Let $\mathfrak{L}=-\triangle+V$ be a Schr\"{o}dinger operator on $\mathbb{R}^{n}(n>3), $ where $\triangle$ is the Laplacian on $\mathbb{R}^{n}$ and $V\neq 0$ is a nonnegative locally integrable function. The problems about the Schr\"{o}dinger operators $\mathfrak{L}$ have been well studied(see [1-3] for example). Especially, Fefferman [1], Shen [2] and Zhong [3] developed some basic results.

The commutators generated by the Riesz transforms associated with the Schrödinger operators and BMO functions or Lipschitz functions also attract much attention(see[4-6] for example). Chu [7], consider the boundedness of commutators generalized by the $\rm{BM}{{\rm{O}}_{\mathfrak{L}}}$ functions and the Riesz transform $\nabla(-\triangle+V)^{-1/2}$ on Lebesgue spaces. Mo et al. [8] established the boundedness of commutators generated by the Campanato-type functions and the Riesz transforms associated with Schrödinger operators.

First, let us introduce some notations. A nonnegative locally $L^{q}(\mathbb{R}^{n})$ integrable function $V$ is said to belong to $B_{q}(1<q<\infty)$ if there exists $C=C(q, V)>0$ such that the reverse Hölder's inequality

$\biggl(\dfrac{1}{|B|}\int_{B}V(x)^{q}dx\biggr)^{1/q}\leq C\biggl(\dfrac{1}{|B|}\int_{B}V(x)dx\biggr)$

(1.1)

holds for every ball $B$ in $\mathbb{R}^{n}$.

Let $T_{j, 1}=\nabla(-\triangle+V)^{-1/2}$ or $T_{j, 1}=\pm I$ and $\mathrm{T}_{j, 2}$ be a linear operator which is bounded on $\mathrm{L} ^{p}(\mathbb{R}^{n})$ space, for $j=1, 2, \dots, m$. Let $M_{b}f=bf$, where $b$ is a locally integrable function on $\mathbb{R}^{n}$, then the Toeplitz operator is defined by

$\Theta^{b}=\sum\limits_{j=1}^{m}T_{j, 1}M_{b}T_{j, 2}.$

About the Toeplitz operator, there are some results. Mo et al. [9] established the boundedness of the Toeplitz operator generalized by the singular integral operator with nonsmooth kernel and the generalized fractional. Liu et al. [11] investigated the boundness of the Toeplitz operator related to the generalized fractional integral operator.

The commutator $[b, T](f)=bT(f)-T(bf)$ is a particular case of the Toeplitz operators. Inspired by [7-10], we will consider the boundedness of the Toeplitz operators generated by the Campanato-type functions and Riesz transforms associated with Schödinger operators.

Definition 1.1  Let $f\in L_{loc}(\mathbb{R}^{n})$, then the sharp maximal function associated with $\mathfrak{L}=-\triangle+V$ is defined by

$\begin{array}{cl} M_{\mathfrak{L}}^{\#}(f)(x)=\left\{\begin{array}{ll} \sup\limits_{x\in B}\dfrac{1}{|B|} \int_{B(x, s)}|f(y)-f_{B(x, s)}|dy, \;\; & \mbox{when } s<\rho(x), \\ \sup\limits_{x\in B}\dfrac{1}{|B(x, s)|} \int_{B(x, s)}|f(y)|dy, \;\; & \mbox{when }s\geq\rho(x).\\ \end{array}\right.\\ \end{array}$

Where $\rho$ is define by

$\rho(x)=\sup\biggl\{r>0:\dfrac{1}{r^{n-2}}\int_{B(x, r)}V\leq 1\biggr\}.$

Definition 1.2 [12, 13]  Let $\mathfrak{L}=-\triangle+V$, $p\in(0, \infty)$ and $\beta\in\mathbb{R}$. A function $f\in L_{loc}^{p}(\mathbb{R}^{n})$ is said to be in $\Lambda_{\mathfrak{L}}^{\beta, p}(\mathbb{R}^{n})$, if there exists a nonnegative constant $C$ such that for all $x\in\mathbb{R}^{n}$ and $0<s<\rho(x)\leq r, $

$\biggl\{\dfrac{1}{|B(x, s)|^{1+p\beta}} \int_{B(x, s)}|f(y)-f_{B(x, s)}|^{p}dy\biggr\}^{1/p}+\biggl\{\dfrac{1}{|B(x, r)|^{1+p\beta}} \int_{B(x, r)}|f(y)|^{p}dy\biggr\}^{1/p}\leq C, $

where $f_{B}=\frac{1}{|B|}\int_{B}f(y)dy$ for any ball $B.$ Moreover, the minimal constant $C$ as above is defined for the norm of $f$ in the space $\Lambda_{\mathfrak{L}}^{\beta, p}(\mathbb{R}^{n})$ and denote by $\|f\|_{\Lambda_{\mathfrak{L}}^{\beta, p}(\mathbb{R}^{n})}.$

Remak 1.1 [12]   When $p\in[1, \infty), $ $\Lambda_{\mathfrak{L}}^{0, p}(\mathbb{R}^{n})=BMO_{\mathfrak{L}}(\mathbb{R}^{n}).$ And, when $0\leq\beta<\infty$ and $p_1, p_2\in[1, \infty)$, $\Lambda_{\mathfrak{L}}^{\beta, p_1}(\mathbb{R}^{n})=\Lambda_{\mathfrak{L}}^{\beta, p_2}(\mathbb{R}^{n})$ and $\|f\|_{\Lambda_{\mathfrak{L}}^{\beta, p_1}(\mathbb{R}^{n})}\sim\|f\|_{\Lambda_{\mathfrak{L}}^{\beta, p_2}(\mathbb{R}^{n})}.$ For simplicity, we denote $\Lambda_{\mathfrak{L}}^{\beta, p}(\mathbb{R}^{n})$ by $\Lambda_{\mathfrak{L}}^{\beta}(\mathbb{R}^{n}).$

Lemma 1.1 (see [2, 7])  Suppose that $V\in B_q (n/2\leq q<n)$ satisfies the condition

$\begin{array}{cl} \int_{B(x, R)}\frac{V(y)}{|x-y|^{n-1}}dy\leq\frac{1}{R^{n-1}}\int_{B(x, R)}V(y)dy, \end{array}$

(1.2)

then the kernel $K(x, y)$ of operator $\nabla(-\triangle+V)^{-1/2}$ satisfies the following estimates: there exists a constant $\delta>0$ such that for any nonnegative integrate $i, $

$|K(x, y)|\leq \dfrac{C_i}{\{|1+m(x, V)|x-y|\}^i}\dfrac{1}{|x-y|^{n}}, $

(1.3)

$|K(x+h, y)-K(x, y)|\leq C\dfrac{|h|^{\delta}}{|x-y|^{n+\delta}}, $

(1.4)

for $0<|h|<\frac{|x-y|}{2}.$

Hence, $\nabla(-\triangle+V)^{-1/2}$ is boundedness on $ L^{p}(\mathbb R^n)$ space for $1<p\leq p_{0}, $ where $1/p_{0}=1/q-1/n.$

Throughout this paper, the letter C always remains to denote a positive constant that may vary at each occurrence but is independent of the essential variable.

Theorems 2.1   Let $V\in B_q$ satisfy (1.2) for $n/2\leq q<n$. Let $0\leq\beta<1, $ $b\in\Lambda_{\mathfrak{L}}^{\beta}(\mathbb{R}^{n})$, $1<\tau<\infty$ and $1<s<p_{0}$, where $1/p_{0}=1/q-1/n.$ Suppose that $\Theta^{1}f=0$ for any $f\in L^{r}(\mathbb{R}^n)$ $(1<r<\infty), $ then there exists a constant $C>0$ such that

$M_{\mathfrak{L}}^{\#}(\Theta^{b}f)(x)\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x).$

Theorems 2.2   Let $V\in B_q$ satisfy(1.2) for $n/2\leq q<n$ and $1<p_0<\infty$ satisfy $1/p_{0}=1/q-1/n.$ Suppose that $\Theta^{1}f=0$ for any $f\in L^{\tau}(\mathbb{R}^n)$ $(1<\tau<\infty), $ $0<\beta<1, $ and $b\in\Lambda_{\mathfrak{L}}^{\beta}(\mathbb{R}^{n})$. Then for $1<r<\min\{1/\beta, p_0\}$ and $1/p=1/r-\beta, $ there exists a constant $C>0$, such that

$\|\Theta^{b}f\|_{L^{p}}\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\|f\|_{L^{r}}.$

Theorems 2.3   Let $V\in B_q$ satisfy (1.2) for $n/2\leq q<n$ and $1<p_0<\infty$ satisfy $1/p_{0}=1/q-1/n.$ Suppose that $\Theta^{1}f=0$ for any $f\in L^{\tau}(\mathbb{R}^n)$ $(1<\tau<\infty)$ and $b\in BMO_{\mathfrak{L}}(\mathbb{R}^{n})$, then for $1<p<p_0, $ there exists a constant $C>0$, such that

$\|\Theta^{b}f\|_{L^{p}}\leq C\sum\limits_{j=1}^{m}\|b\|_{BMO_{\mathfrak{L}}}\|f\|_{L^{p}}.$

To prove the theorems, we need the following lemmas.

Lemma 2.1 (see [7])  Let $0<p_0<\infty, $ $p_0\leq p<\infty$ and $\delta>0.$ If $f$ satisfies the condition $M(|f|^{\delta})^{1/\delta}\in L^{p_0}, $ then exists a constant $C>0$ such that

$\|M(|f|^{\delta})^{1/\delta}\|_{L^{p}}\leq C\|M_{\mathfrak{L}}^{\#}(|f|^{\delta})^{1/\delta}\|_{L^{p}}.$

Lemma 2.2 (see [14])   For $1\leq\gamma<\infty$ and $\beta>0, $ let

$M_{\gamma, \beta}(f)(x)=\sup_{B\ni x}\biggl(\dfrac{1} {|B|^{1-\beta\gamma/n}}\int_{B}|f(y)|^{\gamma}dy\biggr)^{1/\gamma}.$

Suppose that $\gamma<p<n/\beta$ and $1/q=1/p-\beta/n, $ then $\|M_{\gamma, \beta}(f)\|_{L^{q}}\leq C\|f\|_{L^{p}}.$

Remark 2.1  When $\beta=0$, we denote $M_{\gamma, \beta}=M_{r}.$ And, it is easy to see that $M_{r}$ is boundedness on $L^{p}(\mathbb{R}^{n}), $ for $1<r<p.$

Lemma 2.3 (see [8])   Let $B=B(x, r)$ and $0<r<\rho(x)$, then

$|b_{2^{k}B}-b_{B}|\leq C k|2^{k}B|^{\beta}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}, \;\mbox{for}\; k=1, 2, \cdots .$

Firstly, let us prove Theorem 2.1.

Fix a ball $B=B(x, r_0)$ and let $2B=B(x, 2r_0).$ We need only to estimate

$\dfrac{1}{|B|}\int_{B}|\Theta^{b}f(y)-(\Theta^{b}f)_{B}|dy.$

Case Ⅰ When $0<r_0 <\rho(x), $ using the condition $\Theta^{1}f=0$, then we have

$\begin{align} & \quad \frac{1}{|B|}\int_{B}{|{{\Theta }^{b}}f(y)-{{({{\Theta }^{b}}f)}_{B}}|dy} \\ & \le \sum\limits_{j=1}^{m}{\frac{1}{|B|}(\int_{B}{|{{T}_{j,1}}{{M}_{(b-{{b}_{B}})}}{{T}_{j,2}}f(y)-{{({{T}_{j,1}}{{M}_{(b-{{b}_{B}})}}{{T}_{j,2}}f)}_{B}}|dy})}. \\ \end{align}$

If $T_{j, 1}=\nabla(-\triangle+V)^{-1/2}, $ then

$\begin{array}{cl} & \dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})}T_{j, 2}f(y)-(T_{j, 1}M_{(b-b_{B})}T_{j, 2}f)_{B}|dy\\ & \leq\dfrac{2}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})}T_{j, 2}f(y)- T_{j, 1}M_{(b-b_{B})\chi_{R^{n}\backslash 2B}}T_{j, 2}f(x)|dy\\ & \leq\dfrac{2}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})\chi_{2B}}T_{j, 2}f(y)|dy+\dfrac{2}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})\chi_{R^{n}\backslash 2B}}T_{j, 2}f(y)-T_{j, 1}M_{(b-b_{B})\chi_{R^{n}\backslash 2B}}T_{j, 2}f(x)|dy\\ & =:I_{1}+I_{2}.\end{array}$

Let $\tau$ and $s$ be as in Theorem 2.1. Then using Hölder's inequality and the $L^{s}$ boundedness of $T_{j, 1}$(Lemma 1.1), we have

$\begin{array}{cl} I_{1} & \leq C\biggl(\dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})\chi_{2B}}T_{j, 2}f(y)|^{s}dy\biggr)^{\frac{1}{s}}\\ & \leq C\biggl(\dfrac{1}{|2B|^{1+\beta s\tau'}}\int_{2B}|b(y)-b_{B}|^{s\tau^{'}}dy\biggr)^{\frac{1}{s\tau^{'}}} \biggl(\dfrac{1}{|2B|^{1-\beta s\tau}}\int_{2B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{\frac{1}{s\tau}}\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

Let's estimate $I_{2}.$ From (1.4), it follows that

$\begin{align} & \quad |{{T}_{j,1}}[(b-{{b}_{B}}){{\chi }_{{{R}^{n}}\backslash 2B}}{{T}_{j,2}}f](y)-{{T}_{j,1}}[(b-{{b}_{B}}){{\chi }_{{{R}^{n}}\backslash 2B}}{{T}_{j,2}}f](x)| \\ & =\left| \int_{{{\left( 2B \right)}^{c}}}{(b(z)-{{b}_{B}}){{T}_{j,2}}f(z)(K(y,z)-K(x,z))dz} \right| \\ & \le C\sum\limits_{k=1}^{\infty }{\int_{2k_{{{r}_{0}}＜\left| z-x \right|\le {{2}^{k+{{1}_{{{r}_{0}}}}}}}^{k}}{\frac{|y-x{{|}^{\delta }}}{|z-x{{|}^{n+\delta }}}|b(z)-{{b}_{B}}||{{T}_{j,2}}f(z)|dz}} \\ & \le C\sum\limits_{k=1}^{\infty }{\frac{r_{0}^{\delta }}{{{({{2}^{k}}{{r}_{0}})}^{n+\delta }}}\int_{|z-x|\le {{2}^{k+1}}{{r}_{0}}}{|b(z)-{{b}_{{{2}^{k+1}}B}}||{{T}_{j,2}}f(z)|dz}} \\ & \quad +C\sum\limits_{k=1}^{\infty }{\frac{r_{0}^{\delta }}{{{({{2}^{k}}{{r}_{0}})}^{n+\delta }}}}\int_{|z-x|\le {{2}^{k+1}}{{r}_{0}}}{|{{b}_{{{2}^{k+1}}B}}-{{b}_{B}}||{{T}_{j,2}}f(z)|dz} \\ & \text{=:}\quad {{H}_{1}}+{{H}_{2}}. \\ \end{align}$

For $H_{1}, $ since $\delta >0, $ by Hölder's inequality, we have

$\begin{array}{cl} H_{1} & \leq C\sum\limits_{k=1}^{\infty}2^{-k\delta}\biggl(\dfrac{1}{|2^{k+1}B|^{1+\beta (s\tau)'}} \int_{2^{k+1}B}|b(y)-b_{2^{k+1}B}|^{(s\tau)'}dy\biggr)^{1/(s\tau)'}\\ & \quad\times \biggl(\dfrac{1}{|2^{k+1}B|^{1-\beta s\tau}}\int_{2^{k+1}B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{1/s\tau}\\&\leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

From Lemma 2.3 and Hölder's inequality, it follows that

$\begin{array}{cl} H_{2} & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\sum\limits_{k=1}^{\infty}k2^{-k\delta} \dfrac{1}{|2^{k+1}B|^{1-\beta}}\int_{2^{k+1}B}|T_{j, 2}f(y)|dy\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\sum\limits_{k=1}^{\infty}k2^{-k\delta} \biggl(\dfrac{1}{|2^{k+1}B|^{1-\beta s\tau}}\int_{2^{k+1}B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{\frac{1}{s\tau}}\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

Thus

$\begin{array}{cl} I_2 \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

So when $T_{j, 1}=\nabla(-\triangle+V)^{-1/2}, $ we conclude that

$\begin{array}{cl} & \dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})}T_{j, 2}f(y)-(T_{j, 1}M_{(b-b_{B})}T_{j, 2}f)_{B}|dy\leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

If $T_{j, 1}=\pm I, $ it is obvious that

$\dfrac{2}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})\chi_{R^{n}\backslash 2B}}T_{j, 2}f(y)|dy=0.$

Thus using the above formula and Hölder's inequality, we conclude

$\begin{array}{cl} & \dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})}T_{j, 2}f(y)-(T_{j, 1}M_{(b-b_{B})}T_{j, 2}f)_{B}|dy\\ & \leq\dfrac{2}{|B|}\int_{B}|T_{j, 1}M_{(b-b_{B})\chi_{2B}}T_{j, 2}f(y)|dy\\ & \leq C\biggl(\dfrac{1}{|B|^{1+\beta (s\tau)'}}\int_{B}|b(y)-b_{B}|^{(s\tau)^{'}}dy\biggr)^{\frac{1}{(s\tau)^{'}}} \biggl(\dfrac{1}{|B|^{1-\beta s\tau}}\int_{B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{\frac{1}{s\tau}}\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x). \end{array}$

Thus for $0<r_0 <\rho(x), $ we conclude that

$M_{\mathfrak{L}}^{\#}(\Theta^{b}f)(x)\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x), $

Case Ⅱ When $r_0 >\rho(x), $ we have

$\frac{1}{|B|}\int_{B}{|}{{\Theta }^{b}}f(y)|dy\le \sum\limits_{j=1}^{m}{(}\frac{1}{|B|}\int_{B}{|}{{T}_{j,1}}{{M}_{b{{\chi }_{2B}}}}{{T}_{j,2}}f(y)|dy+\frac{1}{|B|}\int_{B}{|}{{T}_{j,1}}{{M}_{b{{\chi }_{{{R}^{n}}\backslash 2B}}}}{{T}_{j,2}}f(y)|dy).\text{ }$

If $T_{j, 1}=\nabla(-\triangle+V)^{-1/2}, $ then for $1<\tau<\infty$ and $1<s<p_{0}$ are as in Theorem 2.1, we have

$\begin{array}{cl} & \dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{b\chi_{2B}}T_{j, 2}f(y)|dy\\ & \leq C\biggl(\dfrac{1}{|B|}\int_{B}|T_{j, 1}b\chi_{2B}T_{j, 2}f(y)|^{s}dy\biggr)^{\frac{1}{s}}\\ & \leq\biggl(\dfrac{1}{|B|}\int_{2B}|b(y)|^{s}|T_{j, 2}f(y)|^{s}dy\biggr)^{\frac{1}{s}}\\ & \leq C\biggl(\dfrac{1}{|2B|^{1+\beta s\tau'}}\int_{2B}|b(y)|^{s\tau^{'}}dy\biggr)^{\frac{1}{s\tau^{'}}} \biggl(\dfrac{1}{|2B|^{1-\beta s\tau}}\int_{2B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{\frac{1}{s\tau}}\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x).\end{array}$

Since, when $y\in 2B$ and $z\in 2^{k+1}B\backslash 2^kB, $ we have $|x-z|\sim|y-z|.$ Then, taking $s, \tau$ as above, we get

$\begin{array}{cl} & |T_{j, 1}M_{b\chi_{R^{n}\backslash 2B}}T_{j, 2}f(y)|\\ & \leq \int_{(2B)^{c}}|b(z)||T_{j, 2}f(z)||K(y, z)|dz\\ & \leq C\sum\limits_{k=1}^{\infty}\int_{2^k r_0<|z-x|\leq 2^{k+1}r_0} \dfrac{C_i}{\{1+|z-x|m(x, V)\}^{i}|z-x|^{n}}|b(z)||T_{j, 2}f(z)|dz\\ & \leq C\sum\limits_{k=1}^{\infty}\dfrac{1}{2^{ki}}\biggl(\dfrac{1}{|2^{k+1}B|^{1+\beta (s\tau)'}} \int_{2^{k+1}B}|b(y)|^{(s\tau)'}dy\biggr)^{1/(s\tau)'} \biggl(\dfrac{1}{|2^{k+1}B|^{1-\beta s\tau}}\int_{2^{k+1}B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{1/s\tau}\\ & \leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).\end{array}$

Thus

$\dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{b\chi_{R^{n}\backslash 2B}}T_{j, 2}f(y)|dy\leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n\beta}(T_{j, 2}f)(x).$

If $T_{j, 1}=\pm I, $ then by Hölder's inequality, we obtain

$\begin{array}{cl} & \dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{b\chi_{2B}}T_{j, 2}f(y)|dy\\ & \leq C\biggl(\dfrac{1}{|B|^{1+\beta(s\tau)'}}\int_{B}|b(y)|^{(s\tau)^{'}}dy\biggr)^{\frac{1}{(s\tau)^{'}}} \biggl(\dfrac{1}{|B|^{1-\beta s\tau}}\int_{B}|T_{j, 2}f(y)|^{s\tau}dy\biggr)^{\frac{1}{s\tau}}\leq C\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x).\end{array}$

And

$\dfrac{1}{|B|}\int_{B}|T_{j, 1}M_{b\chi_{R^{n}\backslash 2B}}T_{j, 2}(f)(y)|dy=0.$

Thus for $r_0 >\rho(x), $

$M_{\mathfrak{L}}^{\#}(\Theta^{b}f)(x)\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x) .$

So whenever $0<r_0 <\rho(x)$ or $r_0 >\rho(x), $ we have

$M_{\mathfrak{L}}^{\#}(\Theta^{b}f)(x)\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}M_{s\tau, n \beta}(T_{j, 2}f)(x).$

Now, let us turn to prove Theorem 2.2.

Let $s, \tau$ be as in Theorem 2.1 and satisfy $1<s\tau<p.$ Then applying Theorem 2.1, Lemma 2.1 and Lemma 2.2, we know that

$\begin{array}{cl} \|\Theta^{b}f\|_{L^{p}} & \leq C\|M_{\mathfrak{L}}^{\#}(\Theta^{b}f) \|_{L^{p}}\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\|M_{s\tau, n \beta}(T_{j, 2}f)\|_{L^{p}}\\ & \leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\|T_{j, 2}f\|_{L^{r}}\leq C\sum\limits_{j=1}^{m}\|b\|_{\Lambda_{\mathfrak{L}}^{\beta}}\|f\|_{L^{r}}.\end{array}$

Thus we complete the proof of Theorem 2.1-2.1.

It is obvious that $\Lambda_{\mathfrak{L}}^{0}=BMO_{\mathfrak{L}}.$ Thus from the proof of Theorem 2.1, we have

$M_{\mathfrak{L}}^{\#}(\Theta^{b}f)(x)\leq C\sum\limits_{j=1}^{m}\|b\|_{BMO_{\mathfrak{L}}}M_{s\tau}(T_{j, 2}f)(x).$

Since $M_{s\tau}$ is boundedness on $L^{p}(\mathbb{R}^{n}), $ then

$\begin{array}{cl} \|\Theta^{b}f\|_{L^{p}} & \leq C\|M_{\mathfrak{L}}^{\#}(\Theta^{b}f)\|_{L^{p}}\leq C\sum\limits_{j=1}^{m} \|b\|_{BMO_{\mathfrak{L}}}\|M_{s\tau}(T_{j, 2}f)\|_{L^{p}}\leq C\sum\limits_{j=1}^{m}\|b\|_{BMO_{\mathfrak{L}}}\|f\|_{L^{p}}.\end{array}$

Therefore, we complete the proof of Theorem 2.3.