The Herz-Boundedness of Commutators Related to Schrödinger Operators in the Setting of Heisenberg Group

Abstract

Let ${\mathcal {L}}=-\Delta _{{\mathbb {H}}^{n}}+V$ be the Schrödinger operator on the Heisenberg group ${\mathbb {H}}^{n}$, where $\Delta _{{\mathbb {H}}^{n}}$ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q}$ for $q\ge Q/2$. Suppose that $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, which is larger than $BMO({\mathbb {H}}^{n})$. We prove that the operator $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ is bounded from the Herz space ${\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})$ into ${\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})$. By a maximal estimate, we obtain the boundedness of the commutators $[b,T_{\beta _{1},\beta _{2}}]$ and $[b,T_{\beta }]$ from ${\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})$ into ${\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})$, where $T_{\beta }=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta }$.

1 Introduction

Let ${\mathbb {H}}^{n}$ be the Heisenberg group with the homogeneous dimension $Q=2n+2$. Let ${\mathcal {L}}=-\Delta _{{\mathbb {H}}^{n}}+V$ be the Schrödinger operator on ${\mathbb {H}}^{n}$, where $\Delta _{{\mathbb {H}}^{n}}$ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q}$, $q\ge Q/2$. The Riesz transforms and the fractional integrals associated with ${\mathcal {L}}$ have been studied extensively. Shen [15] proved the $L^{p}$-boundedness of the operators $(-\Delta +V)^{i\gamma }$, $\nabla ^{2}(-\Delta +V)^{-1}$, $\nabla (-\Delta +V)^{-1/2}$, and $\nabla (-\Delta +V)^{-1}\nabla$, where $\Delta$ denotes the Laplace operator on ${\mathbb {R}}^{n}$, i.e., $\Delta =\sum ^{n}_{i=1}\partial ^{2}/\partial x^{2}_{i}$. Yang-Yang-Zhou [19] obtained the endpoint properties of the Riesz transforms and the fractional integrals associated to ${\mathcal {L}}$. For further information, we refer the reader to Guo-Li-Peng [3], Liu [10], Liu-Huang-Dong [11], Tang-Dong [17] and the references therein. For $0\le \beta _{2}\le \beta _{1}<Q/2$, let

$$\begin{aligned} \left\{ \begin{array}{lcc} T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}},\\ T^{*}_{\beta _{1},\beta _{2}}=V^{\beta _{2}}(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}. \end{array} \right. \end{aligned}$$

For $0<\beta _{2}\le \beta _{1}<1$, Sugano [16] obtained the estimates for $T_{\beta _{1},\beta _{2}}$ and $T^{*}_{\beta _{1},\beta _{2}}$ on ${\mathbb {R}}^{n}$. If $\beta _{2}=0$, we know that $T_{\beta _{1},0}=T_{\beta _{1}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}$. In recent years, the fractional integral operator $I_{\alpha }=(-\Delta +V)^{-\alpha }$ has been studied extensively. We refer to Duong-Yan [2], Jiang [4], Tang-Dong [17] and Yang-Yang-Zhou [19] for details. For $\beta _{1}=\beta _{2}=1$ and $\beta _{1}=\beta _{2}=1/2$, $T^{*}_{1,1}=V(-\Delta +V)^{-1}$ and $T^{*}_{1/2,1/2}=V^{1/2}(-\Delta +V)^{-1/2}$ are studied by Shen [15]. We also refer the reader to [14] for the Fefferman-Phong type inequality for $\mathcal L$ on $\mathbb H^{n}$.

As a useful tool in analysis, the commutators related to singular integral operators play an important role in the study of harmonic analysis and partial differential equations. Let $R_{{\mathcal {L}}}=\nabla _{{\mathbb {H}}^{n}}(-\Delta _{{\mathbb {H}}^{n}}+V)^{-1/2}$ be the Riesz transform associated with ${\mathcal {L}}$, $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ and $b\in BMO({\mathbb {H}}^{n})$. Li-Peng [6] extended the results of [3] to the Heisenberg group ${\mathbb {H}}^{n}$. Li-Wan [7] studied the boundedness on generalized Morrey spaces of commutators $[b,R_{{\mathcal {L}}}]$, $[b,T_{\beta _{1},\beta _{2}}]$ and $[b,T^{*}_{\beta _{1},\beta _{2}}]$, see also Liu-Tang [12] and [13]. Bongioanni–Harboure–Salinas [1] introduced a new class of BMO-type spaces denoted by $BMO_{\rho }^{\theta }({\mathbb {R}}^{n})$ as generalizations of $BMO({\mathbb {R}}^{n})$. By use of a maximal estimate, the authors proved that the commutators $[b,R_{{\mathcal {L}}}]$ are bounded on $L^{p}({\mathbb {R}}^{n})$ with $b\in BMO^{\theta }_{\rho }({\mathbb {R}}^{n})$.

In this paper, we investigate the boundedness of commutators related to Schrodinger operators on Herz spaces in the setting of Heisenberg groups. In Sect. 3.1, suppose that $V \in B_{q}$, $q\ge Q/2$. Let $q_{0}=\sup \{q:V\in B_{q}\}$. Therefore, $q_{0}>Q/2$ under the assumption of $V\in B_{q}$. Given $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$. Let $1/p_{0}=(1/q_{0}-1/Q)^{+}$. We investigate the boundedness of the commutators $[b,R_{{\mathcal {L}}}]$ and $[b,R^{*}_{{\mathcal {L}}}]$ on $L^{p}({\mathbb {H}}^{n})$. Let $\beta _{1}=\beta _{2}=\beta$. Li-Wan [8] further extended the $L^{p}$-boundedness of $[b,R_{{\mathcal {L}}}]$ and $[b,T_{\beta ,\beta }]$ on Herz spaces ${\dot{K}}_{q}^{\alpha ,p}({\mathbb {R}}^{n})$ and $K_{q}^{\alpha ,p}({\mathbb {R}}^{n})$, respectively. In Sect. 4, we prove that if the index $(\beta _{1}, \beta _{2}, p_{1}, p_{2}, \alpha )$ satisfies

$$\begin{aligned} \left\{ \begin{array}{lll} 0\le \beta _{2}\le \beta _{1}<Q/2,\\ q/(q-\beta _{2})<p_{1}<Q/(2\beta _{1}-2\beta _{2}),\\ 1/p_{2}=1/p_{1}-(2\beta _{1}-2\beta _{2})/Q,\\ -Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{1}/Q), \end{array} \right. \end{aligned}$$

then $T_{\beta _{1},\beta _{2}}$ and $[b,T_{\beta _{1},\beta _{2}}]$ are bounded from homogeneous Herz spaces ${\dot{K}}^{\alpha ,p}_{p_{1}}({\mathbb {H}}^{n})$ to ${\dot{K}}^{\alpha ,p}_{p_{2}}({\mathbb {H}}^{n})$, respectively. The rest of this paper is organized as follows. In Sect. 2, we state some notations and known results which will be used throughout rest of this paper. In Sect. 3, we extend the results of [1] on Euclidean space to the Heisenberg group ${\mathbb {H}}^{n}$, see Theorem 1 and Theorem 2 for details. Moreover, we prove that the operator $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ and the commutator $[b,T_{\beta _{1},\beta _{2}}]$ are bounded from $L^{p_{1}}({\mathbb {H}}^{n})$ to $L^{p_{2}}({\mathbb {H}}^{n})$, where $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, $0\le \beta _{2}\le \beta _{1}<Q/2$. In Sect. 4, with the help of the $L^{p}$-boundedness of the operator $T_{\beta _{1},\beta _{2}}$ and the commutator $[b,T_{\beta _{1},\beta _{2}}]$, we prove that the commutators $[b,T_{\beta _{1}\beta _{2}}]$ and $[b,T_{\beta }]$ are bounded from Herz spaces ${\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})$ to ${\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})$ with $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, respectively.

In what follows we recall some basic facts for the Heisenberg group ${\mathbb {H}}^{n}$. The Heisenberg group ${\mathbb {H}}^{n}$ is a Lie group with the underlying manifold ${\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\times {\mathbb {R}}$, and the multiplication

$$\begin{aligned} (x,y,t)(x',y',t')=(x+x',y+y',t+t'+2x'y-2xy'). \end{aligned}$$

A basis for the Lie algebra of left-invariant vector fields on ${\mathbb {H}}^{n}$ is given by

$$\begin{aligned} \begin{aligned} \mathrm {X}_{j}=\frac{\partial }{\partial x_{j}}+2y_{j}\frac{\partial }{\partial t},\mathrm {Y}_{j}=\frac{\partial }{\partial y_{j}}+2x_{j}\frac{\partial }{\partial t}, \mathrm {T}=\frac{\partial }{\partial t}, j=1,2,...,n. \end{aligned} \end{aligned}$$

All non-trivial commutation relations are given by $[\mathrm {X}_{j},\mathrm {Y}_{j}]=-4\mathrm {T}$, $j=1,2,...,n$. Then the sub-Laplacian $\Delta _{{\mathbb {H}}^{n}}$ is defined by $\Delta _{{\mathbb {H}}^{n}}=\sum _{j=1}^{n}(\mathrm {X}_{j}^{2}+\mathrm {Y}_{j}^{2})$ and the gradient operator $\nabla _{{\mathbb {H}}^{n}}$ is defined by

$$\begin{aligned} \nabla _{{\mathbb {H}}^{n}}=(\mathrm {X}_{1},\mathrm {X}_{2},...,\mathrm {X}_{n},\mathrm {Y}_{1},\mathrm {Y}_{2},...,\mathrm {Y}_{n}). \end{aligned}$$

The dilation on ${\mathbb {H}}^{n}$ is of the form $\delta _{\lambda }(x,y,t)=(\lambda x,\lambda y,\lambda ^{2} t)$, $\lambda >0$. The Haar measure on ${\mathbb {H}}^{n}$ coincides with the Lebesgue measure on ${\mathbb {R}}^{n}\times {\mathbb {R}}^{n} \times {\mathbb {R}}$. We denote the measure of any measurable set E by |E|. Then $|\delta _{\lambda }E|=\lambda ^{Q}|E|$, where $Q=2n+2$ is called the homogeneous dimension of ${\mathbb {H}}^{n}$.

We define a homogeneous norm function on ${\mathbb {H}}^{n}$ by $|g|=\Big ((|x|^{2}+|y|^{2})+|t|^{2}\Big )^{1/4}$, $g=(x,y,t)\in {\mathbb {H}}^{n}$. This norm satisfies the triangular inequality and leads to a left-invariant distant function $d(g,h)=|g^{-1}h|$. Then the ball of radius r centered at g is given by $B(g,r)=\{h\in {\mathbb {H}}^{n}:|g^{-1}h|<r\}$. The ball B(g, r) is the left translation by g of B(0, r) and we have $|B(g,r)|=\alpha ^{1}r^{Q}$, where $\alpha ^{1}=|B(0,1)|$.

Throughout this paper, the symbol $A\lesssim B$ means $A\le CB$, where C is a universal positive constant and independent of all important parameters. If $A\lesssim B$ and $B\lesssim A$, we denote $A\thickapprox B$.

2 Preliminaries

2.1 Schrödinger Operators and the Auxiliary Function

Definition 1

A nonnegative locally $L^{q}$ integrable function V on ${\mathbb {H}}^{n}$ is said to belong to $B_{q}$, $1<q<\infty$, if there exists a constant $C>0$ such that the reverse Hölder inequality

$$\begin{aligned} \Big (\frac{1}{|B|}\int _{B} V(g)^{q}dg\Big )^{1/q}\le \frac{C}{|B|}\int _{B}V(g)dg \end{aligned}$$

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

In the analysis of Schrödinger operators, the following auxiliary function plays an important role.

Definition 2

For $g\in {\mathbb {H}}^{n}$, the auxiliary function $\rho (\cdot )$ is defined by

$$\begin{aligned} \rho (g)=\sup \limits _{r>0}\Big \{r:\frac{1}{r^{Q-2}}\int _{B(g,r)}V(h)dh\le 1\Big \}. \end{aligned}$$

Now, we give some related lemmas about $\rho (\cdot )$. Lemmas 1–4 have been proved in [10].

Lemma 1

Suppose that $V\in B_{q}$ with $q>Q/2$. The measure V(h)dh is a doubling measure, that is, there exist constants $\lambda \ge 1$ and C such that

$$\begin{aligned} \int _{tB}V(h)dh\le Ct^{Q\lambda }\int _{B}V(h)dh \end{aligned}$$

holds for every ball B(g, r) in ${\mathbb {H}}^{n}$ and $t>1$.

Lemma 2

Suppose that $V\in B_{q}$ with $q>Q/2$. There exists a constant $C>0$ such that for $0<r<R<\infty$,

$$\begin{aligned} \begin{aligned} \frac{1}{r^{Q-2}}\int _{B(g,r)}V(h)dh\le C\Big (\frac{R}{r}\Big )^{Q/q-2}\frac{1}{R^{Q-2}}\int _{B(g,R)}V(h)dh. \end{aligned} \end{aligned}$$

Lemma 3

There exist $C>0$ and $N_{0}>0$ such that

$$\begin{aligned} \begin{aligned} \frac{1}{(1+r/\rho (g))^{N_{0}}}\int _{B(g,r)}V(h)dh\le Cr^{Q-2} . \end{aligned} \end{aligned}$$

Lemma 4

Suppose that $V\in B_{q}$ with $q>Q/2$. There exist constants $C>0$ and $k_{0}>0$ such that

$$\begin{aligned} 1/C(1+|g^{-1}h|/\rho (g))^{-k_{0}}\le \rho (h)/\rho (g)\le C(1+|g^{-1}h|/\rho (g))^{k_{0}/(k_{0}+1)} \end{aligned}$$

(1)

holds for all $g,h\in {\mathbb {H}}^{n}$.

A ball centered at g and with radius $\rho (g)$ is called a critical ball and is denoted by ${\mathcal {Q}}=B(g,\rho (g))$. (1) implies that if $g,h\in {\mathcal {Q}}=B(g,\rho (g))$, then $\rho (g)\le C_{0}\rho (h)$, where the constant $C_{0}$ depends on the constants C and $k_{0}$ in (1).

Lemma 5

Suppose that $V\in B_{q}$ with $q>Q/2$. Let $l_{0}\in {\mathbb {N}}$ and $g\in 2^{k+1}B(g_{0},r)\setminus 2^{k}B(g_{0},r)$. Then

$$\begin{aligned} \begin{aligned} \frac{1}{\left( 1+2^{k}r/\rho (g)\right) ^{N}}\lesssim \frac{1}{\left( 1+2^{k}r/\rho (g_{0})\right) ^{N/(l_{0}+1)}}. \end{aligned} \end{aligned}$$

Proof

The proof is similar to that of [8, Lemma 2.17]. $\square$

Proposition 1

There exists a sequence of points $\left\{ g_{k}\right\} _{k=1}^{\infty }\subset {\mathbb {H}}^{n}$ such that the family of critical balls $\{{\mathcal {Q}}_{k}=B(g_{k},\rho (g_{k}))\}_{k=1}^{\infty }$ satisfies:

• (i) ${\mathbb {H}}^{n}=\cup _{k}{\mathcal {Q}}_{k}$.

• (ii) There exists N such that for every $k\in {\mathbb {N}}$, card$\{j:4{\mathcal {Q}}_{j}\cap 4{\mathcal {Q}}_{k}\ne \varnothing \}\le N$.

Proof

We refer to [9, page 23 and page 24 ] $\square$

2.2 New BMO-Type Spaces $BMO_{\mathcal {\rho }}^{\theta }({\mathbb {H}}^{n})$

As in [1], we define the following new BMO-type space $BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$.

Definition 3

The new BMO-type space $BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$ with $0<\theta <\infty$ is defined as the set of all locally integrable functions b on ${\mathbb {H}}^{n}$ such that

$$\begin{aligned} \frac{1}{|B(g,r)|}\int _{B(g,r)}|b(h)-b_{B}|dh\le C(1+\frac{r}{\rho (g)})^{\theta } \end{aligned}$$

(2)

holds for all $g\in {\mathbb {H}}^{n}$ and $r>0$, where $b_{B}=\frac{1}{|B|}\int _{B}b(h)dh$. The norm for $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, denoted by $[b]_{\theta }$, is given by the infimum of the constants satisfying (2). We define $BMO_{\rho }^{\infty }({\mathbb {H}}^{n})=\cup _{\theta >0}BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$. Clearly, $BMO({\mathbb {H}}^{n})\subset BMO_{\rho }^{\theta }({\mathbb {H}}^{n})\subset BMO_{\rho }^{\theta '}({\mathbb {H}}^{n})$ for $0<\theta <\theta '$, and hence $BMO({\mathbb {H}}^{n})\subset BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$.

The space $BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$ is a generalization of the classical BMO-type space associated with Schrödinger operators which is defined as follows.

Definition 4

The space $BMO_{{\mathcal {L}}}({\mathbb {H}}^{n})$ is defined as the set of all locally $L^{1}$ integrable functions f on ${\mathbb {H}}^{n}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} \int _{B}|f(h)-f_{B}|dh\le C|B|, r<\rho (g),\\ \int _{B}|f(h)|dh\le C|B|, r\ge \rho (g), \end{array} \right. \end{aligned}$$

where C is a constant and $B=B(g,r)$.

Definition 5

The space $BMO_{\rho }^{\theta ,log}({\mathbb {H}}^{n})$ is defined as the set of all functions b such that

$$\begin{aligned} \frac{1}{|B(g,r)|}\int _{B(g,r)}|b(h)-b_{B}|dh\le C\frac{(1+r/\rho (g))^{\theta }}{1+log^{+}(\rho (g)/r)} \end{aligned}$$

holds for all $g\in {\mathbb {H}}^{n}$ and $r>0$. We define $BMO_{\rho }^{\infty ,log}({\mathbb {H}}^{n})=\cup _{\theta >0}BMO_{\rho }^{\theta ,log}({\mathbb {H}}^{n})$.

We state the following lemma concerning the function $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$ which will be used in the proofs of the main results.

Lemma 6

Let $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, $B=B(g,r)$, $\theta >0$. If $1\le s<\infty$, $\theta '=(k_{0}+1)\theta$, and $k_{0}$ is the constant appearing in (1). Then

• (i) for all $g\in {\mathbb {H}}^{n}$ and $r>0$,

  $$\begin{aligned} \left( \frac{1}{|B|}\int _{B}|b(h)-b_{B}|^{s}dh\right) ^{1/s}\lesssim [b]_{\theta }\Big (1+\frac{r}{\rho (g)}\Big )^{\theta '}. \end{aligned}$$

• (ii) For all $g\in {\mathbb {H}}^{n}$ and $k\in {\mathbb {N}}$ with $r>0$,

  $$\begin{aligned} \Big (\frac{1}{|2^{k}B|}\int _{2^{k}B}|b(h)-b_{B}|^{s}dh\Big )^{1/s}\lesssim k[b]_{\theta }\Big (1+\frac{2^{k}r}{\rho (g)}\Big )^{\theta '}. \end{aligned}$$

Proof

The proof of (i) is similar to that of [1, Proposition 1] and the proof of (ii) is similar to that of [1, Lemma 1]. $\square$

For $k\in {\mathbb {Z}}$, let $B_{k}=\{g\in {\mathbb {H}}^{n}:|g|\le 2^{k}\}$ and $E_{k}=B_{k}\setminus B_{k-1}$. Denote by $\chi _{k}$ the characteristic function of $E_{k}$. The Herz spaces are defined as follows.

Definition 6

Let $\alpha \in {\mathbb {H}}^{n}$, $0<p,q\le \infty$. The homogeneous Herz space ${\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})$ is defined by

$$\begin{aligned} {\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})=\Big \{f\in L_{loc}^{q}({\mathbb {H}}^{n}\setminus 0):\ \Vert f\Vert _{{\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})}<\infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{{\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})}=\Big \{\sum _{k\in {\mathbb {Z}}}2^{k\alpha p}\Vert f\chi _{k}\Vert _{L^{q}({\mathbb {H}}^{n})}^{p}\Big \}^{1/p}. \end{aligned}$$

2.3 Estimates of the Kernels

We recall the estimates of kernels. Let $\Gamma (g,h,\tau )$ denote the fundamental solution for the operator $-\Delta _{{\mathbb {H}}^{n}}+i\tau$, where $\tau \in {\mathbb {R}}$. For any $N>0$, there exists $C_{N}>0$ such that

$$\begin{aligned} |\Gamma (g,h,\tau )|\le \frac{C_{N}}{(1+|h^{-1}g||\tau |^{1/2})^{N}}\frac{1}{|h^{-1}g|^{Q-2}}, \end{aligned}$$

$$\begin{aligned} |\nabla _{{\mathbb {H}}^{n},g}\Gamma (g,h,\tau )|\le \frac{C_{N}}{(1+|h^{-1}g||\tau |^{1/2})^{N}}\frac{1}{|h^{-1}g|^{Q-1}}, \end{aligned}$$

(3)

where $\nabla _{{\mathbb {H}}^{n},g}$ denote the gradient for variable g. The estimate (3) still holds for $\nabla _{{\mathbb {H}}^{n},h}$. We remark that the explicit expression of $|\Gamma (g,h)|=|\Gamma (g,h,0)|$ is obtained by Folland: $|\Gamma (g,h)|=\frac{2^{n-2}\Gamma (n/2)^{2}}{\pi ^{n+1}}\frac{1}{|h^{-1}g|^{Q-2}}$.

Let $\Gamma ^{{\mathcal {L}}}(g,h,\tau )$ denote the fundamental solution for the operator $-\Delta _{{\mathbb {H}}^{n}}+V+i\tau$, where $\tau \in {\mathbb {R}}$. For any $N>0$, there exists $C_{N}>0$ such that

$$\begin{aligned} |\Gamma ^{{\mathcal {L}}}(g,h,\lambda )|\le \frac{C_{N}}{(1+|h^{-1}g||\lambda |^{1/2})^{N}(1+|h^{-1}g|/\rho (g))^{N}}\cdot \frac{1}{|h^{-1}g|^{Q-2}}. \end{aligned}$$

Let $R=\nabla _{{\mathbb {H}}^{n}}(-\Delta _{{\mathbb {H}}^{n}})^{-1/2}$ be the classical Riesz transforms and $R^{*}$ be its adjoint operator. The kernels of R and $R^{*}$ are denoted by $R(\cdot ,\cdot )$ and $R^{*}(\cdot ,\cdot )$, respectively. Let $R_{{\mathcal {L}}}=\nabla _{{\mathbb {H}}^{n}}(-\Delta _{{\mathbb {H}}^{n}}+V)^{-1/2}$ be the vector-valued Riesz transform related to ${\mathcal {L}}$ and $R^{*}_{{\mathcal {L}}}$ be its adjoint operator. The kernels of $R_{{\mathcal {L}}}$ and $R^{*}_{{\mathcal {L}}}$ are denoted by $R_{{\mathcal {L}}}(\cdot ,\cdot )$ and $R^{*}_{{\mathcal {L}}}(\cdot ,\cdot )$, respectively. We have

$$\begin{aligned} \left\{ \begin{array}{lcc} R(g,h)=\frac{1}{2\pi }\int _{{\mathbb {R}}^{n}}(-i\tau )^{-1/2}\nabla _{{\mathbb {H}}^{n},g}\Gamma (g,h,\tau )d\tau ,\\ R^{*}(g,h)=\frac{1}{2\pi }\int _{{\mathbb {R}}^{n}}(-i\tau )^{-1/2}\nabla _{{\mathbb {H}}^{n},h}\Gamma (h,g,\tau )d\tau ,\\ R_{{\mathcal {L}}}(g,h)=\frac{1}{2\pi }\int _{{\mathbb {R}}^{n}}(-i\tau )^{-1/2}\nabla _{{\mathbb {H}}^{n},g}\Gamma ^{{\mathcal {L}}}(g,h,\tau )d\tau ,\\ R^{*}_{{\mathcal {L}}}(g,h)=\frac{1}{2\pi }\int _{{\mathbb {R}}^{n}}(-i\tau )^{-1/2}\nabla _{{\mathbb {H}}^{n},h}\Gamma ^{{\mathcal {L}}}(h,g,\tau )d\tau . \end{array} \right. \end{aligned}$$

Definition 7

The operator $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ is defined by $T_{\beta _{1},\beta _{2}}f(g)=\int _{{\mathbb {H}}^{n}}K_{\beta _{1},\beta _{2}}(g,h)f(h)dh$, where $K_{\beta _{1},\beta _{2}}(g,h)=\Gamma (g,h)V^{\beta _{2}}(h)$ and $\Gamma (g,h)=\Gamma (g,h,0)$.

Definition 8

The operator $R_{{\mathcal {L}}}=\nabla _{{\mathbb {H}}^{n}}(-\Delta _{{\mathbb {H}}^{n}}+V)^{-1/2}$ is defined by $R_{{\mathcal {L}}}f(g)=\int _{{\mathbb {H}}^{n}}R_{{\mathcal {L}}}(g,h)f(h)dh$.

Next, we give some lemmas about the estimates for the kernels of $T_{\beta _{1},\beta _{2}}$, $R_{{\mathcal {L}}}$ and $R_{{\mathcal {L}}}^{*}$.

Lemma 7

( [18, Lemma 3.2]) Suppose $V\in B_{q}, q>Q/2$. For any integer $N>0$, there exists $C_{N}>0$ such that

$$\begin{aligned} |K_{\beta _{1},\beta _{2}}(g,h)|\le \frac{C_{N}}{\left\{ 1+|g^{-1}h|/\rho (g)\right\} ^{N}}\cdot \frac{V^{\beta _{2}}(h)}{|g^{-1}h|^{Q-2\beta _{1}}}. \end{aligned}$$

Lemma 8

If $V\in B_{Q}$, then

• (i) for every N, there exists a constant C such that

  $$\begin{aligned} |R_{{\mathcal {L}}}(g,h)|\le \frac{C}{(1+|gz^{-1}|/\rho (g))^{N}}\frac{1}{|gz^{-1}|^{Q}}. \end{aligned}$$

• (ii) For every N and $0<\delta <\min \left\{ 1,1-Q/q_{0}\right\}$, there exists a constant C such that for $|gh^{-1}|<\frac{2}{3}|gz^{-1}|$,

  $$\begin{aligned} |R_{{\mathcal {L}}}(g,z)-R_{{\mathcal {L}}}(h,z)|\le \frac{C}{(1+|gz^{-1}|/\rho (g))^{N}}\frac{|gh^{-1}|^{\delta }}{|gz^{-1}|^{Q+\delta }}. \end{aligned}$$

• (iii) For every $0<\sigma <2-Q/q_{0}$, then

  $$\begin{aligned} |R(g,z)-R_{{\mathcal {L}}}(g,z)|\le \frac{C}{|gz^{-1}|^{Q}}\Big (\frac{|gz^{-1}|}{\rho (z)}\Big )^{\sigma }. \end{aligned}$$

Proof

The proof is similar to that of [15, Theorem 0.4] $\square$

Lemma 9

( [6, Lemma 5]) If $V\in B_{Q/2}$, then

• (i) for every N, there exists a constant C such that

  $$\begin{aligned}&|R_{{\mathcal {L}}}^{*}(g,z)|\le \frac{C(1+|gz^{-1}|/\rho (g))^{-N}}{|gz^{-1}|^{Q-1}}\nonumber \\&\quad \Big (\int _{B(z,|gz^{-1}|/4)}\frac{V(u)}{|uz^{-1}|^{Q-1}}du+\frac{1}{|gz^{-1}|}\Big ). \end{aligned}$$

  (4)

• (ii) For every N and $0<\delta <\min \left\{ 1, 2-Q/q_{0}\right\}$, there exists a constant C such that for $|gh^{-1}|<\frac{2}{3} |gz^{-1}|$,

  $$\begin{aligned} |R_{{\mathcal {L}}}^{*}(g,z)-R_{{\mathcal {L}}}^{*}(h,z)|\le & {} \frac{C(1+|gz^{-1}|/\rho (g))^{-N}|gh^{-1}|^{\delta }}{|gz^{-1}|^{Q-1+\delta }}\nonumber \\&\times \Big (\int _{B(z,|gz^{-1}|/4)}\frac{V(u)}{|uz^{-1}|^{Q-1}}du+\frac{1}{|gz^{-1}|}\Big ). \end{aligned}$$

  (5)

Lemma 10

( [5, Lemma 6.3]) Let $V\in B_{Q/2}$. For every $0<\sigma <2-Q/q_{0}$,

$$\begin{aligned} |R_{{\mathcal {L}}}^{*}(g,z)-R^{*}(g,z)|\le & {} \frac{C}{|gz^{-1}|^{Q-1}}\nonumber \\&\times q\Big (\int _{B(z,|gz^{-1}|/4)}\frac{V(u)}{|uz^{-1}|^{Q-1}}du+\frac{1}{|gz^{-1}|}\Big (\frac{|gz^{-1}|}{\rho (g)}\Big )^{\sigma }\Big ).\nonumber \\ \end{aligned}$$

(6)

Lemma 11

If $V\in B_{q_{0}}$, when $q_{0}>Q$, then $R_{{\mathcal {L}}}$ and $R_{{\mathcal {L}}}^{*}$ are Calderón-Zygmund operators. The term involving V can be dropped from inequalities (4), (5), and (6).

Proof

We refer to [9, page 2]. $\square$

3 Boundedness of the Commutators on $L^{p}({\mathbb {H}}^{n})$

In this section, we consider the boundedness of commutators $[b,R_{{\mathcal {L}}}]$ and $[b,R^{*}_{{\mathcal {L}}}]$ on $L^{p}({\mathbb {H}}^{n})$, where $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$. Furthermore, let $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$. Suppose $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$. We prove the commutator $[b,T_{\beta _{1},\beta _{2}}]$ is bounded from $L^{p_{1}}({\mathbb {H}}^{n})$ to $L^{p_{2}}({\mathbb {H}}^{n})$.

3.1 Technical Lemmas

We give some technical lemmas which will be used in the sequel.

Definition 9

Let $f\in L_{loc}^{q}({\mathbb {H}}^{n})$. Denote by |B| the Lebesgue measure of the ball $B\subset {\mathbb {H}}^{n}$. The Hardy–Littlewood maximal function Mf and its variant $M_{\sigma ,s}f$ are defined by

$$\begin{aligned} \left\{ \begin{array}{lcc} Mf(g)=\sup \limits _{g\in B}\frac{1}{|B|}\int _{B}|f(h)|dh,\\ M_{\sigma ,s}f(g)=\sup \limits _{g\in B}\Big (\frac{1}{|B|^{1-\sigma s/Q}}\int _{B}|f(h)|^{s}dh\Big )^{1/s}. \end{array} \right. \end{aligned}$$

If $\sigma =0$, then $M_{0,s}f(g)$ is denoted by $M_{s}f(g)$.

Definition 10

Let $\gamma >0$ and ${\mathcal {B}}_{\rho ,\gamma }=\{B(h,r):h\in {\mathbb {H}}^{n},r\le \gamma \rho (h)\}$. For $f\in L_{loc}^{1}({\mathbb {H}}^{n})$ and $g\in {\mathbb {H}}^{n}$, we define the following two maximal functions:

$$\begin{aligned} \left\{ \begin{array}{lcc} M_{\rho ,\gamma }(f)(g)=\sup \limits _{g\in B\in {\mathcal {B}}_{\rho ,\gamma }}\frac{1}{|B|}\int _{B}|f(h)|dh,\\ M_{\rho ,\gamma }^{\sharp }(f)(g)=\sup \limits _{g\in B\in {\mathcal {B}}_{\rho ,\gamma }}\frac{1}{|B|}\int _{B}|f(h)-f_{B}|dh. \end{array} \right. \end{aligned}$$

Definition 11

Let ${\mathcal {S}}_{\mathrm {Q}}=\{B(h,r):h\in \mathrm {Q}, r>0\}$ and let $\mathrm {Q}$ be a ball in ${\mathbb {H}}^{n}$. For $f\in L_{loc}^{1}({\mathbb {H}}^{n})$ and $h\in \mathrm {Q}$, we define

$$\begin{aligned} \left\{ \begin{array}{lcc} M_{\mathrm {Q}}(f)(g)=\sup \limits _{g\in B\in {\mathcal {S}}_{\mathrm {Q}}}\frac{1}{|B\cap \mathrm {Q}|}\int _{B\cap \mathrm {Q}}|f(h)|dh,\\ M_{\mathrm {Q}}^{\sharp }(f)(g)=\sup \limits _{g\in B\in {\mathcal {S}}_{\mathrm {Q}}}\frac{1}{|B\cap \mathrm {Q}|}\int _{B\cap \mathrm {Q}}|f(h)-f_{B\cap \mathrm {Q}}|dh. \end{array} \right. \end{aligned}$$

Lemma 12

(Fefferman–Stein-type inequality) For $1<p<\infty$, there exist $\xi$ and $\gamma$ such that if $\{{\mathcal {Q}}_{k}\}_{k=1}^{\infty }$ is a sequence of the balls as those in Proposition 1, then for all $f\in L_{loc}^{1}({\mathbb {H}}^{n})$,

$$\begin{aligned} \int _{{\mathbb {H}}^{n}}|M_{\rho ,\xi }(f)(g)|^{p}dg\lesssim \int _{{\mathbb {H}}^{n}}|M_{\rho ,\gamma }^{\sharp }(f)(g)|^{p}dg +\sum \limits _{k}|{\mathcal {Q}}_{k}|\Big (\frac{1}{|{\mathcal {Q}}_{k}|}\int _{2{\mathcal {Q}}_{k}}|f(g)|dg\Big )^{p}. \end{aligned}$$

Proof

The proof is similar to that of [1, Lemma 2]. $\square$

Lemma 13

Let $V\in B_{Q/2}$, $1/p_{0}=(1/q_{0}-1/Q)^{+}$, and $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$. Then, for any $s>p_{0}'$, there exists a constant C such that

$$\begin{aligned} \frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|[b,R_{{\mathcal {L}}}^{*}]f(g)|dg\le C[b]_{\theta }\inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h) \end{aligned}$$

holds for all $f\in L_{loc}^{s}({\mathbb {H}}^{n})$ and every ball ${\mathcal {Q}}=B(g_{0},\rho (g_{0}))\subset {\mathbb {H}}^{n}$. Moreover, if $q_{0}>Q$, the above estimate also holds for $R_{{\mathcal {L}}}$ instead of $R_{{\mathcal {L}}}^{*}$.

Proof

Let $f\in L^{p}({\mathbb {H}}^{n})$ and ${\mathcal {Q}}=B(g_{0},\rho (g_{0}))$. We write $[b,R_{{\mathcal {L}}}^{*}]f(g):=I-II$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} I:=(b(g)-b_{{\mathcal {Q}}})R_{{\mathcal {L}}}^{*}f(g),\\ II:=R_{{\mathcal {L}}}^{*}((b(g)-b_{{\mathcal {Q}}})f)(g). \end{array} \right. \end{aligned}$$

Next we deal with the average on ${\mathcal {Q}}$ of each term.

For I, it follows from Hölder’s inequality with $s>p_{0}'$ and Lemma 6 that

$$\begin{aligned}&\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|(b(g)-b_{{\mathcal {Q}}})R_{{\mathcal {L}}}^{*}f(g)|dg\\&\quad \le \Big (\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|b(g)-b_{{\mathcal {Q}}}|^{s'}dg\Big )^{1/s'} \Big (\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|R_{{\mathcal {L}}}^{*}f(g)|^{s}dg\Big )^{1/s}\\&\quad \lesssim [b]_{\theta }\Big (\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|R_{{\mathcal {L}}}^{*}f(g)|^{s}dg\Big )^{1/s}. \end{aligned}$$

We set $f(g)=f_{1}(g)+f_{2}(g)$, where $f_{1}(g)=f\chi _{2{\mathcal {Q}}}(g)$ and $f_{2}(g)=f\chi _{(2{\mathcal {Q}})^{c}}(g)$. Then, by use of the $L^{s}({\mathbb {H}}^{n})$-boundedness of $R^{*}_{{\mathcal {L}}}$, we obtain

$$\begin{aligned} \Big (\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|R_{{\mathcal {L}}}^{*}f_{1}(g)|^{s}dg\Big )^{1/s}\lesssim \Big (\frac{1}{|{\mathcal {Q}}|}\int _{2{\mathcal {Q}}}|f(g)|^{s}dg\Big )^{1/s}\lesssim \inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h). \end{aligned}$$

Now, for $g\in {\mathcal {Q}}$, by (4), we have $|R_{{\mathcal {L}}}^{*}f_{2}(g)|\lesssim I_{1}+I_{2}$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} I_{1}:=\int _{|g_{0}z^{-1}|>2\rho (g_{0})}\frac{1}{(1+|gz^{-1}|/\rho (g))^{N}}\frac{|f(z)|}{|gz^{-1}|^{Q}}dz,\\ I_{2}:=\int _{|g_{0}z^{-1}|>2\rho (g_{0})}\frac{1}{(1+|gz^{-1}|/\rho (g))^{N}}\frac{|f(z)|}{|gz^{-1}|^{Q-1}}\int _{B(z,|gz^{-1}|/4)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz. \end{array} \right. \end{aligned}$$

Via a simple computation, for $z\in 2^{k+1}{\mathcal {Q}} \setminus 2^{k}{\mathcal {Q}}$ and $g\in {\mathcal {Q}}$, we can deduce that $|gz^{-1}|\thickapprox |g_{0}z^{-1}|\thickapprox 2^{k}\rho (g_{0})$. With the help of the condition $\rho (g)\thickapprox \rho (g_{0})$, we obtain

$$\begin{aligned} I_{1} \lesssim \sum \limits _{k\ge 1}\frac{1}{(1+2^{k}\rho (g_{0})/\rho (g_{0}))^{N}}\frac{1}{(2^{k}\rho (g_{0}))^{Q}}\int _{|g_{0}z^{-1}|<2^{k}\rho (g_{0})}|f(z)|dz \lesssim \inf \limits _{h\in {\mathcal {Q}}}Mf(h). \end{aligned}$$

For $I_{2}$, we assume $Q/2<q_{0}<Q$. Let $p_{0}'<s<Q$. By Hölder’s inequality and the boundedness of the fractional integral ${\mathcal {I}}_{1}:L^{s'}\mapsto L^{q}$ with $1/s'=1/q-1/Q$, we can get

$$\begin{aligned} I_{2}&\lesssim \sum \limits _{k\ge 1}\frac{1}{(1+2^{k})^{N}}\frac{1}{(2^{k}\rho (g_{0}))^{Q-1}}\int _{|g_{0}z^{-1}|<2^{k+1}\rho (g_{0})}|f(z)| \int _{B(g_{0},2^{k+3}\rho (g_{0}))}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz\\&\lesssim \sum \limits _{k\ge 1}\frac{2^{-Nk}}{(2^{k}\rho (g_{0}))^{Q-1}} \int _{|g_{0}z^{-1}|<2^{k}\rho (g_{0})}|f(z)|{\mathcal {I}}_{1}(V\chi _{B(g_{0},2^{k}\rho (g_{0}))})dz\\&\lesssim \sum \limits _{k\ge 1}\frac{2^{-Nk}}{(2^{k}\rho (g_{0}))^{Q-1-Q(1/s+1/q)}}\inf \limits _{h\in Q}M_{s}f(h) \frac{1}{|B(g_{0},2^{k}\rho (g_{0}))|}\int _{B(g_{0},2^{k}\rho (g_{0}))}|V(z)|dz\\&\lesssim \inf \limits _{h\in Q}M_{s}f(h)\sum \limits _{k\ge 1}\frac{2^{-Nk}}{(2^{k}\rho (g_{0}))^{Q-1}}2^{k(Q/s+Q\lambda -Q/q')}\rho (g_{0})^{Q/s+Q-Q/q'-2} \lesssim \inf \limits _{h\in Q}M_{s}f(h), \end{aligned}$$

where in the last inequality we have taken N large enough.

For II, we split $f(g)=f_{1}(g)+f_{2}(g)$, where $f_{1}(g)=f\chi _{2{\mathcal {Q}}}(g)$ and $f_{2}(g)=f\chi _{(2{\mathcal {Q}})^{c}}(g)$. We select $p_{0}'<{\widetilde{s}}<s$ and denote $m=\frac{{\widetilde{s}}s}{s-{\widetilde{s}}}$. With the help of Hölder’s inequality and Lemma 6, we can deduce that

$$\begin{aligned}&\frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|R_{{\mathcal {L}}}^{*}f_{1}(b-b_{{\mathcal {Q}}})(g)|dg \le \Big (\frac{1}{|{\mathcal {Q}}|}\int _{2{\mathcal {Q}}}|f(g)|^{s}dg\Big )^{1/s} \Big (\frac{1}{|{\mathcal {Q}}|}\int _{2{\mathcal {Q}}}|b(g)-b_{{\mathcal {Q}}}|^{m}dg\Big )^{1/m}\\&\quad \lesssim [b]_{\theta }\inf \limits _{_{h\in {\mathcal {Q}}}}M_{s}f(h). \end{aligned}$$

Now, we consider the term $R_{{\mathcal {L}}}^{*}((b-b_{{\mathcal {Q}}})f_{2})(g)$. Applying (4), we have $|R_{{\mathcal {L}}}^{*}((b-b_{{\mathcal {Q}}})f_{2})(g)|\lesssim II_{1}+II_{2}$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} II_{1}:=\int _{(2{\mathcal {Q}})^{c}}{\frac{1}{(1+|gz^{-1}|/\rho (g))^{N}}}\frac{|f(z)(b-b_{{\mathcal {Q}}})|}{|gz^{-1}|^{Q}}dz,\\ II_{2}:=\int _{(2{\mathcal {Q}})^{c}}{\frac{1}{(1+|gz^{-1}|/\rho (g))^{N}}}\frac{|f(z)(b-b_{{\mathcal {Q}}})|}{|gz^{-1}|^{Q-1}} \int _{B(z,|gz^{-1}|/4)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz. \end{array} \right. \end{aligned}$$

For $1\le {\widetilde{s}}<s$ and $m=\frac{{\widetilde{s}}s}{s-{\widetilde{s}}}$, with the help of Lemma 6, we obtain

$$\begin{aligned}&\left\| f(b-b_{{\mathcal {Q}}})\chi _{B(g_{0},2^{k}\rho (g_{0}))}(z)\right\| _{{\widetilde{s}}} \\&\quad \lesssim \Big (\frac{1}{|2^{k}{\mathcal {Q}}|}\int _{2^{k}{\mathcal {Q}}}|f(z)|^{s}dz\Big )^{1/s} \Big (\frac{1}{|2^{k}{\mathcal {Q}}|}\int _{2^{k}{\mathcal {Q}}}|b(z)-b_{{\mathcal {Q}}}|^{m}dz\Big )^{1/m}\\&\quad \lesssim \inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h)(2^{k}\rho (g_{0}))^{Q/{\widetilde{s}}}k2^{k\theta '}[b]_{\theta }. \end{aligned}$$

(3.1)

Let ${\widetilde{s}}=1$. We have

$$\begin{aligned} II_{1}&\lesssim \sum \limits _{k\ge 1}\frac{1}{(1+2^{k}\rho (g_{0})/\rho (g_{0}))^{N}}\frac{1}{(2^{k}\rho (g_{0}))^{Q}} \int _{|g_{0}z^{-1}|<2^{k}\rho (g_{0})}|b(z)-b_{{\mathcal {Q}}}||f(z)|dz\\&\lesssim \inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h)\sum \limits _{k\ge 1}2^{kQ}\rho (g_{0})^{Q}2^{-kN}(2^{k}\rho (g_{0}))^{-Q}k2^{k\theta '}[b]_{\theta }\\&\lesssim [b]_{\theta }\inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h)\sum \limits _{k\ge 1}k2^{k(\theta '-N)}\lesssim [b]_{\theta }\inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h). \end{aligned}$$

Next, we deal with $II_{2}$. Similar to $I_{2}$, with the help of (3.1), we have

$$\begin{aligned} II_{2}&\lesssim \rho (g_{0})^{-1-Q/\widetilde{q'}}\sum \limits _{k\ge 1}2^{k(1+Q\lambda -Q-N-Q/\widetilde{q'})} \left\| f(b-b_{{\mathcal {Q}}}\chi _{B(g_{0},2^{k}\rho (g_{0}))}(z)\right\| _{{\widetilde{s}}}\\&\lesssim [b]_{\theta }\inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h), \end{aligned}$$

where in the last inequality we choose N large enough. For the case $q_{0}>Q$, we have the result for $R_{{\mathcal {L}}}$ in view of Lemma 8. $\square$

Remark 1

It is easy to check that if the critical ball ${\mathcal {Q}}$ is replaced by $2{\mathcal {Q}}$, Lemma 13 also holds.

Lemma 14

Let $V\in B_{Q/2}$ and $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$. Then for any $s>p_{0}'$ and $\gamma \ge 1$, there exists a constant C such that

$$\begin{aligned} \int _{(2B)^{c}}|R_{{\mathcal {L}}}^{*}(g,z)-R_{{\mathcal {L}}}^{*}(h,z)||b(z)-b_{B}||f(z)|dz\le C[b]_{\theta }\inf \limits _{u\in B}M_{s}f(u) \end{aligned}$$

holds for all f and $g,h\in B=B(g_{0},r)$ with $r<\gamma \rho (g_{0})$. Moreover, if $q_{0}>Q$, the above estimate also holds for $R_{{\mathcal {L}}}(\cdot ,\cdot )$ instead of $R_{{\mathcal {L}}}^{*}(\cdot ,\cdot )$.

Proof

Let ${\mathcal {Q}}=B(g_{0},\gamma \rho (g_{0}))$. For $g\in B(g_{0},r)$ with $r<\gamma \rho (g_{0})$, $z\in (2B)^{c}$, we can deduce that $|gz^{-1}|\thickapprox |g_{0}z^{-1}|\thickapprox 2^{j}r$. Applying (5) and $\rho (g)\thickapprox \rho (g_{0})$, we consider the following four parts:

$$\begin{aligned} \left\{ \begin{array}{lll} I_{1}:=r^{\delta }\int _{{\mathcal {Q}}\setminus 2B}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta }}dz,\\ I_{2}:=r^{\delta }\rho (g_{0})^{N}\int _{{\mathcal {Q}}^{c}}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta +N}}dz,\\ I_{3}:=r^{\delta } \int _{{\mathcal {Q}}\setminus 2B}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta -1}}\int _{B(g_{0},4|g_{0}z^{-1}|)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz,\\ I_{4}:=r^{\delta }\rho (g_{0})^{N} \int _{{\mathcal {Q}}^{c}}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta +N-1}}\int _{B(g_{0},4|g_{0}z^{-1}|)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz. \end{array} \right. \end{aligned}$$

To deal with $I_{1}$, for all $j\le j_{0}$ such that $2^{j_{0}}\ge \gamma \rho (g_{0})/r$, by Lemma 6, we use Hölder’s inequality to obtain

$$\begin{aligned}&I_{1}\lesssim r^{\delta }\sum \limits _{j=2}^{j_{0}}\int _{2^{j}B}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta }}dz\\&\quad \lesssim \frac{1}{r^{Q}}\sum \limits _{j=2}^{j_{0}}\frac{1}{(2^{j})^{Q+\delta }}\int _{2^{j}B}|f(z)||b(z)-b_{B}|dz \lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h)\sum \limits _{j=2}^{\infty }\frac{j}{(2^{j})^{\delta }}\\&\quad \lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h). \end{aligned}$$

For $I_{2}$, using Lemma 6 and choosing $N>\theta '$, we have

$$\begin{aligned} I_{2}&\lesssim r^{\delta }\rho (g_{0})^{N}\int _{2^{j}B}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q+\delta +N}}dz\\&\lesssim r^{\delta }\rho (g_{0})^{N}\sum \limits _{j=j_{0}-1}^{\infty }\int _{2^{j}B}\frac{1}{(2^{k}r)^{Q+\delta +N}}|f(z)||b(z)-b_{B}|dz \lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h). \end{aligned}$$

Next, for $I_{3}$, we assume $Q/2<q_{0}\le Q$. Then

$$\begin{aligned} I_{3}&\lesssim r^{\delta }\sum \limits _{j=2}^{j_{0}}\int _{2^{j}B}\frac{|f(z)||b(z)-b_{B}|}{|g_{0}z^{-1}|^{Q-1+\delta }} \int _{2^{j+2}B}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz\\&\lesssim r^{\delta }\sum \limits _{j=2}^{j_{0}}\int _{2^{j}B}\frac{|f(z)||b(z)-b_{B}|}{|2^{j}r|^{Q-1+\delta }}{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz\\&\lesssim \frac{1}{r^{Q-1}}\sum \limits _{j=2}^{j_{0}}\frac{1}{2^{j(Q-1+\delta )}}\int _{2^{j}B}|f(z)||b(z)-b_{B}|{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz. \end{aligned}$$

Take $p_{0}'<{\widetilde{s}}<s$, $m=\frac{{\widetilde{s}}s}{s-{\widetilde{s}}}$ and q such that $1/q=1/{\widetilde{s}}'+1/Q$. Using Lemma 6 and $j\le j_{0}$, we have

$$\begin{aligned}&\int _{2^{j}B}|f(z)||b(z)-b_{B}|{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz\\&\le \Big (\int _{2^{j}B}|f(z)|^{s}dz\Big )^{1/s}\Big (\int _{2^{j}B}|b(z)-b_{B}|^{m}dz\Big )^{1/m} \Big (\int _{2^{j}B}|{\mathcal {I}}_{1}(V)(z)|^{{\widetilde{s}}'}dz\Big )^{1/{\widetilde{s}}'}\\&\lesssim |2^{j}B|^{1/{\widetilde{s}}}\inf \limits _{h\in B}M_{s}f(h)j[b]_{\theta }(1+2^{j}r/\rho (g_{0}))^{\theta '}\Big (\int _{2^{j+2}B}|V(z)|^{q}dz\Big )^{1/q}\\&\lesssim |2^{j}B|^{1/{\widetilde{s}}}j[b]_{\theta }\inf \limits _{h\in B}M_{s}f(h)\rho (g_{0})^{Q/q-2}. \end{aligned}$$

For all $j\le j_{0}$ such that $2^{j_{0}}\ge \gamma \rho (g_{0})/r$, since $Q/{\widetilde{s}}=Q+1-Q/q$ and $2-Q/q>0$, then

$$\begin{aligned} I_{3}&\lesssim \frac{1}{r^{Q-1}}\sum \limits _{j=2}^{j_{0}}\frac{1}{2^{j(Q-1+\delta )}}|2^{j}B|^{1/{\widetilde{s}}}j[b]_{\theta } \inf \limits _{h\in B}M_{s}f(h)\rho (g_{0})^{Q/q-2}\\&\lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h)\Big (\frac{r}{\rho (g_{0})}\Big )^{2-Q/q}2^{j_{0}(2-Q/q)}\sum \limits _{j=2}^{j_{0}}\frac{j}{2^{j\delta }} \lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h). \end{aligned}$$

Finally, to deal with $I_{4}$, we get

$$\begin{aligned} I_{4}&\lesssim \frac{\rho (g_{0})^{N}}{r^{Q+N-1}}\sum \limits _{j=j_{0}-1}^{\infty }\frac{1}{2^{j(Q+\delta +N-1)}} \int _{2^{j}B}|f(z)||b(z)-b_{b}|{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz. \end{aligned}$$

For $j>j_{0}$, we have

$$\begin{aligned}&\int _{2^{j}B}|f(z)||b(z)-b_{B}|{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz\\&\lesssim |2^{j}B|^{1/{\widetilde{s}}}j[b]_{\theta }\inf \limits _{h\in B}M_{s}f(h)(1+2^{j}r/\rho (g_{0}))^{\theta '}\Big (\int _{2^{j+2}B}|V(z)|^{q}dz\Big )^{1/q}\\&\lesssim [b]_{\theta } \inf \limits _{h\in B}M_{s}f(h)j\frac{(2^{j}r)^{Q/{\widetilde{s}}+\theta '}}{\rho (g_{0})^{\theta '}}\Big (\int _{2^{j+2}B}|V(z)|^{q}dz\Big )^{1/q}. \end{aligned}$$

Noticing

$$\begin{aligned} \Big (\int _{2^{j+2}B}|V(z)|^{q}dz\Big )^{1/q}&\lesssim (2^{j}r)^{-Q/q'}\Big (\frac{2^{j}r}{\rho (g_{0})}\Big )^{Q\lambda }\int _{{\mathcal {Q}}}V(z)dz\\&\lesssim (2^{j}r)^{Q\lambda -Q/q'}\frac{r^{Q\lambda -Q/q'}}{\rho (g_{0})^{Q\lambda -Q+2}}, \end{aligned}$$

we obtain

$$\begin{aligned}&I_{4}\lesssim \frac{\rho (g_{0})^{N}}{r^{Q-1+N}}\sum \limits _{j=j_{0}-1}^{\infty }\frac{j[b]_{\theta }}{2^{j(Q-1+\delta +N)}} \inf \limits _{h\in B}M_{s}f(h)\frac{(2^{j}r)^{Q/{\widetilde{s}}+\theta '}}{\rho (g_{0})^{\theta '}} (2^{j}r)^{Q\lambda -Q/q'}\frac{r^{Q\lambda -Q/q'}}{\rho (g_{0})^{Q\lambda -Q+2}}\\&\quad \lesssim [b]_{\theta }\inf \limits _{h\in B}M_{s}f(h), \end{aligned}$$

where we take N large enough such that $N+Q+\delta -2-\theta '-Q\lambda >0$. Now, suppose $q_{0}>Q$. In order to obtain the estimate of $R_{{\mathcal {L}}}(\cdot ,\cdot )$, we apply Lemma 8 (ii) to get

$$\begin{aligned} \int _{(2B)^{c}}|R_{{\mathcal {L}}}(g,z)-R_{{\mathcal {L}}}(h,z)||b(z)-b_{B}||f(z)|dz\lesssim I_{1}+I_{2}\lesssim [b]_{\theta }\inf _{h\in B}M_{s}f(h). \end{aligned}$$

This completes the proof of Lemma 14. $\square$

3.2 The $L^{p}({\mathbb {H}}^{n})$ Boundedness of $[b,R_{{\mathcal {L}}}]$ and $[b,R_{{\mathcal {L}}}^{*}]$

Theorem 1

Suppose that $V\in B_{Q/2}$, $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$ and $p_{0}$ such that $1/p_{0}=(1/q_{0}-1/Q)^{+}$.

• (i) If $1<p<p_{0}$, then $\Vert [b,R_{{\mathcal {L}}}]f\Vert _{L^{p}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }\Vert f\Vert _{L^{p}({\mathbb {H}}^{n})}$ for all $f\in L^{p}({\mathbb {H}}^{n})$.

• (ii) If $p_{0}'<p<\infty$, then $\Vert [b,R_{{\mathcal {L}}}^{*}]f\Vert _{L^{p}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }\Vert f\Vert _{L^{p}({\mathbb {H}}^{n})}$ for all $f\in L^{p}({\mathbb {H}}^{n})$.

Proof

We only prove (ii), and (i) holds by duality. Because $f\in L^{p}({\mathbb {H}}^{n})$ with $p_{0}'<p<\infty$, according to Lemma 13, we have $[b,R_{{\mathcal {L}}}^{*}]f\in L_{loc}^{1}({\mathbb {H}}^{n})$. Applying Lemmas 12 and 13 with $p_{0}'<p<\infty$ and Remark 1, we get

$$\begin{aligned}&\left\| [b,R_{{\mathcal {L}}}^{*}]\right\| _{L^{p}({\mathbb {H}}^{n})}^{p}\\&\quad \le \int _{{\mathbb {H}}^{n}}|M_{\rho ,\beta }([b,R_{{\mathcal {L}}}^{*}]f)(g)|^{p}dg\\&\quad \lesssim \int _{{\mathbb {H}}^{n}}|M_{\rho ,\gamma }^{\sharp }([b,R_{{\mathcal {L}}}^{*}]f)(g)|^{p}dg+ \sum \limits _{k}|{\mathcal {Q}}_{k}|\Big (\frac{1}{|{\mathcal {Q}}_{k}|}\int _{2{\mathcal {Q}}_{k}}|([b,R_{{\mathcal {L}}}^{*}]f)(g)|dg\Big )^{p}\\&\quad \lesssim \int _{{\mathbb {H}}^{n}}|M_{\rho ,\gamma }^{\sharp }([b,R_{{\mathcal {L}}}^{*}]f)(g)|^{p}dg+[b]_{\theta }^{p}\left\| f\right\| _{L^{p}({\mathbb {H}}^{n})}^{p}. \end{aligned}$$

For the term $\int _{{\mathbb {H}}^{n}}|M_{\rho ,\gamma }^{\sharp }([b,R_{{\mathcal {L}}}^{*}]f)(g)|^{p}dg$, we write $[b,R_{{\mathcal {L}}}^{*}]f(g):=B_{1}(g)-B_{2}(g)$, where

$$\begin{aligned} \left\{ \begin{array}{lll} B_{1}(g):=(b(g)-b_{B})R_{{\mathcal {L}}}^{*}f(g),\\ B_{2}(g):=R_{{\mathcal {L}}}^{*}((b(g)-b_{B})f)(g). \end{array} \right. \end{aligned}$$

This implies that $\frac{1}{|B|}\int _{B}|[b,R_{{\mathcal {L}}}^{*}]f(g)-([b,R_{{\mathcal {L}}}^{*}]f)_{B}|dg\lesssim I+II$, where

$$\begin{aligned} \left\{ \begin{array}{lll} I:=\frac{1}{|B|}\int _{B}|B_{1}(g)-(B_{1})_{B}|dg,\\ II:=\frac{1}{|B|}\int _{B}|B_{2}(g)-(B_{2})_{B}|dg. \end{array} \right. \end{aligned}$$

For I, let $s>p_{0}'$. An application of Lemma 6 and Hölder’s inequality gives

$$\begin{aligned} I&\lesssim \Big (\frac{1}{|B|}\int _{B}|b(g)-b_{B}|^{s'}dg\Big )^{1/s'}\Big (\frac{1}{|B|}\int _{B}|R_{{\mathcal {L}}}^{*}f(g)|^{s}dg\Big )^{1/s} \lesssim [b]_{\theta }M_{s}(R_{{\mathcal {L}}}^{*}f). \end{aligned}$$

For II, let $g\in {\mathbb {H}}^{n}$ and $B=B(g_{0},r)$ with $r<\gamma \rho (g_{0})$ such that $g\in B$. We split $f=f_{1}+f_{2}$ with $f_{1}=f\chi _{2B}$. We further divide II into two parts as $II:=II_{1}+II_{2}$, where

$$\begin{aligned} \left\{ \begin{array}{lll} II_{1}:=\frac{1}{|B|}\int _{B}|R_{{\mathcal {L}}}^{*}((b-b_{B})f_{1})(g)-(R_{{\mathcal {L}}}^{*}((b-b_{B})f_{1})_{B}|dg,\\ II_{2}:=\frac{1}{|B|}\int _{B}|R_{{\mathcal {L}}}^{*}((b-b_{B})f_{2})(g)-(R_{{\mathcal {L}}}^{*}((b-b_{B})f_{2})_{B}|dg. \end{array} \right. \end{aligned}$$

For $II_{1}$, let $p_{0}'<{\widetilde{s}}<s$ and $m=\frac{{\widetilde{s}}s}{s-{\widetilde{s}}}$. Then

$$\begin{aligned} II_{1}&\le \frac{1}{|B|}\int _{B}|R_{{\mathcal {L}}}^{*}((b-b_{B})f_{1})(g)|dg\\&\lesssim \Big (\frac{1}{|B|}\int _{B}|R_{{\mathcal {L}}}^{*}((b-b_{B})f_{1})(g)|^{{\widetilde{s}}}dg\Big )^{1/{\widetilde{s}}} \Big (\frac{1}{|B|}\int _{B}1^{{\widetilde{s}}'}dg\Big )^{1/{\widetilde{s}}'}\\&\le \Big (\frac{1}{|B|}\int _{B}|b(g)-b_{B}|^{m}dg\Big )^{1/m}\Big (\frac{1}{|B|}\int _{B}|f(g)|^{s}dg\Big )^{1/s} \lesssim [b]_{\theta }M_{s}f(g). \end{aligned}$$

(3.2)

For $II_{2}$, it follows from Lemma 14 that

$$\begin{aligned} II_{2}&\lesssim \frac{1}{|B|^{2}}\int _{B}\int _{B}|(R_{{\mathcal {L}}}^{*}(b-b_{B})f_{2})(u)-R_{{\mathcal {L}}}^{*}((b-b_{B})f_{2})(z)|dudz \lesssim [b]_{\theta }M_{s}f(g). \end{aligned}$$

(3.3)

Finally, the $L^{p}$-boundedness of $M_{s}$ implies that

$$\begin{aligned} |M_{\rho ,\gamma }^{\sharp }([b,R_{{\mathcal {L}}}^{*}]f)(g)|\lesssim [b]_{\theta } (M_{s}R_{{\mathcal {L}}}^{*}f(g)+M_{s}f(g))\lesssim [b]_{\theta }\left\| f\right\| _{L^{p}({\mathbb {H}}^{n})}. \end{aligned}$$

This completes the proof of Theorem 1. $\square$

Theorem 2

Let $V\in B_{Q/2}$ and $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$. Then

• (i) $[b,R_{{\mathcal {L}}}^{*}]: L^{\infty }({\mathbb {H}}^{n}) \mapsto BMO_{{\mathcal {L}}}({\mathbb {H}}^{n})$ if and only if $b\in BMO_{\rho }^{\infty ,log}({\mathbb {H}}^{n})$.

• (ii) If $V\in B_{Q}$, the above result is also true for $[b,R_{{\mathcal {L}}}]$.

Proof

We only prove (i), and the proof of (ii) is similar to that of (i). Let $f\in L^{\infty }({\mathbb {H}})$ and ${\mathcal {Q}}=B(g_{0},\rho (g_{0}))$. Due to Lemma 13, we have

$$\begin{aligned} \frac{1}{|{\mathcal {Q}}|}\int _{{\mathcal {Q}}}|[b,R^{*}_{{\mathcal {L}}}]f(g)|dg\lesssim [b]_{\theta }\inf \limits _{h\in {\mathcal {Q}}}M_{s}f(h)\lesssim [b]_{\theta }\left\| f\right\| _{\infty }. \end{aligned}$$

In order to deal with the oscillations, let $B=B(g_{0},r)$ with $r<\rho (g_{0})$. From Lemma 13, we know the function $[b,R^{*}_{{\mathcal {L}}}]f$ belongs to $L_{loc}^{1}({\mathbb {H}}^{n})$.

We write $[b,R^{*}_{{\mathcal {L}}}]f:=I-II-III$, where

$$\begin{aligned} \left\{ \begin{array}{lll} I:=(b(g)-b_{B})R^{*}_{{\mathcal {L}}}f(g),\\ II:=R^{*}_{{\mathcal {L}}}(f_{1}(b(g)-b_{B}))(g),\\ III:=R^{*}_{{\mathcal {L}}}(f_{2}(b(g)-b_{B}))(g). \end{array} \right. \end{aligned}$$

We have done the estimates of II and III in (3.2) and (3.3). Therefore, both terms are bounded by $[b]_{\theta }\left\| f\right\| _{\infty }$.

To deal with I, we fix $h\in B$ and write

$$\begin{aligned} I:=&(b(g)-b_{B})R^{*}_{{\mathcal {L}}}f_{1}(g)+(b(g)-b_{B})(R^{*}_{{\mathcal {L}}}f_{2}(g)-R^{*}_{{\mathcal {L}}}f_{2}(u))\\&\quad +(b(g)-b_{B})R^{*}_{{\mathcal {L}}}f_{2,1}(u)+(b(g)-b_{B})R^{*}_{{\mathcal {L}}}f_{2,2}(u), \end{aligned}$$

where $f_{2}=f_{2,1}(u)+f_{2,2}(u)$ with $f_{2,2}(u)=f\chi _{4{\mathcal {Q}}\setminus 2B}$ and ${\mathcal {Q}}=B(g_{0},\rho (g_{0}))$. Denote by $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ the each term oscillation.

Notice that $R^{*}_{{\mathcal {L}}}f_{2,1}(u)$ and $R^{*}_{{\mathcal {L}}}f_{2,2}(u)$ are finite for any $u\in B$ since $f\in L^{\infty }({\mathbb {H}}^{n})$ and

$$\begin{aligned} \int _{(2B)^{c}}|R^{*}_{{\mathcal {L}}}(u,z)|dz<\infty . \end{aligned}$$

(3.4)

Next, we consider the boundedness of $I_{1}$, $I_{2}$ and $I_{3}$ under the condition that $b\in BMO^{\infty }_{\rho }({\mathbb {H}}^{n})$. For $I_{1}$, selecting s such that $R^{*}_{{\mathcal {L}}}$ is bounded on $L^{s}({\mathbb {H}}^{n})$, we have

$$\begin{aligned} I_{1}&\le \Big (\frac{1}{|B|}\int _{B}|b(g)-b_{B}|^{s'}dg\Big )^{1/s'}\Big (\frac{1}{|B|}\int _{2B}|f(g)|^{s}dg\Big )^{1/s} \lesssim [b]_{\theta }\left\| f\right\| _{\infty }. \end{aligned}$$

For the estimate of $I_{2}$, we consider the term $R^{*}_{{\mathcal {L}}}f_{2}(g)-R^{*}_{{\mathcal {L}}}f_{2}(u)$. Note that when $V\in B_{Q}$, then $|R^{*}_{{\mathcal {L}}}f_{2}(g)-R^{*}_{{\mathcal {L}}}f_{2}(u)|<\infty$. For $V\in B_{q}$ with $Q/2\le q<Q$, according to (5), we only need to estimate

$$\begin{aligned} \left\{ \begin{array}{lll} A_{1}:=r^{\delta }\int _{{\mathcal {Q}}\setminus 2B}\frac{|f(z)|}{|g_{0}z^{-1}|^{Q+\delta -1}}\int _{B(g_{0},4|g_{0}z^{-1}|)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz,\\ A_{2}:=r^{\delta }\rho (g_{0})^{N}\int _{{\mathcal {Q}}^{c}}\frac{|f(z)|}{|g_{0}z^{-1}|^{Q+\delta +N-1}}\int _{B(g_{0},4|g_{0}z^{-1}|)}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz. \end{array} \right. \end{aligned}$$

Similar to $I_{3}$ and $I_{4}$ in Lemma 14, we have

$$\begin{aligned}&A_{1}\lesssim r^{\delta }\sum \limits _{j=2}^{j_{0}}\int _{2^{j}B}\frac{|f(z)|}{|g_{0}z^{-1}|^{Q-1+\delta }}\int _{2^{j+2}B}\frac{V(u)}{|uz^{-1}|^{Q-1}}dudz\\&\quad \lesssim \frac{\left\| f\right\| _{\infty }}{r^{Q-2}}\sum \limits _{j=2}^{j_{0}}2^{-j(Q-2+\delta )}\int _{4{\mathcal {Q}}}V(z)dz \lesssim \left\| f\right\| _{\infty } \end{aligned}$$

and

$$\begin{aligned} A_{2}&\lesssim \left\| f\right\| _{\infty }\frac{\rho (g_{0})^{N}}{r^{Q-1+N}}\sum \limits _{j=j_{0}-1}^{\infty }2^{-j(Q-1+\delta +N)} \int _{2^{j}B}{\mathcal {I}}_{1}(V\chi _{2^{j+2}B})(z)dz\\&\lesssim \left\| f\right\| _{\infty }\frac{\rho (g_{0})^{N-Q\lambda }}{r^{Q-2+N-Q\lambda }} \sum \limits _{j=j_{0}-1}^{\infty }2^{-j(Q-2+\delta +N-Q\lambda )}\int _{{\mathcal {Q}}}V(z)dz \lesssim \left\| f\right\| _{\infty }, \end{aligned}$$

which gives $|R^{*}_{{\mathcal {L}}}f_{2}(g)-R^{*}_{{\mathcal {L}}}f_{2}(u)|\lesssim \left\| f\right\| _{\infty }$.

Applying Lemma 6 and Hölder’s inequality, we get

$$\begin{aligned} I_{2}&\lesssim \frac{1}{|B|}\int _{B}|b(g)-b_{B}|dg\cdot \left\| f\right\| _{\infty } \le \Big (\frac{1}{|B|}\int _{B}|b(g)-b_{B}|^{s}dg\Big )^{1/s}\cdot \left\| f\right\| _{\infty }\\&\lesssim [b]_{\theta }\left\| f\right\| _{\infty }. \end{aligned}$$

According to (3.4), we can get $I_{3}\lesssim [b]_{\theta }\left\| f\right\| _{\infty }$ .

Finally, the theorem will follow if and only if there exists a constant $C_{b}$ such that for any $B\in {\mathcal {B}}_{\rho ,1}$ and $u\in B$,

$$\begin{aligned} \frac{1}{|B|}\Big (\int _{B}|b(z)-b_{B}|dz\Big )\Big |\int _{4{\mathcal {Q}}\setminus 2B}R^{*}_{{\mathcal {L}}}(u,z)f(z)dz\Big | \le C_{b}\left\| f\right\| _{\infty }. \end{aligned}$$

(3.5)

However, the kernel $R^{*}(u,z)$ is added or subtracted, (3.5) is true, and only if the estimate (3.6) is true.

$$\begin{aligned} \frac{1}{|B|}\Big (\int _{B}|b(z)-b_{B}|dz\Big )\Big |\int _{4{\mathcal {Q}}\setminus 2B}R^{*}(u,z)f(z)dz\Big | \le C_{b}\left\| f\right\| _{\infty }. \end{aligned}$$

(3.6)

In fact, with the help of (6), we can deduce that $\int _{4{\mathcal {Q}}}|R^{*}_{{\mathcal {L}}}(u,z)-R^{*}(u,z)|dz$ is bounded independently of the critical ball ${\mathcal {Q}}$ and

$$\begin{aligned} \int _{4{\mathcal {Q}}}\frac{1}{|uz^{-1}|^{Q}}\Big (\frac{|uz^{-1}|}{\rho (u)}\big )^{\sigma }dz \lesssim \rho (g_{0})^{-\sigma }\int _{4{\mathcal {Q}}}\frac{1}{|g_{0}z^{-1}|^{Q-\sigma }}dz\lesssim 1. \end{aligned}$$

Assume $V\in B_{q}$ for $Q/2<q<Q$. Let $1/q=1/s-1/Q$. We have

$$\begin{aligned}&\int _{4Q}\frac{1}{|uz^{-1}|^{Q-1}}\int _{B(z,|uz^{-1}|/4)}\frac{V(w)}{|wz^{-1}|^{Q-1}}dwdz\\&\quad \lesssim \Big (\int _{Q}\Big |\frac{1}{|uz^{-1}|^{Q-1}}\Big |^{s'}dz\Big )^{1/s'}\Big (\int _{4{\mathcal {Q}}}|V(w)|^{q}dw\Big )^{1/q} \lesssim 1. \end{aligned}$$

Notice, so far, we have only used $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$. Now, we assume that b satisfies the stronger condition $b\in BMO_{\rho }^{\infty ,log}({\mathbb {H}}^{n})$, since $|\int _{4{\mathcal {Q}}\setminus 2B}R^{*}(u,z)f(z)dz|\lesssim \left\| f\right\| _{\infty }$ and

$$\begin{aligned}&\frac{1}{|B|}\int _{B}|b(z)-b_{B}|dz\int _{4{\mathcal {Q}}\setminus 2B}|R^{*}(u,z)f(z)|dz\lesssim \cdot \frac{C\left\| f\right\| _{\infty }(1+r/\rho (g_{0}))^{\theta }}{(1+log^{+}(\rho (g)/r))}\\&\quad \le \left\| f\right\| _{\infty }\cdot C(1+r/\rho (g_{0}))^{\theta }, \end{aligned}$$

we obtain the conclusion (3.6), thus proving the boundedness of the commutator $[b,R_{{\mathcal {L}}}^{*}]$.

On the other hand, we suppose that the commutator $[b,R_{{\mathcal {L}}}^{*}]$ is bounded with $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$, then (3.6) holds for each component $R^{*}_{i}$, $i=1,...,Q$ of $R^{*}$ and for any $f\in L^{\infty }({\mathbb {H}}^{n})$. Choosing $f=sg(uz^{-1})$, and adding over i, inequality (3.6) implies

$$\begin{aligned} \frac{1}{|B|}\int _{B}|b(z)-b_{B}|dz\int _{4{\mathcal {Q}}\setminus 2B}\frac{\sum _{i=1}^{Q}|z_{i}u_{i}^{-1}|}{|zu^{-1}|^{Q-1}}dz\le C_{b}. \end{aligned}$$

Since $|zu^{-1}|\thickapprox |zg_{0}^{-1}|$, performing the integration, the inequality $\frac{1}{|B|}\int _{B}|b(z)-b_{B}|dz\le \frac{C_{b}}{1+log(\rho (g_{0})/r)}$ holds for any $B\in {\mathcal {B}}_{\rho ,1}$. Since $b\in BMO_{\rho }^{\infty }({\mathbb {H}}^{n})$, we conclude that $b\in BMO_{\rho }^{\infty ,log}({\mathbb {H}}^{n})$. This completes the proof of Theorem 2. $\square$

3.3 The $L^{p}({\mathbb {H}}^{n})$ Boundedness of the Commutator $[b,T_{\beta _{1},\beta _{2}}]$

Theorem 3

Suppose that $V\in B_{q}$ for $q>Q/2$. Let $0\le \beta _{2}\le \beta _{1}<Q/2$. If $1<(q/\beta _{2})'<p_{1}<Q/(2\beta _{1}-2\beta _{2})$ and $1/p_{2}=1/p_{1}-(2\beta _{1}-2\beta _{2})/Q$, then

$$\begin{aligned} \Vert T_{\beta _{1},\beta _{2}}f\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\lesssim \Vert f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}, \end{aligned}$$

where $(q/\beta _{2})'$ is the conjugate of $q/\beta _{2}$.

Proof

We only prove the following inequality:

$$\begin{aligned} I:=|(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}(V^{\beta _{2}}f)(g)|\lesssim M_{2(\beta _{1}-\beta _{2}),(q/\beta _{2})'}f(g). \end{aligned}$$

(3.7)

Let $r=\rho (g)$, $B=B(g_{0},r)$. With the help of Lemma 7, we use Hölder’s inequality to deduce that

$$\begin{aligned} \begin{aligned} I\lesssim&\sum \limits _{k=-\infty }^{+\infty }\int _{2^{k}B\setminus 2^{k-1}B}\frac{1}{(1+|g^{-1}h|\rho (g)^{-1})^{N}}\cdot \frac{1}{|g^{-1}h|^{Q-2\beta _{1}}}\cdot V^{\beta _{2}}(h)|f(h)|dh\\ \lesssim&\sum \limits _{k=-\infty }^{+\infty }\frac{(2^{k}r)^{2\beta _{2}}}{(1+2^{k})^{N}} \cdot \Big (\frac{1}{|2^{k}B|}\int _{2^{k}B}|V(h)|dh\Big )^{\beta _{2}}\cdot M_{2(\beta _{1}-\beta _{2}),(q/(q-\beta _{2}))}f(g). \end{aligned} \end{aligned}$$

For $k\ge 1$, because V(h)dh is a doubling measure, we have

$$\begin{aligned} \begin{aligned} \frac{(2^{k}r)^{2}}{|2^{k}B|}\int _{2^{k}B}V(h)dh&\lesssim \frac{(2^{k}r)^{2}}{(2^{k}r)^{Q}}\int _{B}V(h)dh \lesssim 2^{N_{0}}. \end{aligned} \end{aligned}$$

Take N large enough. We can get

$$\begin{aligned} \begin{aligned} \sum \limits _{k=1}^{+\infty }\frac{2^{N_{0}}}{(1+2^{k})^{N}}M_{2(\beta _{1}-\beta _{2}),q/(q-\beta _{2})}f(g) \lesssim M_{2(\beta _{1}-\beta _{2}),q/(q-\beta _{2})}f(g). \end{aligned} \end{aligned}$$

For $k\le 0$, Lemma 2 implies that

$$\begin{aligned} \begin{aligned} \frac{(2^{k}r)^{2}}{|2^{k}B|}\int _{2^{k}B}V(h)dh&\lesssim \frac{1}{(2^{k}r)^{Q-2}}\int _{B}V(h)dh\lesssim (2^{k})^{2-Q/q}. \end{aligned} \end{aligned}$$

Taking N large enough, we obtain

$$\begin{aligned} \begin{aligned}&\sum \limits _{k=-\infty }^{0}\frac{1}{(1+2^{k})^{N}}\cdot (2^{k})^{2-Q/q}M_{2(\beta _{1}-\beta _{2}),q/(q-\beta _{2})}f(g) \lesssim M_{2(\beta _{1}-\beta _{2}),q/(q-\beta _{2})}f(g). \end{aligned} \end{aligned}$$

Finally, it holds

$$\begin{aligned} |(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}(V^{\beta _{2}}f)(g)|\lesssim M_{2(\beta _{1}-\beta _{2}),(q/\beta _{2})'}f(g). \end{aligned}$$

By (3.7) and the boundedness of the fractional maximal function, we obtain Theorem 3. $\square$

Theorem 4

Suppose $V\in B_{q}$, $q\ge Q/2$ and $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, $0<\theta <\infty$. If $0\le \beta _{2}\le \beta _{1}<Q/2$, $q/(q-\beta _{2})<p_{1}<Q/(2\beta _{1}-2\beta _{2})$, $1/p_{2}=1/p_{1}-(2\beta _{1}-2\beta _{2})/Q$, then

$$\begin{aligned} \Vert [b,T_{\beta _{1},\beta _{2}}]f\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }\Vert f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned}$$

Proof

Because $\beta _{2}\le \beta _{1}$, we can decompose the operator $(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ as

$$\begin{aligned} (-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{\beta _{2}-\beta _{1}}(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{2}}V^{\beta _{2}}. \end{aligned}$$

Denote by $L^{\beta _{2}-\beta _{1}}$ and $T_{\beta _{2}}$ the operators $(-\Delta _{{\mathbb {H}}^{n}}+V)^{\beta _{2}-\beta _{1}}$ and $(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{2}}V^{\beta _{2}}$, respectively. Then we can get

$$\begin{aligned}{}[b,T_{\beta _{1},\beta _{2}}]f(g)=[b,L^{\beta _{2}-\beta _{1}}]T_{\beta _{2}}f(g)+L^{\beta _{2}-\beta _{1}}[b,T_{\beta _{2}}]f(g). \end{aligned}$$

By Theorem 3, we obtain

$$\begin{aligned} \begin{aligned} \Vert [b,T_{\beta _{1},\beta _{2}}]f\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}&\le \Vert [b,L^{\beta _{2}-\beta _{1}}]T_{\beta _{2}}f(g)\Vert _{L^{p_{2}}({\mathbb {H}}^{n})} +\Vert L^{\beta _{2}-\beta _{1}}[b,T_{\beta _{2}}]f(g)\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\\&\lesssim [b]_{\theta }\Vert T_{\beta _{2}}f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}+\Vert [b,T_{\beta _{2}}]f(g)\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\\&\lesssim [b]_{\theta }\Vert f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}+[b]_{\theta }\Vert f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }\Vert f\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned} \end{aligned}$$

This completes the proof. $\square$

4 Boundedness of the Commutators on Herz Spaces ${\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})$

In this section, we consider the boundedness of the operator $T_{\beta _{1},\beta _{2}}=(-\Delta _{{\mathbb {H}}^{n}}+V)^{-\beta _{1}}V^{\beta _{2}}$ and the commutator $[b,T_{\beta _{1},\beta _{2}}]$ on Herz spaces ${\dot{K}}_{q}^{\alpha ,p}({\mathbb {H}}^{n})$, where $0\le \beta _{2}\le \beta _{1}<Q/2$.

Theorem 5

Suppose that $V\in B_{q}$, $q\ge Q/2$. Let $0<p<\infty$, $0\le \beta _{2}\le \beta _{1}<Q/2$. If $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{1}/Q)$ and $q/(q-\beta _{2})<p_{1}<Q/(2\beta _{1}-2\beta _{2})$ with $1/p_{2}=1/p_{1}-(2\beta _{1}-2\beta _{2})/Q$, then

$$\begin{aligned} \Vert T_{\beta _{1},\beta _{2}}f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}\lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}. \end{aligned}$$

Proof

We write $f(h){=}\!\sum \limits _{j=-\infty }^{\infty }\! f(h)\chi _{E_{j}}(h){=}\!\sum \limits _{j=-\infty }^{\infty }\! f_{j}(h)$, which gives $\Vert T_{\beta _{1},\beta _{2}}f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p} \lesssim M_{1}+M_{2}+M_{3}$, where

$$\begin{aligned} \left\{ \begin{array}{lll} M_{1}:=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2}\Vert T_{\beta _{1},\beta _{2}}f_{j}\chi _{k}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p},\\ M_{2}:=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=k-1}^{k+1}\Vert T_{\beta _{1},\beta _{2}}f_{j}\chi _{k}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p},\\ M_{3}:=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=k+2}^{+\infty }\Vert T_{\beta _{1},\beta _{2}}f_{j}\chi _{k}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p}. \end{array} \right. \end{aligned}$$

For $M_{2}$, by Theorem 3, we obtain $M_{2}\lesssim \sum \limits _{k=-\infty }^{\infty }2^{k\alpha p}\Big (\sum \limits _{j=k-1}^{k+1}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p} \lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}.$

Now we estimate $M_{1}$. Via a simple computation, for $g\in E_{k}$ and $h\in E_{j}$ with $j\le k-2$, we can deduce that $|gh^{-1}|\thickapprox 2^{k}$. Applying Lemmas 7, 5, and 3, we use Hölder’s inequality to get

$$\begin{aligned} \begin{aligned} \Vert \chi _{k}T_{\beta _{1},\beta _{2}}f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}&\lesssim \frac{(2^{k})^{2\beta _{1}-Q}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)}} \Big (\int _{E_{k}}|\int _{E_{j}}V^{\beta _{2}}(h)f(h)dh|^{p_{2}}dg\Big )^{1/p_{2}}\\&\lesssim \frac{\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-N_{0}\beta _{2}}} \cdot \frac{|E_{j}|^{1-1/p_{2}-2\beta _{1}/Q}}{|E_{k}|^{1-1/p_{2}-2\beta _{1}/Q}}. \end{aligned} \end{aligned}$$

Take N large enough. For $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{1}/Q)$, we have

$$\begin{aligned} \begin{aligned} M_{1}&\lesssim \sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2}\frac{\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-N_{0}\beta _{2}}} \cdot \frac{|E_{j}|^{1-1/p_{2}-2\beta _{1}/Q}}{|E_{k}|^{1-1/p_{2}-2\beta _{1}/Q}}\Big )^{p}\\&\lesssim \sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2}\frac{|E_{j}|^{1-1/p_{2}-2\beta _{1}/Q}}{|E_{k}|^{1-1/p_{2}-2\beta _{1}/Q}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p}\\&\lesssim \sum \limits _{k=-\infty }^{\infty } \Big (\sum \limits _{j=-\infty }^{k-2}2^{(j-k)Q(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p}. \end{aligned} \end{aligned}$$

Below two cases are considered.

Case 1:$p\le 1$. By the p-triangle inequality,

$$\begin{aligned} \Big (\sum \limits _{k=-\infty }^{\infty }|a_{k}|\Big )^{r}\le \sum \limits _{k=-\infty }^{\infty }|a_{k}|^{r}, 0<r<1, \end{aligned}$$

(4.1)

we have

$$\begin{aligned}&M_{1}\lesssim \sum \limits _{k=-\infty }^{\infty } \sum \limits _{j=-\infty }^{k-2}2^{(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p}\\&\quad \lesssim \sum \limits _{j=-\infty }^{\infty }2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p} \lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}. \end{aligned}$$

Case 2:$p> 1$. Using the Hölder inequality, we can get

$$\begin{aligned} \begin{aligned} M_{1}\lesssim&\sum \limits _{k=-\infty }^{\infty }\Big \{(\sum \limits _{j=-\infty }^{k-2}2^{1/2(j-k)Qp'(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)})^{1/p'}\\&\times (\sum \limits _{j=-\infty }^{k-2}2^{1/2(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p})^{1/p}\Big \}^{p}\\ \lesssim&\sum \limits _{j=-\infty }^{\infty }2^{j\alpha p} \sum \limits _{k=j+2}^{\infty }2^{1/2(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p}\\ \lesssim&\sum \limits _{j=-\infty }^{\infty }2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p} \lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}. \end{aligned} \end{aligned}$$

For $M_{3}$, because $g\in E_{k}$, $h\in E_{j}$, and $j\ge k+2$, then $|gh^{-1}|\thickapprox 2^{j}$. Similar to $M_{1}$, we have

$$\begin{aligned} \begin{aligned} \Vert \chi _{k}T_{\beta _{1},\beta _{2}}f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}&\lesssim \frac{(2^{j})^{2\beta _{1}-Q}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)}} \Big (\int _{E_{k}}|\int _{E_{j}}V^{\beta _{2}}(h)f(h)dh|^{p_{2}}dg\Big )^{1/p_{2}}\\&\lesssim \frac{\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)-N_{0}\beta _{2}}} \cdot \frac{|E_{k}|^{1/p_{2}}}{|E_{j}|^{1/p_{2}}}. \end{aligned} \end{aligned}$$

Because $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{2}/Q)$, we use (4.1) and Hölder’s inequality to deduce that

$$\begin{aligned} \begin{aligned} M_{3}&\lesssim \sum \limits _{k=-\infty }^{\infty }\Big (\sum \limits _{j=k+2}^{\infty }2^{(k-j)Q(1/p_{2}+\alpha /Q)}2^{j\alpha }\Vert f\Vert _{L^{p_{1}({\mathbb {H}}^{n})}}\Big )^{p} \lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}. \end{aligned} \end{aligned}$$

Finally, we get $\Vert T_{\beta _{1},\beta _{2}}f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}\lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}$. $\square$

Theorem 6

Suppose that $V\in B_{q}$, $q\ge Q/2$ and $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, $0<\theta <\infty$. Let $0<p<\infty$, $0\le \beta _{2}\le \beta _{1}<Q/2$. If $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{1}/Q)$ and $q/(q-\beta _{2})<p_{1}<Q/(2\beta _{1}-2\beta _{2})$ with $1/p_{2}=1/p_{1}-(2\beta _{1}-2\beta _{2})/Q$, then

$$\begin{aligned} \Vert [b,T_{\beta _{1},\beta _{2}}]f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }\Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}. \end{aligned}$$

Proof

We decompose f as follows.

$$\begin{aligned} f(h)=\sum \limits _{j=-\infty }^{\infty }f(h)\chi _{E_{j}}(h)=\sum \limits _{j=-\infty }^{\infty }f_{j}(h), \end{aligned}$$

which implies that $\left\| [b,T_{\beta _{1},\beta _{2}}]f \right\| _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p} \lesssim I+II+III$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} I=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2} \Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p},\\ II=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=k-1}^{k+1} \Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p},\\ III=\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=k+2}^{\infty } \Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\Big )^{p}.\\ \end{array} \right. \end{aligned}$$

Applying Theorem 4, we can get $II\lesssim [b]_{\theta }^{p}\sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=k-1}^{k+1}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p} \lesssim [b]_{\theta }^{p}\Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}$.

For I, we write $b(g)-b(h):=(b(g)-b_{B})-(b(h)-b_{B})$. By Hölder’s inequality and the fact that $V\in B_{q}$, we apply Lemmas 7 and 5 to deduce that

$$\begin{aligned} \begin{aligned}&\Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})} \lesssim \frac{(2^{k})^{2\beta _{1}-Q}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)}}\\&\quad \Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}. \end{aligned} \end{aligned}$$

Now we consider the term $\Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}$. We have

$$\begin{aligned} \begin{aligned}&\Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}\\&\quad \lesssim \Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh \\&\quad +|E_{k}|^{1/p_{2}}\int _{E_{j}}|(b(h)-b_{B})V^{\beta _{2}}(h)|f(h)|dh, \end{aligned} \end{aligned}$$

which implies that $\Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\lesssim I_{1}+I_{2}$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} I_{1}=\frac{1}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)}}\cdot \frac{1}{(2^{k})^{Q-2\beta _{1}}} \Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh,\\ I_{2}=\frac{1}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)}}\cdot \frac{1}{(2^{k})^{Q-2\beta _{1}}} |E_{k}|^{1/p_{2}}\int _{E_{j}}|(b(h)-b_{B})V^{\beta _{2}}(h)f(h)|dh. \end{array} \right. \end{aligned}$$

For $I_{1}$, we use the Hölder inequality to deduce that

$$\begin{aligned} \begin{aligned} \int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh&\le \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s}\cdot \Big (\int _{E_{j}}|f(h)|^{s/(s-\beta _{2})}dh\Big )^{(s-\beta _{2})/s}\\&\lesssim |E_{j}|^{1-1/p-\beta _{2}}\cdot \Big (\int _{E_{j}}V(h)dh\Big )^{\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned} \end{aligned}$$

Note that $\Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\lesssim (k-j)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}}$. Then, we have

$$\begin{aligned} \begin{aligned}&\Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh\\&\lesssim (k-j)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}} |E_{j}|^{1-1/p_{1}-\beta _{2}}\Big (\int _{E_{j}}V(h)dh\Big )^{\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\\&\lesssim (k-j)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}} |E_{j}|^{1-1/p_{1}-\beta _{2}}\\&(1+2^{k}/\rho (0))^{N_{0}\beta _{2}}(2^{j})^{(Q-2)\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned} \end{aligned}$$

Therefore, we can get

$$\begin{aligned}&I_{1}\lesssim \frac{(k-j)[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}} \cdot \frac{|E_{j}|^{1-1/p_{1}-2\beta _{2}/Q}}{|E_{k}|^{1-1/p_{2}-2\beta _{1}/Q}}\\&\quad \lesssim \frac{(k-j)[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}2^{(j-k)Q(1-1/p_{2}-2\beta _{1}/Q)}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}}. \end{aligned}$$

For the term $I_{2}$, denote $m=sp_{1}/(p_{1}(s-\beta _{2})-s)$. We use Hölder’s inequality to obtain

$$\begin{aligned} \begin{aligned}&\int _{E_{j}}|(b(h)-b_{B})V^{\beta _{2}}(h)f(h)|dh\\&\quad \le \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s} \cdot \Big (\int _{E_{j}}|(b(h)-b_{B})f(h)|^{s/(s-\beta _{2})}dh\Big )^{(s-\beta _{2})/s}\\&\quad \lesssim \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s}\cdot \Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})} \Big (\frac{1}{|E_{j}|}\int _{E_{j}}|b(h)-b_{B}|^{m}dh\Big )^{1/m}\\&\quad \lesssim [b]_{\theta }(1+2^{j}/\rho (0))^{N_{0}\beta _{2}+\theta '}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}|E_{j}|^{1-1/p_{1}-2\beta _{2}/Q}, \end{aligned} \end{aligned}$$

which gives

$$\begin{aligned}&I_{2}\lesssim \frac{[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}} \cdot \frac{|E_{j}|^{1-1/p_{1}-2\beta _{2}/Q}}{|E_{k}|^{1-1/p_{2}-2\beta _{1}/Q}}\\&\quad \lesssim \frac{[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}2^{(j-k)Q(1-1/p_{2}-2\beta _{1}/Q)}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}}. \end{aligned}$$

Take N large enough. Because $-Q/p_{2}<\alpha <Q(-1/p_{2}+1-2\beta _{1}/Q)$, we obtain

$$\begin{aligned} \begin{aligned} I&\lesssim \sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2}\frac{(k-j)[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}2^{(j-k)Q(1-1/p_{2}-2\beta _{1}/Q)}}{(1+2^{k}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}}\Big )^{p}\\&\lesssim [b]_{\theta }^{p}\sum \limits _{k=-\infty }^{\infty } \Big (\sum \limits _{j=-\infty }^{k-2}\Big ((k-j)2^{(j-k)Q(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p}. \end{aligned} \end{aligned}$$

We still divide the proof into two cases.

Case 1: $p\le 1$. By (4.1), we have

$$\begin{aligned} \begin{aligned} I&\lesssim [b]_{\theta }^{p}\sum \limits _{k=-\infty }^{\infty }\sum \limits _{j=-\infty }^{k-2} (k-j)^{p}2^{(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p}\\&\lesssim [b]_{\theta }^{p}\sum \limits _{j=-\infty }^{\infty }2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p} \lesssim [b]_{\theta }^{p}\Vert f_{j}\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}. \end{aligned} \end{aligned}$$

Case 2: $p>1$. The Hölder inequality implies that

$$\begin{aligned} I\lesssim&[b]_{\theta }^{p}\sum \limits _{k=-\infty }^{\infty } \Big \{(\sum \limits _{j=-\infty }^{k-2}(k-j)^{p'}2^{1/2(j-k)Qp'(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}\Big )^{1/p'}\\&\times (2^{1/2(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p}\Big )^{1/p}\Big \}^{p}\\ \lesssim&[b]_{\theta }^{p}\sum \limits _{j=-\infty }^{\infty }2^{j\alpha p}\sum \limits _{k=j+2}^{\infty } 2^{1/2(j-k)Qp(1-1/p_{2}-2\beta _{1}/Q-\alpha /Q)}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p}\\ \lesssim&[b]_{\theta }^{p}\sum \limits _{j=-\infty }^{\infty }2^{j\alpha p}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}^{p} \lesssim [b]_{\theta }^{p}\Vert f_{j}\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}. \end{aligned}$$

For III, note that when $g\in E_{k}$, $y\in E_{j}$, and $j\ge k+2$, then $|gh^{-1}|\thickapprox 2^{j}r$. Using the following decomposition: $b(g)-b(h):=(b(g)-b_{B})-(b(h)-b_{B})$ and Lemmas 7 and 5, we can get

$$\begin{aligned}&\Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\lesssim \frac{(2^{j})^{2\beta _{1}-Q}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)}}\\&\quad \Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}. \end{aligned}$$

Now we consider the term $\Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}$. We have

$$\begin{aligned} \begin{aligned}&\Big (\int _{E_{k}}\Big |\int _{E_{j}}(b(g)-b(h))V^{\beta _{2}}(h)f(h)dh\Big |^{p_{2}}dg\Big )^{1/p_{2}}\\&\lesssim \Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh\\&\quad +|E_{k}|^{1/p_{2}}\int _{E_{j}}|(b(h)-b_{B})V^{\beta _{2}}(h)f(h)|dh, \end{aligned} \end{aligned}$$

which implies that $\Vert \chi _{k}[b,T_{\beta _{1},\beta _{2}}]f_{j}\Vert _{L^{p_{2}}({\mathbb {H}}^{n})}\lesssim III_{1}+III_{2}$, where

$$\begin{aligned} \left\{ \begin{array}{lcc} III_{1}=\frac{1}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)}}\cdot \frac{1}{(2^{j})^{Q-2\beta _{1}}} \Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh,\\ III_{2}=\frac{1}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)}}\cdot \frac{1}{(2^{j})^{Q-2\beta _{1}}} |E_{k}|^{1/p_{2}}\int _{E_{j}}|(b(h)-b_{B})V^{\beta _{2}}(h)f(h)|dh. \end{array} \right. \end{aligned}$$

For $III_{1}$, we use the Hölder inequality to deduce that

$$\begin{aligned} \begin{aligned} \int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh&\le \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s}\cdot \Big (\int _{E_{j}}|f(h)|^{s/(s-\beta _{2})}dh\Big )^{(s-\beta _{2})/s}\\&\lesssim |E_{j}|^{1-1/p_{1}-\beta _{2}}\cdot \Big (\int _{E_{j}}V(h)dh\Big )^{\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned} \end{aligned}$$

Note that $\Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\lesssim (j-k)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}}$. Then, we have

$$\begin{aligned} \begin{aligned}&\Big (\int _{E_{k}}|b(g)-b_{B}|^{p_{2}}dg\Big )^{1/p_{2}}\int _{E_{j}}|V^{\beta _{2}}(h)f(h)|dh\\&\quad \lesssim (j-k)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}} |E_{j}|^{1-1/p_{1}-\beta _{2}}\Big (\!\int _{E_{j}}\! V(h)dh\!\Big )^{\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\\&\quad \lesssim (j-k)[b]_{\theta }(1+2^{k}/\rho (0))^{\theta '}|E_{k}|^{1/p_{2}} |E_{j}|^{1-1/p_{1}-\beta _{2}}\\&\quad (1+2^{k}/\rho (0))^{N_{0}\beta _{2}}(2^{j})^{(Q-2)\beta _{2}}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}. \end{aligned} \end{aligned}$$

Therefore, we can get

$$\begin{aligned} III_{1}\lesssim \frac{(j-k)[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}} \cdot \frac{|E_{k}|^{1/p_{2}}}{|E_{j}|^{1/p_{2}}}. \end{aligned}$$

For $III_{2}$, denote $m=sp_{1}/(p_{1}(s-\beta _{2})-s)$. The Hölder inequality implies that

$$\begin{aligned} \begin{aligned}&\int _{E_{j}}|b(h)-b_{B}|V^{\beta _{2}}(h)|f(h)|dh\\&\quad \le \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s} \cdot \Big (\int _{E_{j}}|(b(h)-b_{B})f(h)|^{s/(s-\beta _{2})}dh\Big )^{(s-\beta _{2})/s}\\&\quad \le \Big (\int _{E_{j}}V(h)^{s}dh\Big )^{\beta _{2}/s}\cdot \Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})} \Big (\frac{1}{|E_{j}|}\int _{E_{j}}|b(h)-b_{B}|^{m}dh\Big )^{1/m}\\&\quad \lesssim [b]_{\theta }(1+2^{j}/\rho (0))^{N_{0}\beta _{2}+\theta '}\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}|E_{j}|^{1-1/p_{1}-2\beta _{2}/Q}, \end{aligned} \end{aligned}$$

which gives

$$\begin{aligned} III_{2}\lesssim \frac{[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}} \cdot \frac{|E_{k}|^{1/p_{2}}}{|E_{j}|^{1/p_{2}}}. \end{aligned}$$

Since N is large enough and $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta _{1}/Q)$, we obtain

$$\begin{aligned} \begin{aligned} III&\lesssim \sum \limits _{k=-\infty }^{\infty }2^{k\alpha p} \Big (\sum \limits _{j=-\infty }^{k-2}\frac{(j-k)[b]_{\theta }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}2^{(k-j)Q/p_{2}}}{(1+2^{j}/\rho (0))^{N/(l_{0}+1)-\theta '-N_{0}\beta _{2}}}\Big )^{p}\\&\lesssim [b]_{\theta }^{p}\sum \limits _{k=-\infty }^{\infty } \Big (\sum \limits _{j=k+2}^{\infty }(j-k)2^{(k-j)Q(1/p_{2}+\alpha /Q)}2^{j\alpha }\Vert f_{j}\Vert _{L^{p_{1}}({\mathbb {H}}^{n})}\Big )^{p}. \end{aligned} \end{aligned}$$

Similar to I, it holds $III\lesssim [b]_{\theta }^{p}\Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}^{p}$. Finally, we get

$$\begin{aligned} \Vert [b,T_{\beta _{1},\beta _{2}}]f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}\lesssim [b]_{\theta }^{p}\Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}. \\ \end{aligned}$$

$\square$

Let $\beta _{2}=0$. The following result is an immediate consequence of Theorem 6.

Corollary 1

Suppose that $V\in B_{q}$, $q\ge Q/2$ and $b\in BMO_{\rho }^{\theta }({\mathbb {H}}^{n})$, $0<\theta <\infty$. Let $0<p<\infty$, $0< \beta <Q/2$. If $-Q/p_{2}<\alpha <Q(1-1/p_{2}-2\beta /Q)$ and $1<p_{1}<Q/2\beta$ with $1/p_{2}=1/p_{1}-2\beta /Q$, then

$$\begin{aligned} \Vert [b,T_{\beta }]f\Vert _{{\dot{K}}_{p_{2}}^{\alpha ,p}({\mathbb {H}}^{n})}\lesssim \Vert f\Vert _{{\dot{K}}_{p_{1}}^{\alpha ,p}({\mathbb {H}}^{n})}\cdot [b]_{\theta }. \end{aligned}$$