness of the commutator $$[b,T_{\beta}]$$, where $$T_{\beta }=V^{\alpha}\nabla\mathcal{L}^{-\beta}$$ and the function $$b\in \operatorname{BMO}_{\theta }(\rho )$$. We note that the space $$\operatorname{BMO}_{\theta }(\rho )$$ is larger than the space $$\operatorname{BMO}(\mathbb{R}^{n})$$.

For $$s>1$$, a nonnegative locally $$L^{s}$$-integrable function V is said to belong to $$\mathit{RH}_{s}$$ if there exists a constant $$C>0$$ such that the reverse Hölder inequality

$$\biggl(\frac{1}{ \vert B \vert } \int _{B}V(y)^{s}\,dy \biggr)^{1/s}\leq \frac{C}{ \vert B \vert } \int _{B}V(y)\,dy$$

holds for every ball $$B\subset \mathbb{R}^{n}$$. It is obvious that $$\mathit{RH}_{s_{1}}\subseteq \mathit{RH}_{s_{2}}$$ for $$s_{1}\geq s_{2}$$.

As in , for a given potential $$V\in \mathit{RH}_{s}$$ with $$s>n/2$$, we will use the auxiliary function $$\rho (x)$$ defined as

$$\rho (x)=\sup \biggl\{ r>0: \frac{1}{r^{n-2}} \int _{B(x,r)}V(y)\,dy\leq 1 \biggr\} , \quad x\in \mathbb{R}^{n}.$$

It is well known that $$0<\rho (x)<\infty$$ for any $$x\in \mathbb{R} ^{n}$$.

Let $$\mathcal{L}=-\Delta +V$$ be the Schrödinger operator on $$\mathbb{R}^{n}$$, where $$V\in \mathit{RH}_{s}$$ with $$s>n/2$$ and $$n\geq 3$$. We know $$\mathcal{L}$$ generates a $$(C_{0})$$ semigroup $$\{e^{-t \mathcal{L}}\}_{t>0}$$. The maximal function with respect to the semigroup $$\{e^{-t\mathcal{L}}\}_{t>0}$$ is defined by $$M^{\mathcal{L}}f(x)= \sup_{t>0}|e^{-t\mathcal{L}}f(x)|$$. The Hardy space associated with $$\mathcal{L}$$ is defined as follows (see [3, 4]).

### Definition 1

We say that f is an element of $$H_{\mathcal{L}}^{1}(\mathbb{R}^{n})$$ if the maximal function $$M^{\mathcal{L}}f$$ belongs to $$L^{1}( \mathbb{R}^{n})$$. The quasi-norm of f is defined by

$$\Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}= \bigl\Vert M^{\mathcal{L}}f \bigr\Vert _{L^{1}(\mathbb{R}^{n})}.$$

### Definition 2

Let $$1< q\leq \infty$$. A measurable function a is called an $$(1,q)_{\rho }$$-atom related to the ball $$B(x_{0},r)$$ if $$r<\rho (x _{0})$$ and the following conditions hold:

1. (1)

$$\operatorname{supp} a\subset B(x_{0},r)$$;

2. (2)

$$\|a\|_{L^{q}(\mathbb{R}^{n})}\leq |B(x_{0},r)|^{1/q-1}$$;

3. (3)

$$\int _{B(x_{0},r)}a(x)\,dx=0$$ if $$r<\rho (x_{0})/4$$.

The space $$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$$ admits the following atomic decomposition (see [3, 4]).

### Proposition 1

Let $$f\in L^{1}(\mathbb{R}^{n})$$. Then $$f\in H_{\mathcal{L}}^{1}( \mathbb{R}^{n})$$ if and only if f can be written as $$f=\sum_{j} \lambda _{j}a_{j}$$, where $$a_{j}$$ are $$(1,q)_{\rho }$$- atoms, $$\sum_{j}|\lambda _{j}|<\infty$$, and the sum converges in the $$H_{\mathcal{L}}^{1}(\mathbb{R}^{n})$$ quasi-norm. Moreover

$$\Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}\sim \inf \biggl\{ \sum _{j} \vert \lambda _{j} \vert \biggr\} ,$$

where the infimum is taken over all atomic decompositions of f into $$(1,q)_{\rho }$$- atoms.

Following , the space $$\operatorname{BMO}_{\theta }(\rho )$$ with $$\theta \geq 0$$ is defined as the set of all locally integrable functions b such that

$$\frac{1}{ \vert B(x,r) \vert } \int _{B(x,r)} \bigl\vert b(y)-b_{B} \bigr\vert \,dy \leq C \biggl(1+\frac{r}{ \rho (x)} \biggr)^{\theta }$$

for all $$x\in \mathbb{R}^{n}$$ and $$r>0$$, where $$b_{B}=\frac{1}{|B|} \int _{B}b(y)\,dy$$. A norm for $$b\in \operatorname{BMO}_{\theta }(\rho )$$, denoted by $$[b]_{\theta }$$, is given by the infimum of the constants in the inequalities above. Clearly, $$\operatorname{BMO}\subset \operatorname{BMO}_{\theta }(\rho )$$.

We consider the operator

$$T_{\beta }=V^{\alpha }\nabla \mathcal{L}^{-\beta },\quad \frac{1}{2}\leq \beta \leq 1, \beta -\alpha =\frac{1}{2}.$$

The boundedness of operator $$T_{1/2}$$ and its commutator have been researched under the condition $$V\in \mathit{RH}_{s}$$ for $$n/2< s< n$$. In , Shen showed that $$T_{1/2}$$ is bounded on $$L^{p}(\mathbb{R} ^{n})$$ for $$1< p< p_{0}$$, $$\frac{1}{p_{0}}=\frac{1}{s}-\frac{1}{n}$$. For $$b\in \operatorname{BMO}(\mathbb{R}^{n})$$, Guo, Li and Peng  investigated the $$L^{p}$$-boundedness of commutator $$[b,T_{1/2}]$$ for $$1< p< p_{0}$$; Li and Peng  studied the boundedness of $$[b, T_{1/2}]$$ from $$H_{\mathcal{L}}^{1}(\mathbb{R}^{n})$$ into weak $$L^{1}(\mathbb{R}^{n})$$. When $$b\in \operatorname{BMO}_{\theta }(\rho )$$, Bongioanni, Harboure and Salinas  obtained the $$L^{p}$$-boundedness of $$[b,T_{1/2}]$$ and Liu, Sheng and Wang  proved that $$[b,T_{1/2}]$$ is bounded from $$H_{\mathcal{L}}^{1}(\mathbb{R}^{n})$$ to weak $$L^{1}(\mathbb{R}^{n})$$. More boundedness of commutator $$[b,T_{1/2}]$$ can be found in  and .

For $$1/2<\beta \leq 1$$, $$\beta -\alpha =1/2$$, $$n/2< s< n$$, Sugano  established the estimate for $$T^{*}_{\beta }$$ (the adjoint operator of $$T_{\beta }$$), and proved that there exists a constant C such that

$$\bigl\vert T^{*}_{\beta }f(x) \bigr\vert \leq C M\bigl( \vert f \vert ^{{p}'_{\alpha }}\bigr) (x)^{1/{p}'_{\alpha }}$$

for all $$f\in C_{0}^{\infty }(\mathbb{R}^{n})$$, where $$\frac{1}{{p} _{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}$$, and $$\frac{1}{{p}_{ \alpha }}+ \frac{1}{{p}'_{\alpha }}=1$$. Then, by the boundedness of maximal function, we get

### Theorem 1

Suppose $$V\in \mathit{RH}_{s}$$ with $$n/2< s< n$$. Let $$1/2< \beta \leq 1$$, $$\frac{1}{p _{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}$$. Then

$$\bigl\Vert T^{*}_{\beta }f \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}$$

for $$p'_{\alpha }< p\leq \infty$$, and by duality we get

$$\Vert T_{\beta }f \Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{L^{p}(\mathbb{R} ^{n})}$$

for $$1\leq p< p_{\alpha }$$.

Inspired by the above results, in the present work, we are interested in the boundedness of $$[b,T_{\beta }]$$. Our main results are as follows.

### Theorem 2

Suppose $$V\in \mathit{RH}_{s}$$ with $$n/2< s< n$$. Let $$1/2< \beta \leq 1$$, $$b \in \operatorname{BMO}_{\theta }(\rho )$$. Then,

$$\bigl\Vert \bigl[b,T^{*}_{\beta }\bigr](f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}$$

for $$p'_{\alpha }< p< \infty$$, and

$$\bigl\Vert [b,T_{\beta }](f) \bigr\Vert _{L^{p}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{L^{p}( \mathbb{R}^{n})}$$

for $$1< p< p_{\alpha }$$, where $$\frac{1}{p_{\alpha }}= \frac{\alpha +1}{s}-\frac{1}{n}$$.

### Theorem 3

Suppose $$V\in \mathit{RH}_{s}$$ with $$n/2< s< n$$. Let $$1/2<\beta \leq 1$$, $$b\in \operatorname{BMO} _{\theta }(\rho )$$. Then,

$$\bigl\Vert [b,T_{\beta }](f) \bigr\Vert _{WL^{1}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{H_{ \mathcal{L}}^{1}(\mathbb{R}^{n})}.$$

In this paper, we shall use the symbol $$A\lesssim B$$ to indicate that there exists a universal positive constant c, independent of all important parameters, such that $$A\leq cB$$. $$A\sim B$$ means that $$A\lesssim B$$ and $$B\lesssim A$$.

## Some preliminaries

We recall some important properties concerning the auxiliary function $$\rho (x)$$ which have been proved by Shen . Throughout this section we always assume $$V\in \mathit{RH}_{s}$$ with $$n/2< s< n$$.

### Proposition 2

There exist constants C and $$k_{0}\geq 1$$ such that

$$C^{-1}\rho (x) \biggl(1+\frac{ \vert x-y \vert }{\rho (x)} \biggr)^{-k_{0}}\leq \rho (y)\leq C\rho (x) \biggl(1+\frac{ \vert x-y \vert }{\rho (x)} \biggr)^{\frac{k _{0}}{1+k_{0}}}$$

for all $$x,y \in \mathbb{R}^{n}$$.

Assume that $$Q=B(x_{0},\rho (x_{0}))$$, for any $$x\in Q$$, then Proposition 2 tells us that $$\rho (x)\sim \rho (y)$$, if $$|x-y|< C\rho (x)$$. It is easy to get the following result from Proposition 2.

### Lemma 1

Let $$k\in \mathbb{N}$$ and $$x\in 2^{k+1}B(x_{0},r)\setminus 2^{k}B(x _{0},r)$$. Then we have

$$\frac{1}{ (1+\frac{2^{k}r}{\rho (x)} )^{N}}\lesssim \frac{1}{ (1+\frac{2^{k}r}{\rho (x_{0})} )^{N/(k_{0}+1)}}.$$

### Lemma 2

There exists a constant $$l_{0}>0$$ such that

$$\frac{1}{r^{n-2}} \int _{B(x,r)}V(y)\,dy\lesssim \biggl(1+ \frac{r}{\rho (x)} \biggr)^{l_{0}}.$$

The following finite overlapping property was given by Dziubański and Zienkiewicz in .

### Proposition 3

There exists a sequence of points $$\{x_{k}\}_{k=1}^{\infty }$$ in $$\mathbb{R}^{n}$$, so that the family of critical balls $$Q_{k}=B(x_{k}, \rho (x_{k}))$$, $$k\geq 1$$, satisfies

1. (i)

$$\bigcup_{k} Q_{k}=\mathbb{R}^{n}$$.

2. (ii)

There exists $$N=N(\rho )$$ such that for every $$k\in N$$, $$\operatorname{card}\{j: 4Q_{j}\cap 4Q_{k}\}\leq N$$.

For $$\alpha >0$$, $$g\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})$$ and $$x\in \mathbb{R}^{n}$$, we introduce the following maximal functions:

$$M_{\rho ,\alpha }g(x)=\sup_{x\in B\in \mathcal{B}_{\rho ,\alpha }} \frac{1}{ \vert B \vert } \int _{B} \bigl\vert g(y) \bigr\vert \,dy,$$

and

$$M^{\sharp }_{\rho ,\alpha }g(x)= \sup_{x\in B\in \mathcal{B}_{\rho ,\alpha }} \frac{1}{ \vert B \vert } \int _{B} \bigl\vert g(y)-g _{B} \bigr\vert \,dy,$$

where $$\mathcal{B}_{\rho ,\alpha }=\{B(z,r): z\in \mathbb{R}^{n} \text{ and } r\leq \alpha \rho (y)\}$$.

The following Fefferman–Stein type inequality can be found in .

### Proposition 4

For $$1< p<\infty$$, then there exist δ and γ such that if $$\{Q_{k}\}_{k}$$ is a sequence of balls as in Proposition 3 then

$$\int _{\mathbb{R}^{n}} \bigl\vert M_{\rho ,\delta }g(x) \bigr\vert ^{p}\,dx\lesssim \int _{\mathbb{R}^{n}} \bigl\vert M^{\sharp }_{\rho ,\gamma }g(x) \bigr\vert ^{p}\,dx +\sum_{k} \vert Q_{k} \vert \biggl(\frac{1}{ \vert Q_{k} \vert } \int _{2Q_{k}} \vert g \vert \biggr)^{p}$$

for all $$g\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$$.

We have the following result for the function $$b\in \operatorname{BMO}_{\theta }( \rho )$$.

### Lemma 3

()

Let $$1\leq s<\infty$$, $$b\in \operatorname{BMO}_{\theta }(\rho )$$, and $$B=B(x,r)$$. Then

$$\biggl(\frac{1}{ \vert 2^{k}B \vert } \int _{2^{k}B} \bigl\vert b(y)-b_{B} \bigr\vert ^{s}\,dy \biggr)^{1/s} \lesssim [b]_{\theta }k \biggl(1+ \frac{2^{k}r}{\rho (x)} \biggr)^{ \theta '}$$

for all $$k\in \mathbb{N}$$, with $$r>0$$, where $$\theta '=(k_{0}+1) \theta$$ and $$k_{0}$$ is the constant appearing in Proposition 2.

We give an estimate of fundamental solutions; this result can be found in . We denote by $$\varGamma (x,y,\lambda )$$ the fundamental solution of $$-\Delta +(V(x)+i\lambda )$$, and then $$\varGamma (x,y,\lambda )=\varGamma (y,x,-\lambda )$$.

### Lemma 4

Assume that $$-\Delta u+(V(x)+i\lambda )u=0$$ in $$B(x_{0},2R)$$ for some $$x_{0}\in \mathbb{R}^{n}$$. Then, there exists a $$k'_{0}$$ such that

$$\biggl( \int _{B(x_{0}, R)} \vert \nabla u \vert ^{t}\,dx \biggr)^{1/t}\lesssim R^{n/s-2} \biggl(1+\frac{R}{\rho (x_{0})} \biggr)^{k'_{0}} \sup_{B(x_{0},2R)} \vert u \vert ,$$

where $$1/t=1/s-1/n$$.

Suppose $$\mathcal{W}_{\beta }= \nabla \mathcal{L}^{-\beta }$$. Let $$\mathcal{W}_{\beta }^{*}$$ be the adjoint operator of $$\mathcal{W} _{\beta }$$, K and $$K^{*}$$ be the kernels of $$\mathcal{W}_{\beta }$$ and $$\mathcal{W}_{\beta }^{*}$$ respectively, then $$K(x,z)=K^{*}(z,x)$$, and we have the following estimates.

### Lemma 5

Suppose $$1/2<\beta \leq 1$$.

1. (i)

For every N there exists a constant $$C_{N}$$ such that

$$\bigl\vert K^{*}(x,z) \bigr\vert \leq \frac{C_{N}}{ (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}} \frac{1}{ \vert x-z \vert ^{n-2\beta }} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr).$$

Moreover, the inequality above also holds with $$\rho (x)$$ replaced by $$\rho (z)$$.

2. (ii)

For every N and $$0<\delta <\min \{1,2-n/q_{0}\}$$ there exists a constant $$C_{N}$$ such that

\begin{aligned} \bigl\vert K^{*}(x,z) -K^{*}(y,z) \bigr\vert &\leq \frac{C_{N}}{ (1+ \frac{ \vert x-z \vert }{\rho (x)} )^{N}} \\ &\quad {} \times \frac{ \vert x-y \vert ^{\delta }}{ \vert x-z \vert ^{n-2\beta +\delta }} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr) \end{aligned}

whenever $$|x-y|<\frac{1}{16}|x-z|$$. Moreover, the inequality above also holds with $$\rho (x)$$ replaced by $$\rho (z)$$.

### Proof

The proof of (i) can be found in , page 449. Let us prove (ii). By (6) of  we know

$$K (x,z)= \textstyle\begin{cases} \frac{1}{2\pi }\int _{\mathbb{R}}(-i\tau )^{-\beta }\nabla _{x}\varGamma (x,z,\tau )\,d \tau , &\text{for } \frac{1}{2}< \beta < 1, \\ \nabla _{x}\varGamma (x,z,0),& \text{for } \beta =1. \end{cases}$$

Then

$$\bigl\vert K^{*} (x,z)-K^{*} (y,z) \bigr\vert \lesssim \int _{-\infty }^{\infty } \vert \tau \vert ^{- \beta } \bigl\vert \nabla _{z}\varGamma (z,x,\tau )-\nabla _{z} \varGamma (z,y,\tau ) \bigr\vert \,d \tau$$

for $$\frac{1}{2}<\beta <1$$ and

$$\bigl\vert K^{*} (x,z)-K^{*} (y,z) \bigr\vert \lesssim \bigl\vert \nabla _{z}\varGamma (z,x,0)- \nabla _{z}\varGamma (z,y,0) \bigr\vert$$

for $$\beta =1$$.

Fix $$x,z\in \mathbb{R}^{n}$$ and let $$R=|x-z|/8$$, $$1/t=1/s-1/n$$, $$\delta =2-n/s>0$$. For any $$|x-y|< R/2$$, it follows from the Morrey embedding theorem (see ) and Lemma 4 that

\begin{aligned} & \bigl\vert \nabla _{z}\varGamma (z,x,\tau )-\nabla _{z} \varGamma (z,y,\tau ) \bigr\vert \\ &\quad \lesssim \vert x-y \vert ^{1-n/t} \biggl( \int _{B(x,R)} \bigl\vert \nabla _{u}\nabla _{z} \varGamma (z,u,\tau ) \bigr\vert ^{t}\,du \biggr)^{1/t} \\ &\quad \lesssim \vert x-y \vert ^{1-n/t}R^{(n/s)-2} \biggl(1+ \frac{R}{\rho (x)} \biggr)^{k _{0}}\sup_{u\in B(x,2R)} \bigl\vert \nabla _{z} \varGamma (z,u,\tau ) \bigr\vert . \end{aligned}

It follows from [11, p. 428] that

\begin{aligned} &\sup_{u\in B(x,2R)} \bigl\vert \nabla _{z} \varGamma (z,u, \tau ) \bigr\vert \\ &\quad \lesssim \frac{C_{k_{1}}}{(1+ \vert \tau \vert ^{1/2} \vert z-u \vert )^{k_{1}} (1+\frac{ \vert z-u \vert }{ \rho (z)} )^{k_{1}}} \frac{1}{ \vert z-u \vert ^{n-2}} \\ &\qquad {} \times \biggl( \int _{B(z, \vert z-u \vert /4)}\frac{V(\xi )}{ \vert z-\xi \vert ^{n-1}}\,d \xi +\frac{1}{ \vert z-u \vert } \biggr). \end{aligned}

Then, by the fact that $$6R\leq |z-u|\leq 10R$$, we get

\begin{aligned} & \bigl\vert \nabla _{z}\varGamma (z,x,\tau )-\nabla _{z} \varGamma (z,y,\tau ) \bigr\vert \\ &\quad \lesssim \frac{ \vert x-y \vert ^{\delta }}{ \vert x-z \vert ^{n-2+\delta }} \frac{C_{N}}{(1+ \vert \tau \vert ^{1/2} \vert x-z \vert )^{N} (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}} \\ &\qquad {} \times \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert z-\xi \vert ^{n-1}}\,d \xi +\frac{1}{ \vert x-z \vert } \biggr). \end{aligned}

Thus, for $$\beta =1$$,

\begin{aligned} \bigl\vert K^{*} (x,z)-K^{*} (y,z) \bigr\vert &\lesssim \bigl\vert \nabla _{z}\varGamma (z,x,0)- \nabla _{z}\varGamma (z,y,0) \bigr\vert \\ &\lesssim \frac{ \vert x-y \vert ^{\delta }}{ \vert x-z \vert ^{n-2+\delta }} \frac{C_{N}}{ (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V( \xi )}{ \vert z-\xi \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr). \end{aligned}

Note that

$$\int _{-\infty }^{\infty } \frac{ \vert \tau \vert ^{-\beta }\,d \tau }{(1+ \vert \tau \vert ^{1/2} \vert x-z \vert )^{k}}\lesssim \vert x-z \vert ^{2 \beta -2}.$$

Then, for $$\frac{1}{2}<\beta <1$$, we have

\begin{aligned} \bigl\vert K^{*} (x,z)- K^{*} (y,z) \bigr\vert &\lesssim \frac{ \vert x-y \vert ^{\delta }}{ \vert x-z \vert ^{n+ \delta -2\beta }} \\ &\quad {}\times \frac{C_{N}}{ (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}} \biggl( \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi +\frac{1}{ \vert x-z \vert } \biggr). \end{aligned}

By Lemma 2, we know that the inequality above also holds with $$\rho (x)$$ replaced by $$\rho (z)$$. □

## Proof of main results

Before proving Theorem 2, we need to give some necessary lemmas.

### Lemma 6

Let $$V\in \mathit{RH}_{s}$$ with $$n/2< s< n$$, $$\frac{1}{{p}_{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}$$, and $$b\in \operatorname{BMO}_{\theta }(\rho )$$. Then, for any $${p}'_{\alpha }< t<\infty$$, we have

$$\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl[b, T^{*}_{\beta } \bigr]f \bigr\vert \lesssim [b]_{\theta } \inf_{y\in Q}M_{t}f(y)$$

for all $$f\in L^{t}_{\mathrm{loc}}(\mathbb{R}^{n})$$ and every ball $$Q=B(x_{0}, \rho (x_{0}))$$.

### Proof

Let $$f\in L^{t}_{\mathrm{loc}}(\mathbb{R}^{n})$$ and $$Q=B(x_{0},\rho (x _{0}))$$. We consider

$$\bigl[b,T^{*}_{\beta }\bigr]f=(b-b_{Q})T^{*}_{\beta }f-T^{*}_{\beta } \bigl(f(b-b_{Q})\bigr).$$
(1)

By Hölder’s inequality with $$t>{p}'_{\alpha }$$ and Lemma 3,

\begin{aligned} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert (b-b_{Q})T^{*}_{\beta }f \bigr\vert &\lesssim \biggl( \frac{1}{ \vert Q \vert } \int _{Q} \vert b-b_{Q} \vert ^{t'} \biggr)^{1/t'} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T^{*}_{\beta }f \bigr\vert ^{t} \biggr)^{1/t} \\ &\lesssim [b]_{\theta } \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T^{*}_{\beta }f \bigr\vert ^{t} \biggr)^{1/t}. \end{aligned}

Write $$f=f_{1}+f_{2}$$ with $$f_{1}=f\chi _{2Q}$$. By Theorem 1, we know that $$T^{*}_{\beta }$$ is bounded on $$L^{t}(\mathbb{R}^{n})$$ with $$t> {p}'_{\alpha }$$, and then

$$\biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T^{*}_{\beta }f_{1} \bigr\vert ^{t} \biggr)^{1/t} \lesssim \biggl( \frac{1}{ \vert Q \vert } \int _{2Q} \vert f \vert ^{t} \biggr)^{1/t}\lesssim \inf_{y\in Q}M_{t}f(y).$$

For $$x\in Q$$, using (i) in Lemma 5, we get

$$\bigl\vert T^{*}_{\beta }f_{2}(x) \bigr\vert = \biggl\vert \int _{(2Q)^{c}}V(z)^{\alpha }K^{*}(x,z)f(z)\,dz \biggr\vert \lesssim I_{1}(x)+I_{2}(x),$$

where

$$I_{1}(x)\lesssim \int _{(2Q)^{c}}\frac{ \vert f(z) \vert }{ (1+\frac{ \vert x-z \vert }{ \rho (x)} )^{N}} \frac{V(z)^{\alpha }}{ \vert x-z \vert ^{n-2\beta +1}}\,dz$$

and

$$I_{2}(x)\lesssim \int _{(2Q)^{c}}\frac{ \vert f(z) \vert }{ (1+\frac{ \vert x-z \vert }{ \rho (x)} )^{N}} \frac{V(z)^{\alpha }}{ \vert x-z \vert ^{n-2\beta }} \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi \,dz.$$

To deal with $$I_{2}(x)$$, note that $$\rho (x)\sim \rho (x_{0})$$ and $$|x-z|\sim |x_{0}-z|$$ for $$x\in Q$$. We split $$(2Q)^{c}$$ into annuli to obtain

$$I_{2}(x)\lesssim \sum_{k\geq 2} \frac{2^{-kN}(2^{k}\rho (x_{0}))^{2 \beta }}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q} \bigl\vert f(z) \bigr\vert V(z)^{\alpha } \mathcal{I}_{1}(V\chi _{2^{k}Q}) (z)\,dz.$$

Observe that $$\frac{1}{{p}'_{\alpha }}+\frac{\alpha }{s}+\frac{1}{q _{1}}=1$$, $$\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n}$$, $$t> {p}'_{\alpha }$$, and $$\beta -\alpha =1/2$$. Then by Hölder’s inequality and the boundedness of fractional integral $$\mathcal{I}_{1}: L^{s}\rightarrow L^{q_{1}}$$ with $$\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n}$$, we get

\begin{aligned} I_{2}(x)&\lesssim \sum_{k\geq 2}{2^{-kN}} {\bigl(2^{k}\rho (x_{0})\bigr)^{2 \beta }} \biggl( \frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q} \bigl\vert f(z) \bigr\vert ^{ {p}'_{\alpha }}\,dz \biggr)^{1/{{p}'_{\alpha }}} \\ &\quad {}\times \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q}V(z)^{s}\,dz \biggr) ^{\alpha /s} \biggl( \frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q} \bigl\vert \mathcal{I}_{1}(V\chi _{2^{k}Q}) (z) \bigr\vert ^{q_{1}}\,dz \biggr)^{1/{q_{1}}} \\ &\lesssim \sum_{k\geq 2}{2^{-kN}} { \bigl(2^{k}\rho (x_{0})\bigr)^{2\beta +n/s-n/ {q_{1}}}} \biggl( \frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q}V(z)^{s}\,dz \biggr) ^{\alpha /s} \\ &\quad {}\times \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q}V(z)^{s}\,dz \biggr) ^{1/s}\inf _{y\in Q}M_{t}f(y). \end{aligned}

Then, since $$V\in \mathit{RH}_{s}$$, from Lemma 2 and $$2\beta +n(1/s-1/ {q_{1}})-2\alpha -2=0$$, we get

\begin{aligned} I_{2}(x)&\lesssim \sum_{k\geq 2}2^{-kN} \bigl(2^{k}\rho (x_{0})\bigr)^{2\beta +n(1/s-1/ {q_{1}})-2\alpha -2} \bigl(1+2^{k}\bigr)^{(\alpha +1) l_{0}}\inf_{y\in Q}M_{t}f(y) \\ &\lesssim \inf_{y\in Q}M_{t}f(y). \end{aligned}
(2)

For $$I_{1}(x)$$, we split $$(2Q)^{c}$$ into annuli to obtain

$$I_{1}(x)\lesssim \sum_{k\geq 1} \frac{2^{-kN}(2^{k}\rho (x_{0}))^{2 \beta -1}}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q} \bigl\vert f(z) \bigr\vert V(z)^{\alpha }\,dz.$$

By Hölder’s inequality with $$\frac{1}{{p}'_{\alpha }}+\frac{ \alpha }{s}+\frac{1}{q_{1}}=1$$, $$t> {p}'_{\alpha }$$, $$\beta -\alpha =1/2$$, and Lemma 2, we get

\begin{aligned} I_{1}(x)&\lesssim \sum_{k\geq 1}{2^{-kN}} {\bigl(2^{k}\rho (x_{0})\bigr)^{2 \beta -1}} \biggl( \frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q} \bigl\vert f(z) \bigr\vert ^{ {p}'_{\alpha }}\,dz \biggr)^{1/{{p}'_{\alpha }}} \\ &\quad {}\times \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q}V(z)^{s}\,dz \biggr) ^{\alpha /s} \\ &\lesssim \sum_{k\geq 1}\frac{2^{-kN}}{(2^{k}\rho (x_{0}))^{1-2 \beta }} \biggl( \frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q}V(z)\,dz \biggr) ^{\alpha }\inf _{y\in Q}M_{t}f(y) \\ &\lesssim \sum_{k\geq 1}2^{-kN} \bigl(1+2^{k}\bigr)^{\alpha l_{0}}\inf_{y \in Q}M_{t}f(y) \lesssim \inf_{y\in Q}M_{t}f(y). \end{aligned}
(3)

To deal with the second term of (1), we write again $$f=f_{1}+f_{2}$$. Choosing $${p}'_{\alpha }<\bar{t}<t$$ and denoting $$\nu =\frac{\bar{t} t}{t-\bar{t}}$$, using the boundedness of $$T_{\beta }^{*}$$ on $$L^{\bar{t}}(\mathbb{R}^{n})$$ and applying Hölder’s inequality,

\begin{aligned} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T_{\beta }^{*}f_{1}(b-b_{Q}) \bigr\vert &\lesssim \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T_{\beta }^{*}f_{1}(b-b_{Q}) \bigr\vert ^{\bar{t}} \biggr)^{1/\bar{t}} \\ &\lesssim \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f_{1}(b-b_{Q}) \bigr\vert ^{\bar{t}} \biggr) ^{1/\bar{t}} \\ &\lesssim \biggl(\frac{1}{ \vert Q \vert } \int _{2Q} \vert f \vert ^{t} \biggr)^{1/t} \biggl(\frac{1}{ \vert Q \vert } \int _{2Q} \vert b-b_{q} \vert ^{\nu } \biggr)^{1/\nu } \\ &\lesssim [b]_{\theta }\inf_{y\in Q}M_{t}f(y). \end{aligned}

For the remaining term, we have

$${I}'_{1}(x)\lesssim \int _{(2Q)^{c}}\frac{ \vert f(z)(b-b_{Q}) \vert }{ (1+\frac{ \vert x-z \vert }{ \rho (x)} )^{N}} \frac{V(z)^{\alpha }}{ \vert x-z \vert ^{n-2\beta +1}}\,dz$$

and

$${I}'_{2}(x)\lesssim \int _{(2Q)^{c}}\frac{ \vert f(z)(b-b_{Q}) \vert }{ (1+\frac{ \vert x-z \vert }{ \rho (x)} )^{N}} \frac{V(z)^{\alpha }}{ \vert x-z \vert ^{n-2\beta }} \int _{B(z, \vert x-z \vert /4)}\frac{V(\xi )}{ \vert \xi -z \vert ^{n-1}}\,d\xi \,dz.$$

Since $$1\leq {p}'_{\alpha }< t$$, we can choose such that $${p}'_{\alpha }<\bar{t} <t$$. Let $$\nu =\frac{\bar{t} t}{t-\bar{t}}$$, and then by Hölder’s inequality and Lemma 3, we get

\begin{aligned} & \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q} \bigl\vert f(z) \bigl(b(z)-b _{Q} \bigr) \bigr\vert ^{{p}'_{\alpha }}\,dz \biggr)^{1/{{p}'_{\alpha }}} \\ &\quad \lesssim \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k+1}Q} \bigl\vert f(z) \bigl(b(z)-b _{Q} \bigr) \bigr\vert ^{\bar{t}}\,dz \biggr)^{1/\bar{t}} \\ &\quad \lesssim \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q} \bigl\vert f(z) \bigr\vert ^{t}\,dz \biggr) ^{1/t} \\ &\qquad {}\times \biggl(\frac{1}{(2^{k}\rho (x_{0}))^{n}} \int _{2^{k}Q} \bigl\vert \bigl(b(z)-b _{Q}\bigr) \bigr\vert ^{\nu }\,dz \biggr)^{1/\nu } \\ &\quad \lesssim k2^{k\theta '}[b]_{\theta }\inf_{y\in Q}M_{t}f(y). \end{aligned}
(4)

Then, similar to the estimate of (3), we get

$${I}'_{1}(x) \lesssim \sum_{k\geq 1}2^{-kN} \bigl(1+2^{k}\bigr)^{\alpha l_{0}}k2^{k \theta '}[b]_{\theta } \inf_{y\in Q}M_{t}f(y) \lesssim [b]_{\theta } \inf_{y\in Q}M_{t}f(y).$$

By (4) and similar to the estimate of (2), we can get

$${I}'_{2}(x) \lesssim [b]_{\theta }\inf _{y\in Q}M_{t}f(y).$$

This completes the proof of Lemma 6. □

### Lemma 7

Let $$V\in \mathit{RH}_{s}$$ for $$n/2< s< n$$, $$\frac{1}{{p}_{\alpha }}=\frac{\alpha +1}{s}-\frac{1}{n}$$, and $$b\in \operatorname{BMO}_{\theta }(\rho )$$. Then, for any $${p}'_{\alpha }< t<\infty$$ and $$\gamma \geq 1$$ we have

$$\int _{(2B)^{c}} \bigl\vert K^{*}(x,z)-K^{*}(y,z) \bigr\vert V(z)^{\alpha } \bigl\vert b(z)-b_{B} \bigr\vert \bigl\vert f(z) \bigr\vert \,dz \lesssim [b]_{\theta }\inf _{u\in B}M_{t}f(u),$$

for all f and $$x,y\in B=B(x_{0},r)$$ with $$r<\gamma \rho (x_{0})$$.

### Proof

Denote $$Q=B(x_{0},\gamma \rho (x_{0}))$$. By Lemma 5 and since in our situation $$\rho (x)\sim \rho (x_{0})$$ and $$|x-z|\sim |x _{0}-z|$$, we need to estimate the following four terms:

\begin{aligned}& J_{1}=r^{\delta } \int _{Q\setminus 2B}\frac{ \vert f(z) \vert V(z)^{\alpha } \vert b(z)-b _{B} \vert }{ \vert x_{0}-z \vert ^{n-2\beta +\delta +1}}\,dz, \\& J_{2}=r^{\delta }\rho (x_{0})^{N} \int _{Q^{c}}\frac{ \vert f(z) \vert V(z)^{\alpha } \vert b(z)-b_{B} \vert }{ \vert x_{0}-z \vert ^{n-2\beta +\delta +1+N}}\,dz, \\& J_{3}=r^{\delta } \int _{Q\setminus 2B}\frac{ \vert f(z) \vert V(z)^{\alpha } \vert b(z)-b _{B} \vert }{ \vert x_{0}-z \vert ^{n-2\beta +\delta }} \int _{B(x_{0},4 \vert x_{0}-z \vert )} \frac{V(u)}{ \vert u-z \vert ^{n-1}}\,du\,dz, \end{aligned}

and

$$J_{4}=r^{\delta }\rho (x_{0})^{N} \int _{Q^{c}}\frac{ \vert f(z) \vert V(z)^{\alpha } \vert b(z)-b_{B} \vert }{ \vert x_{0}-z \vert ^{n-2\beta +\delta +N}} \int _{B(x_{0},4 \vert x_{0}-z \vert )} \frac{V(u)}{ \vert u-z \vert ^{n-1}}\,du\,dz.$$

Splitting into annuli, we have

$$J_{1}\lesssim \sum_{j=2}^{j_{0}}2^{-j\delta } \bigl(2^{j}r\bigr)^{2\beta -1} \frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha }\,dz,$$

where $$j_{0}$$ is the least integer such that $$2^{j_{0}}\geq \gamma \rho (x_{0})/r$$. By Hölder’s inequality with $$\frac{1}{{p}'_{ \alpha }}+\frac{\alpha }{s}+\frac{1}{q_{1}}=1$$, $$t> {p}'_{\alpha }$$, similar to the estimate of (4), we have

\begin{aligned} &\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha }\,dz \\ &\quad \lesssim \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl( \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b _{B} \bigr\vert \bigr)^{{p}'_{\alpha }} \,dz \biggr)^{1/{p}'_{\alpha }} \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B}V(z)^{s} \,dz \biggr)^{\alpha /s} \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\alpha }[b]_{\theta }\inf _{y\in B}M_{t}f(y) \biggl(1+\frac{2^{j}r}{\rho (x_{0})} \biggr)^{\theta '+l_{0}\alpha } \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{1-2\beta }[b]_{\theta }\inf _{u\in B}M_{t}f(u). \end{aligned}

Then, using $$\beta -\alpha =1/2$$, we get

$$J_{1} \lesssim [b]_{\theta }\inf_{u\in B}M_{t}f(u).$$

To deal with $$I_{2}$$, we split into annuli and get

$$J_{2}\lesssim \biggl(\frac{\rho (x_{0})}{r} \biggr)^{N}\sum _{j=j_{0}-1} ^{\infty }2^{-j(\delta +N)} \bigl(2^{j}r\bigr)^{2\beta -1}\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha }\,dz.$$

Notice that

\begin{aligned} &\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha }\,dz \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\alpha }[b]_{\theta }\inf _{y\in B}M_{t}f(y) \biggl(1+\frac{2^{j}r}{\rho (x_{0})} \biggr)^{\theta '+l_{0}\alpha } \\ &\quad \lesssim j2^{j(\theta '+l_{0}\alpha )} \biggl(\frac{\rho (x_{0})}{r} \biggr) ^{-(\theta '+l_{0}\alpha )} \bigl(2^{j}r\bigr)^{1-2\beta }[b]_{\theta }\inf _{u \in B}M_{t}f(u). \end{aligned}

Then, taking $$N>\theta '+l_{0}\alpha$$, we get

$$J_{2} \lesssim [b]_{\theta }\inf_{u\in B}M_{t}f(u).$$

For $$J_{3}$$, splitting into annuli, we obtain

$$J_{3}\lesssim \sum_{j=2}^{j_{0}}2^{-j\delta } \bigl(2^{j}r\bigr)^{2\beta } \frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha } \mathcal{I}_{1}(V \chi _{2^{j+2}B}) (z) \,dz.$$

By Hölder’s inequality with $$\frac{1}{{p}'_{\alpha }}+\frac{ \alpha }{s}+\frac{1}{q_{1}}=1$$, similar to the estimate of (2), we get

\begin{aligned} &\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha } \mathcal{I}_{1}(V \chi _{2^{j+2}B}) (z)\,dz \\ &\quad \lesssim \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl( \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b _{B} \bigr\vert \bigr)^{{p}'_{\alpha }} \,dz \biggr)^{1/{p}'_{\alpha }} \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B}V(z)^{s} \,dz \biggr)^{\alpha /s} \\ &\qquad {} \times \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert \mathcal{I}_{1}(V \chi _{2^{j+2}B}) (z) \bigr\vert ^{q_{1}} \,dz \biggr)^{1/{q_{1}}} \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\alpha +n(1/s-1/q_{1})}[b]_{\theta } \inf_{y\in B}M_{t}f(y) \biggl(1+\frac{2^{j}r}{\rho (x_{0})} \biggr)^{ \theta '+l_{0}\alpha } \\ &\qquad {}\times \biggl(\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B}V(z)^{s}\,dz \biggr)^{1/s} \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\beta }[b]_{\theta }\inf _{y\in B}M_{t}f(y) \biggl(1+\frac{2^{j}r}{\rho (x_{0})} \biggr)^{\theta '+l_{0}(\alpha +1)} \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\beta }[b]_{\theta }\inf _{u\in B}M_{t}f(u). \end{aligned}

Then

$$J_{3}\lesssim [b]_{\theta }\inf_{u\in B}M_{t}f(u).$$

Finally, for $$J_{4}$$ we have

\begin{aligned} J_{4}&\lesssim \biggl(\frac{\rho (x_{0})}{r} \biggr)^{N}\sum _{j_{0}-1} ^{\infty }2^{-j(\delta +N)} \bigl(2^{j}r\bigr)^{2\beta } \\ &\quad {} \times \frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{ \alpha }\mathcal{I}_{1}(V \chi _{2^{j+2}B}) (z) \,dz. \end{aligned}

Notice that

\begin{aligned} &\frac{1}{ \vert 2^{j}B \vert } \int _{2^{j}B} \bigl\vert f(z) \bigr\vert \bigl\vert b(z)-b_{B} \bigr\vert V(z)^{\alpha } \mathcal{I}_{1}(V \chi _{2^{j+2}B}) (z)\,dz \\ &\quad \lesssim j\bigl(2^{j}r\bigr)^{-2\beta }[b]_{\theta }\inf _{y\in B}M_{t}f(y) \biggl(1+\frac{2^{j}r}{\rho (x_{0})} \biggr)^{\theta '+l_{0}(\alpha +1)} \\ &\quad \lesssim j2^{j(\theta '+l_{0}(\alpha +1))} \biggl(\frac{\rho (x _{0})}{r} \biggr)^{-\theta '-l_{0}(\alpha +1)} \bigl(2^{j}r\bigr)^{-2\beta }[b]_{ \theta }\inf _{u\in B}M_{t}f(u). \end{aligned}

We choose N large enough such that $$N>\theta '+l_{0}(\alpha +1)$$, and then

$$J_{4}\lesssim [b]_{\theta }\inf_{u\in B}M_{t}f(u),$$

which finishes the proof of Lemma 7. □

Now we are in a position to give the proof of Theorem 2.

### Proof of Theorem 2

We will prove part (i), and (ii) follows by duality. We start with a function $$f\in L^{p}(\mathbb{R} ^{n})$$ with $$p'_{\alpha }< p<\infty$$, and by Lemma 6 we have $$[b,T_{\beta }^{*}]f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$$.

By Proposition 3 and Lemma 6 with $$p'_{\alpha }< t< p< \infty$$, we have

\begin{aligned} \bigl\Vert \bigl[b,T_{\beta }^{*}\bigr]f \bigr\Vert ^{p}_{L^{p}(\mathbb{R}^{n})}&\lesssim \int _{\mathbb{R}^{n}} \bigl\vert M_{\rho ,\delta }\bigl[b,T_{\beta }^{*} \bigr]f \bigr\vert ^{p}\,dx \\ &\lesssim \int _{\mathbb{R}^{n}} \bigl\vert M^{\sharp }_{\rho ,\gamma } \bigl[b,T _{\beta }^{*}\bigr]f \bigr\vert ^{p}\,dx+ \sum_{k} \vert Q_{k} \vert \biggl( \frac{1}{ \vert Q_{k} \vert } \int _{2Q_{k}}\bigl\vert \bigl[b,T_{\beta }^{*}\bigr]f\bigr\vert \biggr)^{p} \\ &\lesssim \int _{\mathbb{R}^{n}} \bigl\vert M^{\sharp }_{\rho ,\gamma } \bigl[b,T _{\beta }^{*}\bigr]f \bigr\vert ^{p}\,dx+[b]^{p}_{\theta } \sum_{k} \int _{2Q_{k}} \bigl\vert M_{t}(f) \bigr\vert ^{p}\,dx. \end{aligned}

By Proposition 2 and the boundedness of $$M_{t}$$ on $$L^{p}(\mathbb{R}^{n})$$, the second term is controlled by $$[b]^{p} _{\theta }\|f\|^{p}_{L^{p}(\mathbb{R}^{n})}$$. Then, we only need to consider the first term.

Our goal is to find a point-wise estimate of $$M_{\rho ,\gamma }[b,T _{\beta }^{*}]f$$. Let $$x\in \mathbb{R}^{n}$$ and $$B=B(x_{0},r)$$ with $$r<\gamma \rho (x_{0})$$ such that $$x\in B$$. Write $$f=f_{1}+f_{2}$$ with $$f_{1}=f\chi _{2B}$$, then

$$\bigl[b,T_{\beta }^{*}\bigr]f =(b-b_{B})T_{\beta }^{*}f-T_{\beta }^{*} \bigl(f_{1}(b-b _{B})\bigr)-T_{\beta }^{*} \bigl(f_{2}(b-b_{B})\bigr).$$

Then, we need to control the mean oscillation on B of each term that we call $$\mathcal{O}_{1}$$, $$\mathcal{O}_{2}$$ and $$\mathcal{O}_{3}$$.

Let $$t>p'_{\alpha }$$, then, by Hölder’s inequality and Lemma 3, we get

\begin{aligned} \mathcal{O}_{1}&\lesssim \frac{1}{ \vert B \vert } \int _{B} \bigl\vert (b-b_{B})T_{\beta } ^{*}f \bigr\vert \\ &\lesssim \biggl(\frac{1}{ \vert B \vert } \int _{B} \vert b-b_{B} \vert ^{t'} \biggr)^{1/t'} \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\beta }^{*}f \bigr\vert ^{t} \biggr)^{1/t} \\ &\lesssim [b]_{\theta }M_{t}T_{\beta }^{*}f(x_{0}), \end{aligned}

since $$r<\gamma \rho (x_{0})$$.

To estimate $$\mathcal{O}_{2}$$, let $$p'_{\alpha }<\bar{t}<t$$ and $$\nu =\frac{\bar{t}t}{t-\bar{t}}$$. Then

\begin{aligned} \mathcal{O}_{2}&\lesssim \frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\beta }^{*} \bigl((b-b _{B})f_{1} \bigr) \bigr\vert \\ &\lesssim \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert T_{\beta }^{*} \bigl((b-b_{B})f _{1} \bigr) \bigr\vert ^{\bar{t}} \biggr)^{1/\bar{t}} \\ &\lesssim \biggl(\frac{1}{ \vert B \vert } \int _{B} \bigl\vert (b-b_{B})f_{1} \bigr\vert ^{\bar{t}} \biggr) ^{1/\bar{t}} \\ &\lesssim \biggl(\frac{1}{ \vert B \vert } \int _{B} \vert b-b_{B} \vert ^{\nu } \biggr)^{1/ \nu } \biggl(\frac{1}{ \vert B \vert } \int _{2B} \vert f \vert ^{t} \biggr)^{1/t} \\ &\lesssim [b]_{\theta }M_{t}f(x_{0}). \end{aligned}

For $$\mathcal{O}_{3}$$, note that $$\inf_{y\in B}M_{t}f(y)\leq M_{t}f(x _{0})$$, and so by Lemma 7 we get

\begin{aligned} \mathcal{O}_{3}&\lesssim \frac{1}{ \vert B \vert ^{2}} \int _{B} \int _{B} \bigl\vert T_{ \beta }^{*} \bigl((b-b_{B})f_{2} \bigr) (x)-T_{\beta }^{*} \bigl((b-b_{B})f _{2} \bigr) (y) \bigr\vert \,dx\,dy \\ &\lesssim [b]_{\theta }M_{t}f(x_{0}). \end{aligned}

Thus, we have showed that

$$\bigl\vert M^{\sharp }_{\rho ,\gamma }\bigl[b,T_{\beta }^{*} \bigr]f \bigr\vert \lesssim [b]_{\theta } \bigl( M_{t}T_{\beta }^{*}f(x)+M_{t}f(x) \bigr).$$

Since $$t< p$$, we obtain the desired result. □

### Proof of Theorem 3

Let $$f\in H_{\mathcal{L}}^{1}( \mathbb{R}^{n})$$. By Proposition 1, we can write $$f= \sum_{j=-\infty }^{\infty }\lambda _{j}a_{j}$$, where each $$a_{j}$$ is a $$(1,q)_{\rho }$$-atom with $$1< q< {p}_{\alpha }$$, $$\frac{1}{{p}_{\alpha }}=\frac{ \alpha +1}{q_{0}}-\frac{1}{n}$$ and $$\sum_{j=-\infty }^{\infty }| \lambda _{j}|\leq 2\|f\|_{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}$$. Suppose $$\operatorname{supp} a_{j}\subset B_{j}=B(x_{j},r_{j})$$ with $$r_{j}<\rho (x _{j})$$. Write

\begin{aligned}{} [b,T_{\beta }]f(x)&= \sum_{j=-\infty }^{\infty } \lambda _{j}[b,T_{ \beta }] a_{j}(x)\chi _{8B_{j}}(x) \\ &\quad {} +\sum_{j:r_{j}\geq {\rho (x_{j})}/{4}}\lambda _{j} \bigl(b(x)-b_{B _{j}} \bigr)T_{\beta }a_{j}(x)\chi _{(8B_{j})^{c}}(x) \\ &\quad {} +\sum_{j:r_{j}< {\rho (x_{j})}/{4}}\lambda _{j} \bigl(b(x)-b_{B_{j}} \bigr)T_{\beta }a_{j}(x)\chi _{(8B_{j})^{c}}(x) \\ &\quad {} -\sum_{j=-\infty }^{\infty }\lambda _{j} T_{\beta }\bigl((b-b_{B_{j}})a _{j}\bigr) (x)\chi _{(8B_{j})^{c}}(x) \\ &= \sum_{i=1}^{4}\sum _{j=-\infty }^{\infty }\lambda _{j}A_{ij}(x). \end{aligned}

Note that

$$\biggl( \int _{B_{j}} \bigl\vert a_{j}(x) \bigr\vert ^{q}\,dx \biggr)^{1/q}\lesssim \vert B_{j} \vert ^{ \frac{1}{q}-1}.$$

By Hölder’s inequality, for $$1< q< {p}_{\alpha }$$, and using Theorem 2 we get

\begin{aligned} \Vert A_{1,j} \Vert _{L^{1}(\mathbb{R}^{n})} &\lesssim \biggl( \int _{8B _{j}} \bigl\vert [b,T_{\beta }] a_{j}(x) \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}}r _{j}^{\frac{n}{q'}} \\ &\lesssim [b]_{\theta }r_{j}^{\frac{n}{q'}} \biggl( \int _{B_{j}} \bigl\vert a _{j}(x) \bigr\vert ^{q}\,dx \biggr)^{1/{q}} \\ &\lesssim [b]_{\theta } \vert B_{j} \vert ^{\frac{1}{q'}+\frac{1}{q}-1} \lesssim [b]_{\theta }. \end{aligned}

Thus

\begin{aligned} \Biggl\Vert \sum_{j=-\infty }^{\infty }\lambda _{j}A_{1j} \Biggr\Vert _{L^{1}( \mathbb{R}^{n})}&\lesssim \sum _{j=-\infty }^{\infty } \vert \lambda _{j} \vert \Vert A_{1j} \Vert _{L^{1}(\mathbb{R}^{n})} \\ &\lesssim [b]_{\theta }\sum_{j=-\infty }^{\infty } \vert \lambda _{j} \vert \lesssim [b]_{\theta } \Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}. \end{aligned}

And so

$$\Biggl\vert \Biggl\{ x\in \mathbb{ R}^{n}: \Biggl\vert \sum _{j=-\infty }^{\infty }\lambda _{j}A_{1j} \Biggr\vert >\frac{\lambda }{4} \Biggr\} \Biggr\vert \lesssim \frac{[b]_{ \theta }}{\lambda } \Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}.$$

Since $$z\in B_{j}$$, $$x\in 2^{k}B_{j}\setminus 2^{k-1}B_{j}$$, we have $$|x-z|\sim |x-x_{j}|\sim 2^{k}r_{j}$$, and by Lemma 1 we get

$$\frac{1}{ (1+\frac{ \vert x-z \vert }{\rho (x)} )^{N}}\lesssim \frac{1}{ (1+\frac{2^{k}r_{j}}{\rho (x_{j})} )^{\frac{N}{k_{0}+1}}}.$$

By Hölder’s inequality, Lemmas 2 and 3, we get

\begin{aligned} &\frac{1}{ \vert 2^{k}B_{j} \vert } \int _{2^{k}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{\alpha }\,dx \\ &\quad \lesssim \biggl(\frac{1}{ \vert 2^{k}B_{j} \vert } \int _{2^{k}B_{j}} \bigl\vert b(x)-b _{B_{j}} \bigr\vert ^{(\frac{s}{\alpha })'}\,dx \biggr)^{1/(\frac{s}{\alpha })'} \biggl(\frac{1}{ \vert 2^{k}B_{j} \vert } \int _{2^{k}B_{j}}V(x)^{s}\,dx \biggr)^{ \alpha /s} \\ &\quad \lesssim k[b]_{\theta } \biggl(1+\frac{2^{k}r_{j}}{\rho (x_{j})} \biggr) ^{\theta '} \biggl(\frac{1}{ \vert 2^{k}B_{j} \vert } \int _{2^{k}B_{j}}V(x)\,dx \biggr) ^{\alpha } \\ &\quad \lesssim k[b]_{\theta }\bigl(2^{k}r_{j} \bigr)^{-2\alpha } \biggl(1+\frac{2^{k}r _{j}}{\rho (x_{j})} \biggr)^{\theta '+l_{0}\alpha }. \end{aligned}
(5)

Note that $$\frac{1}{{p}'_{\alpha }}+\frac{\alpha }{s}+ \frac{1}{q_{1}}=1$$, $$\frac{1}{q_{1}}=\frac{1}{s}-\frac{1}{n}$$, so by Hölder’s and Hardy–Littlewood–Sobolev’s inequalities and using the fact that $$V\in \mathit{RH}_{s}$$, we obtain

\begin{aligned} &\frac{1}{ \vert 2^{k} B_{j} \vert } \int _{2^{k}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{\alpha }\bigl(\mathcal{I}_{1}(V\chi _{2^{k}B}) (x) \bigr)\,dx \\ &\quad \lesssim \biggl(\frac{1}{ \vert 2^{k} B_{j} \vert } \int _{2^{k}B_{j}} \bigl\vert b(x)-b _{B_{j}} \bigr\vert ^{{p}'_{\alpha }}\,dx \biggr)^{1/{p}'_{\alpha }} \biggl(\frac{1}{ \vert 2^{k} B_{j} \vert } \int _{2^{k}B_{j}}V(x)^{s}\,dx \biggr)^{ \alpha /s} \\ &\qquad {} \times \biggl(\frac{1}{ \vert 2^{k} B_{j} \vert } \int _{2^{k}B_{j}}\bigl( \mathcal{I}_{1}(V\chi _{2^{k}B_{j}}) (x)\bigr)^{q_{1}}\,dx \biggr)^{1/q_{1}} \\ &\quad \lesssim [b]_{\theta }k \bigl\vert 2^{k}B_{j} \bigr\vert ^{1/s-1/q_{1}} \biggl(1+\frac{2^{k}r _{j}}{\rho (x_{j})} \biggr)^{\theta '} \biggl(\frac{1}{ \vert 2^{k} B_{j} \vert } \int _{2^{k}B_{j}}V(x)^{s}\,dx \biggr)^{(\alpha +1)/s} \\ &\quad \lesssim [b]_{\theta }k\bigl(2^{k}r_{j} \bigr)^{-2\alpha -1} \biggl(1+\frac{2^{k}r _{j}}{\rho (x_{j})} \biggr)^{\theta '+(\alpha +1)l_{0}}. \end{aligned}
(6)

Recall $$\int _{B_{j}}|a_{j}(y)|\,dy\lesssim 1$$, $$\beta -\alpha = \frac{1}{2}$$ and $$r_{j}/\rho (x_{j})\geq 1/4$$. Then, taking N large enough such that $$\frac{N}{k_{0}+1}>\theta '+l_{0}(\alpha +1)$$, we get

\begin{aligned} & \bigl\Vert A_{2,j}(x) \bigr\Vert _{L^{1}(\mathbb{R}^{n})} \\ &\quad \lesssim \sum_{k\geq 4}\frac{1}{ (1+ \frac{2^{k}r_{j}}{\rho (x)} )^{N}} \frac{1}{(2^{k}r_{j})^{n-2 \beta +1}} \int _{2^{k}B_{j}\setminus 2^{k-1}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{ \alpha }\,dx \int _{B_{j}} \bigl\vert a_{j}(z) \bigr\vert \,dz \\ &\qquad {} +\sum_{k\geq 4}\frac{1}{ (1+\frac{2^{k}r_{j}}{\rho (x)} ) ^{N}}\frac{1}{(2^{k}r_{j})^{n-2\beta }} \\ &\qquad {} \times \int _{2^{k}B_{j}\setminus 2^{k-1}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{ \alpha }\bigl(\mathcal{I}_{1}(V\chi _{2^{k}B}) (x) \bigr)\,dx \int _{B_{j}} \bigl\vert a_{j}(z) \bigr\vert \,dz \\ &\quad \lesssim [b]_{\theta }\sum_{k\geq 4} \frac{k(2^{k}r_{j})^{2\beta -1}}{ (1+\frac{2^{k}r_{j}}{\rho (x_{j})} )^{\frac{N}{k_{0}+1}}} \bigl(2^{k}r_{j}\bigr)^{-2\alpha } \biggl(1+\frac{2^{k}r_{j}}{\rho (x_{j})} \biggr) ^{\theta '+l_{0}\alpha } \\ &\qquad {} +[b]_{\theta }\sum_{k\geq 4}\frac{(2^{k}r_{j})^{2\beta }}{ (1+\frac{2^{k}r _{j}}{\rho (x_{j})} )^{\frac{N}{k_{0}+1}}} \bigl(2^{k}r_{j}\bigr)^{-2 \alpha -1} \biggl(1+ \frac{2^{k}r_{j}}{\rho (x_{j})} \biggr)^{\theta '+( \alpha +1)l_{0}} \\ &\quad \lesssim [b]_{\theta }\sum_{k\geq 3} \frac{k}{(2^{k})^{\frac{N}{k _{0}+1}-\theta '-l_{0}\alpha }} +[b]_{\theta }\sum_{k\geq 3} \frac{k}{(2^{k})^{\frac{N}{k _{0}+1}-\theta '-l_{0}(\alpha +1)}} \\ &\quad \lesssim [b]_{\theta }. \end{aligned}

Thus

$$\Biggl\Vert \sum_{j=-\infty }^{\infty }\lambda _{j}A_{2j} \Biggr\Vert _{L^{1}( \mathbb{R}^{n})}\lesssim [b]_{\theta } \Vert f \Vert _{H_{\mathcal{L}}^{1}( \mathbb{R}^{n})}.$$

Therefore

$$\Biggl\vert \Biggl\{ x\in \mathbb{ R}^{n}: \Biggl\vert \sum _{j=-\infty }^{\infty }\lambda _{j}A_{2j} \Biggr\vert >\frac{\lambda }{4} \Biggr\} \Biggr\vert \lesssim \frac{[b]_{ \theta }}{\lambda } \Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}.$$

When $$x\in 2^{k}B_{j}\setminus 2^{k-1}B_{j}$$, and $$z\in B_{j}$$, by Lemmas 5 and 1, we have

\begin{aligned} \bigl\vert K(x,z) -K(x,x_{j}) \bigr\vert &\lesssim \frac{1}{ (1+\frac{2^{k}r_{j}}{ \rho (x_{j})} )^{N/(k_{0}+1)}}\frac{r_{j}^{\delta }}{(2^{k}r _{j})^{n+\delta -2\beta +1}} \\ &\quad {} +\frac{1}{ (1+\frac{2^{k}r_{j}}{\rho (x_{j})} )^{N/(k _{0}+1)}}\frac{r_{j}^{\delta }}{(2^{k}r_{j})^{n+\delta -2\beta }} \mathcal{I}_{1}(V\chi _{2^{k}B_{j}}) (z), \end{aligned}

where $$\delta =2-n/s>0$$. Thus, by the vanishing condition of $$a_{j}$$, together with (5) and (6), we have

\begin{aligned} & \bigl\Vert A_{3,j}(x) \bigr\Vert _{L^{1}(\mathbb{R}^{n})} \\ &\quad \lesssim \sum_{k\geq 4} \int _{2^{k}B_{j}\setminus 2^{k-1}B_{j}} \bigl\vert b(x)-b _{B_{j}} \bigr\vert V(x)^{\alpha } \int _{B_{j}} \bigl\vert K_{\alpha }(x,z)-K_{\alpha }(x,x _{j}) \bigr\vert \bigl\vert a_{j}(z) \bigr\vert \,dz\,dx \\ &\quad \lesssim \sum_{k\geq 3}\frac{1}{ (1+\frac{2^{k}r_{j}}{\rho (x _{j})} )^{\frac{N}{k_{0}+1}}} \frac{r_{j}^{\delta }}{(2^{k}r_{j})^{n+ \delta -2\beta +1}} \int _{2^{k+1}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{\alpha }\,dx \int _{B_{j}} \bigl\vert a_{j}(z) \bigr\vert \,dz \\ &\qquad {} +\sum_{k\geq 3}\frac{1}{ (1+\frac{2^{k}r_{j}}{\rho (x_{j})} ) ^{\frac{N}{k_{0}+1}}}\frac{r_{j}^{\delta }}{(2^{k}r_{j})^{(n+\delta -2 \beta )}} \\ & \qquad {} \times \int _{2^{k+1}B_{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert V(x)^{\alpha }\mathcal{I} _{1}(V\chi _{2^{k}B_{j}}) (x)\,dx \int _{B_{j}} \bigl\vert a_{j}(z) \bigr\vert \,dz \\ &\quad \lesssim [b]_{\theta }\sum_{k\geq 3} \frac{1}{ (1+\frac{2^{k}r _{j}}{\rho (x_{j})} )^{\frac{N}{k_{0}+1}-\theta '-l_{0}\alpha }}\frac{k}{2^{k \delta }} +[b]_{\theta }\sum _{k\geq 3}\frac{1}{ (1+\frac{2^{k}r _{j}}{\rho (x_{j})} )^{\frac{N}{k_{0}+1}-\theta '-l_{0}(\alpha +1)}}\frac{k}{2^{k\delta }} \lesssim [b]_{\theta }. \end{aligned}

So that

$$\Biggl\vert \Biggl\{ x\in \mathbb{ R}^{n}: \Biggl\vert \sum _{j=-\infty }^{\infty }\lambda _{j}A_{3j} \Biggr\vert >\frac{\lambda }{4} \Biggr\} \Biggr\vert \lesssim \frac{[b]_{ \theta }}{\lambda } \Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}.$$

Now let us deal with the last part. Since $$r_{j}\leq \rho (x_{j})$$, we get

\begin{aligned} \bigl\Vert (b-b_{B_{j}})a_{j} \bigr\Vert _{L^{1}(\mathbb{R}^{n})} &\leq \biggl( \int _{B _{j}} \bigl\vert b(x)-b_{B_{j}} \bigr\vert ^{q'}\,dx \biggr)^{1/q'} \biggl( \int _{B_{j}} \bigl\vert a_{j}(x) \bigr\vert ^{q}\,dx \biggr) ^{1/q} \\ &\lesssim [b]_{\theta } \biggl(1+\frac{r_{j}}{\rho (x_{j})} \biggr) ^{\theta '}\lesssim [b]_{\theta }. \end{aligned}

Note that

\begin{aligned} \bigl\vert A_{4j}(x) \bigr\vert &\leq \sum _{j=-\infty }^{\infty } \vert \lambda _{j} \vert T_{\beta }\bigl( \bigl\vert (b-b _{B_{j}})a_{j} \bigr\vert \bigr) (x)\chi _{(8B_{j})^{c}}(x) \\ &\leq T_{\beta } \Biggl(\sum_{j=-\infty }^{\infty } \bigl\vert \lambda _{j}(b-b _{B_{j}})a_{j} \bigr\vert \Biggr) (x). \end{aligned}

By Theorem 1, we know $$T_{\beta }$$ is bounded from $$L^{1}(\mathbb{R}^{n})$$ into weak $$L^{1}(\mathbb{R}^{n})$$. Then

\begin{aligned} & \Biggl\vert \Biggl\{ x\in \mathbb{R}^{n}: \Biggl\vert \sum _{j=-\infty }^{\infty }\lambda _{j} A_{4j} \Biggr\vert >\frac{\lambda }{4} \Biggr\} \Biggr\vert \\ &\quad \leq \Biggl\vert \Biggl\{ x\in \mathbb{R}^{n}: \Biggl\vert T_{\beta } \Biggl( \sum_{j=-\infty }^{\infty } \bigl\vert \lambda _{j}(b-b_{B_{j}})a_{j} \bigr\vert \Biggr) (x) \Biggr\vert >\frac{\lambda }{4} \Biggr\} \Biggr\vert \\ &\quad \lesssim \frac{1}{\lambda } \Biggl\Vert \sum_{j=-\infty }^{\infty } \bigl\vert \lambda _{j}(b-b_{B_{j}})a_{j} \bigr\vert \Biggr\Vert _{L^{1}(\mathbb{R}^{n})} \\ &\quad \lesssim \frac{1}{\lambda }\sum_{j=-\infty }^{\infty } \vert \lambda _{j}| \bigl\Vert (b-b_{B_{j}})a_{j} \bigr\Vert _{L^{1}(\mathbb{R}^{n})} \\ &\quad \lesssim \frac{[b]_{\theta }}{\lambda } \Biggl(\sum_{j=-\infty } ^{\infty } \vert \lambda _{j} \vert \Biggr)\lesssim \frac{[b]_{\theta }}{\lambda } \Vert f \Vert _{H_{\mathcal{L}}^{1}(\mathbb{R}^{n})}. \end{aligned}

Thus,

\begin{aligned} & \Biggl\vert \Biggl\{ x\in \mathbb{R}^{n}: \Biggl\vert \sum _{i=1}^{4}\sum _{j=- \infty }^{\infty }\lambda _{j} A_{ij} \Biggr\vert >\lambda \Biggr\} \Biggr\vert \\ &\quad \lesssim \sum_{i=1}^{4} \Biggl\vert \Biggl\{ x\in \mathbb{R}^{n}: \Biggl\vert \sum _{j=-\infty }^{\infty }\lambda _{j} A_{ij} \Biggr\vert > \frac{\lambda }{4} \Biggr\} \Biggr\vert \\ &\quad \lesssim \frac{[b]_{\theta }}{\lambda } \Vert f \Vert _{H_{\mathcal{L}}^{1}( \mathbb{R}^{n})}. \end{aligned}

□

## Conclusion

In this paper, we established the $$L^{p}$$-boundedness of commutator operators $$[b,T_{\beta }]$$ and $$[b,T^{*}_{\beta }]$$, where $$T_{ \beta }=V^{\alpha }\nabla \mathcal{L}^{-\beta }$$, $$\frac{1}{2}< \beta \leq 1$$, $$\beta -\alpha =\frac{1}{2}$$, and $$b\in \operatorname{BMO}_{\theta }(\rho )$$, which is larger than the space $$\operatorname{BMO}(\mathbb{R}^{n})$$. At the endpoint, we show that the operator $$[b,T_{\beta }]$$ is bounded from Hardy space $$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$$ continuously into weak $$L^{1}(\mathbb{R}^{n})$$. These results enrich the theory of Schrödinger operator.

## References

1. Zhong, J.: Harmonic analysis for some Schrödinger type operators. Dissertation, Princeton University (1993)

2. Shen, Z.: $$L^{p}$$ estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)

3. Dziubański, J., Zienkiewicz, J.: Hardy spaces $$H^{1}$$ associated to Schrödinger operators with potentials potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15, 279–296 (1999)

4. Dziubański, J., Zienkiewicz, J.: $$H^{p}$$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98, 5–38 (2003)

5. Sugano, S.: Estimates for the operators $$V^{\alpha }(-\Delta +V)^{- \beta }$$ and $$V^{\alpha }\nabla (-\Delta +V)^{-\beta }$$ with certain non-negative potentials V. Tokyo J. Math. 21, 441–452 (1998)

6. Li, P., Mo, Y., Zhang, C.: A compactness criterion and application to the commutators associated with Schrödinger operators. Math. Nachr. 288, 235–248 (2015)

7. Li, P., Wan, X., Zhang, C.: Schrödinger type operators on generalized Morrey spaces. J. Inequal. Appl. 2015, 229 (2015)

8. Li, P., Wan, X.: The boundedness of commutators associated with Schrödinger operators on Herz spaces. J. Inequal. Appl. 2016, 172 (2016)

9. Hu, Y., Wang, Y.: Hardy type estimates for Riesz transforms associated with Schrödinger operators. Anal. Math. Phys. 9, 275–287 (2019)

10. Bongioanni, B., Harboure, E., Salinas, O.: Commutators of Riesz transforms related to Schrödinger operators. J. Fourier Anal. Appl. 17, 115–134 (2011)

11. Guo, Z., Li, P., Peng, L.: $$L^{p}$$ boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341, 421–432 (2008)

12. Li, P., Peng, L.: Endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators. Bull. Aust. Math. Soc. 82, 367–389 (2010)

13. Liu, Y., Sheng, J., Wang, L.: Weighted endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators. Abstr. Appl. Anal. 2013, 281562 (2013)

14. Liu, Y., Sheng, J.: Some estimates for commutators of Riesz transforms associated with Schrödinger operators. J. Math. Anal. Appl. 419, 298–328 (2014)

15. Liu, Y., Huang, J., Dong, J.: Commutators of Calderón–Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators. Sci. China Math. 56, 1895–1913 (2013)

16. Gilberg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

### Acknowledgements

The authors are very grateful to the anonymous referees and the editor for their insightful comments and suggestions.

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Correspondence to Yue Hu.

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