Schrödinger type operators on generalized Morrey spaces

Abstract

In this paper we introduce a class of generalized Morrey spaces associated with the Schrödinger operator $L=-\Delta+V$. Via a pointwise estimate, we obtain the boundedness of the operators $V^{\beta_{2}}(-\Delta +V)^{-\beta_{1}}$ and their dual operators on these Morrey spaces.

1 Introduction

The investigation of Schrödinger operators on the Euclidean space $\mathbb{R}^{n}$ with nonnegative potentials which belong to the reverse Hölder class has attracted attention of many authors. Shen [1] studied the Schrödinger operator $L=-\Delta+ V$, assuming the nonnegative potential V belongs to the reverse Hölder class $B_{q}$, $q\geq\frac{n}{2}$. In [1], Shen proved the $L^{p}$-boundedness of the operators $(-\Delta+V)^{i\gamma}$, $\nabla^{2}(-\Delta+V)^{-1}$, $\nabla (-\Delta+V)^{-1/2}$ and $\nabla(-\Delta+V)^{-1}\nabla$. For further information, we refer the reader to Guo et al. [2], Liu [3], Liu et al. [4, 5], Tang and Dong [6], Yang et al. [7, 8] and the references therein.

The purpose of this paper is to generalize the results of Shen [1] and Sugano [9] to a class of Morrey spaces associated with L, denoted by $L_{\alpha,\theta,V}^{p,q,\lambda}(\mathbb {R}^{n})$. See Definition 2.8 below. The significance of these spaces is that for particular choices of the parameters p, q, λ, θ and α, one obtains many classical function spaces (see Table 1).

Table 1 Special cases of $\pmb{L^{p,q,\lambda}_{\alpha,\beta, V}}$

In Section 3, let T be one of the Schrödinger type operators $\nabla(-\Delta+V)^{-1}\nabla$, $\nabla(-\Delta +V)^{-1/2}$ and $(-\Delta+V)^{-1/2}\nabla$. With the help of the $L^{p}$-boundedness of T, it is easy to verify that T is bounded on $L_{\alpha,\theta, V}^{p,q, \lambda}(\mathbb{R}^{n})$. For $b\in \mathit{BMO}(\mathbb{R}^{n})$, we can also obtain the boundedness of the commutator $[b, T]$ on $L_{\alpha,\theta, V}^{p,q, \lambda}(\mathbb {R}^{n})$. See Theorems 3.2 and 3.3. For $\theta=0$, $p=q$ and $0<\lambda<1$, $L_{\alpha,0, V}^{p,p, \lambda }(\mathbb{R}^{n})$ becomes the spaces $L_{\alpha,V}^{p,\lambda }(\mathbb{R}^{n})$ introduced by Tang and Dong [6]. Hence, the results are generalizations of Theorems 1 and 2 in [6].

In recent years, the fractional integral operator $I_{\alpha}=(-\Delta +V)^{-\alpha}$ has been studied extensively. We refer to Duong and Yan [14], Jiang [15], Tang and Dong [6] and Yang et al. [7] for details. Suppose that $V\in B_{s}$, $s\geq\frac {n}{2}$. For $0\leq\beta_{2}\leq\beta_{1}<\frac{n}{2}$, let

$$ \textstyle\begin{cases} T_{\beta_{1},\beta_{2}}=:V^{\beta_{2}}(- \Delta+V)^{-\beta_{1}}, \\ T^{\ast}_{\beta_{1},\beta_{2}}=:(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}. \end{cases} $$

Sugano [9] obtained the weighted estimates for $T_{\beta_{1}, \beta_{2}}$, $T^{\ast}_{\beta_{1},\beta_{2}}$, $0<\beta_{2}\leq\beta_{1}<1$. If $\beta_{2}=0$, we can see that $T_{\beta_{1},0}=I_{\beta_{1}}$. So $T_{\beta_{1},\beta_{2}}$ and $T_{\beta_{1},\beta_{2}}^{\ast}$ can be seen as generalizations of $I_{\alpha}$. Moreover, for $(\beta _{1}, \beta_{2})=(1,1)$ and $(1/2,1/2)$, $T_{1,1}^{\ast}=(-\Delta +V)^{-1}V$ and $T_{1/2,1/2}^{\ast}=(-\Delta+V)^{-1/2}V^{1/2}$, respectively, which are studied by Shen [1] thoroughly. In Section 4, assume that $1< p_{1}<\infty$, $1< p_{2}<{s}/{\beta_{2}}$ and $1< q<\infty$. If the index $(q, \beta _{1},\beta_{2},\lambda,\alpha,\theta)$ satisfies

$$ \textstyle\begin{cases} {1}/{p_{2}}={1}/{p_{1}}-{2( \beta_{1}-\beta_{2})}/{n}, \\ \alpha\in(-\infty,0] \quad \mbox{and} \quad \lambda\in(0,n), \\ {\lambda}/{q}-{1}/{p_{1}}+{2\beta_{1}}/{n}< \theta< {\lambda }/{q}+1-{1}/{p_{1}}, \end{cases} $$

we prove that $T_{\beta_{1},\beta_{2}}$ is bounded from $L^{p_{1},q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})$ to $L^{p_{2},q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})$. Specially, we know that $(-\Delta+V)^{-1}V$ and $(-\Delta +V)^{-1/2}V^{1/2}$ are bounded on $L_{\alpha,\theta, V}^{p,q, \lambda }(\mathbb{R}^{n})$. See Theorems 4.7 and 4.8 for details.

In the research of harmonic analysis and partial differential equations, the commutators play an important role. If T is a Calderón-Zygmund operator, $b\in \mathit{BMO}(\mathbb{R}^{n})$, the $L^{p}$-boundedness of $[b,T ]$ was first discovered by Coifman et al. [16]. Later, Strömberg [14] gave a simple proof, adopting the idea of relating commutators with the sharp maximal operator of Fefferman and Stein. In 2008, Guo et al. [2] introduced a condition $H(m)$ and obtained $L^{p}$-boundedness of the commutator of Riesz transforms associated with L, where $b\in \mathit{BMO}(\mathbb{R}^{n})$. For further information, we refer to Liu [17], Liu et al. [4, 5], Yang et al. [8] and the references therein.

In Section 5, by the boundedness of $I_{\alpha}$ and $(-\Delta+V)^{-\beta}V^{\beta}$, we can deduce that the commutators $[b, T_{\beta_{1},\beta_{2}}]$ and $[b, T^{\ast}_{\beta_{1},\beta_{2}}]$ are bounded from $L^{p_{1}}(\mathbb{R}^{n})$ to $L^{p_{2}}(\mathbb {R}^{n})$ (see Theorem 5.1). Theorem 5.1 together with Lemmas 4.1 and 2.7 can be used to prove that the commutators $[b, T_{\beta_{1},\beta_{2}}]$ and $[b, T^{\ast}_{\beta_{1},\beta_{2}}]$ are bounded from $L_{\alpha,\theta,V}^{p_{1},q,\lambda}(\mathbb{R}^{n})$ to $L_{\alpha,\theta,V}^{p_{2},q,\lambda}(\mathbb{R}^{n})$, respectively (see Theorems 5.2 and 5.3).

Remark 1.1

Unlike the setting of the Lebesgue spaces, it is well known that the dual of $L^{p,\lambda}(\mathbb{R}^{n})$ is not $L^{p',-\lambda }(\mathbb{R}^{n})$. Hence, after obtaining Theorem 4.7, we cannot deduce Theorem 4.8 via the method of duality used by Guo et al. [2].

2 Preliminaries

2.1 Schrödinger operator and the auxiliary function

In this paper, we consider the Schrödinger differential operator $L=-\Delta +V$ on $\mathbb{R}^{n}$, $n\geq3$, where V is a nonnegative potential belonging to the reverse Hölder class $B_{s}$, $s\geq\frac {n}{2}$, which is defined as follows.

Definition 2.1

Let V be a nonnegative function.

• (i) We say $V\in B_{s}$, $s>1$, if there exists $C>0$ such that for every ball $B\subset\mathbb{R}^{n}$, the reverse Hölder inequality

  $$\biggl(\frac{1}{|B|}\int_{B}V^{s}(x)\, dx \biggr)^{\frac{1}{s}}\lesssim \biggl(\frac{1}{|B|}\int_{B}V(x) \, dx \biggr) $$

  holds.

• (ii) We say $V\in B_{\infty}$ if there exists a constant C such that for every ball $B\subset\mathbb{R}^{n}$,

  $$\|V\|_{L^{\infty}(B)}=\frac{1}{|B|}\int_{B}V(x)\, dx. $$

Remark 2.2

Assume $V\in B_{s}$, $1< s<\infty$. Then $V(y)\, dy$ is a doubling measure. Namely, there exists a constant $C_{0}$ such that for any $r>0$ and $y\in\mathbb{R}^{n}$,

$$ \int_{B(x,2r)}V(y)\, dy\lesssim C_{0} \int_{B(x,r)}V(y)\, dy. $$

(2.1)

Definition 2.3

(Shen [1])

For $x\in\mathbb{R}^{n}$, the function $m_{V}(x)$ is defined as

$$ \frac{1}{m_{V}(x)}=:\sup \biggl\{ r>0 :\frac{1}{r^{n-2}}\int_{B(x,r)}V(y) \, dy\leq1 \biggr\} . $$

Remark 2.4

The function $m_{V}$ reflects the scale of V essentially, but behaves better. It is deeply studied in Shen [1] and plays a crucial role in our proof. We list a property of $m_{V}$ which will be used in the sequel and refer the reader to Guo et al. [2] for the details.

We state some notations and properties of $m_{V}$.

Lemma 2.5

(Lemma 1.4 in [1])

Suppose that $V \in B_{s}$ with $s\geq\frac {n}{2}$. Then there exist positive constants C and $k_{0}$ such that

• (a) if $|x-y|\leq\frac{C}{m_{V}(x)}$, $m_{V}(x)\sim m_{V}(y)$;

• (b) $m_{V}(y)\lesssim(1+|x-y| m_{V}(x))^{k_{0}}m_{V}(x)$;

• (c) $m_{V}(y)\geq{Cm_{V}(x)}/\{1+|x-y|m_{V}(x)\}^{k_{0}/(k_{0}+1)}$.

Lemma 2.6

(Lemma 1.2 in [1])

Suppose that $V\in B_{s}$, $s>\frac{n}{2}$. There exists a constant C such that for $0< r< R<\infty$,

$$\frac{1}{r^{n-2}}\int_{B(x,r)}V(y)\, dy\lesssim \biggl( \frac{R}{r} \biggr)^{\frac{n}{s}-2}\cdot\frac{1}{R^{n-2}}\int _{B(x,R)}V(y)\, dy. $$

Lemma 2.7

(Lemma 2.3 in [2])

Suppose $V\in B_{s}$, $s>\frac{n}{2}$. Then, for any $N>\log_{2}C_{0}+1$, there exists a constant $C_{N}$ such that for any $x\in\mathbb{R}^{n}$ and $r>0$,

$$\frac{1}{(1+rm_{V}(x))^{N}}\int_{B(x,r)}V(y)\, dy\lesssim C_{N}r^{n-2}. $$

2.2 Generalized Morrey spaces associated with L

Suppose that $V\in B_{s}$, $s>1$. Let $L=-\Delta+V$ be the Schrödinger operator. Now we introduce a class of generalized Morrey spaces associated with L. For $k\in\mathbb{Z}$, let $E_{k}=B(x_{0},2^{k}r)\backslash B(x_{0},2^{k-1}r)$ and $\chi_{k}$ be the characteristic function of $E_{k}$.

Definition 2.8

Suppose that $V\in B_{s}$, $s>1$. Let $p\in[1,+\infty)$, $q\in [1,+\infty)$, $\alpha\in(-\infty,+\infty)$ and $\lambda\in (0,n)$, $\theta\in(-\infty,+\infty)$. For $f\in L_{\mathrm{loc}}^{q}(\mathbb {R}^{n})$, we say $f\in L_{\alpha,\theta,V}^{p,q,\lambda}(\mathbb {R}^{n})$ provided that

$$\|f\|^{q}_{L_{\alpha,\theta,V}^{p,q,\lambda}(\mathbb{R}^{n})}=\sup_{B(x_{0},r)\subset\mathbb{R}^{n}} \frac{(1+rm_{V}(x_{0}))^{\alpha }}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \|\chi_{k}f\| ^{q}_{L^{p}(\mathbb{R}^{n})}< \infty, $$

where $B(x_{0},r)$ denotes a ball centered at $x_{0}$ and with radius r.

Proposition 2.9

• (i) For $\alpha_{1}>\alpha_{2}$, $L_{\alpha_{1},\theta, V}^{p,q,\lambda}(\mathbb{R}^{n})\subseteq L_{\alpha_{2},\theta, V}^{p,\lambda,q}(\mathbb{R}^{n})$.

• (ii) If $\theta=0$, $p=q$ and $\alpha<0$, $L^{p,\lambda }(\mathbb{R}^{n}) \subset L_{\alpha,\theta,V}^{p,q, \lambda }(\mathbb{R}^{n})$.

• (iii) If $\theta=0$, $p=q$ and $\alpha>0$, $L_{\alpha,\theta ,V}^{p,q, \lambda}(\mathbb{R}^{n}) \subset L^{p,\lambda}(\mathbb{R}^{n})$.

2.3 Calderón-Zygmund operators

We say that an operator T taking $C_{c}^{\infty}(\mathbb{R}^{n})$ into $L_{\mathrm{loc}}^{1}(\mathbb {R}^{n})$ is called a Calderón-Zygmund operator if

• (a) T extends to a bounded linear operator on $L^{2}(\mathbb {R}^{n})$;

• (b) there exists a kernel K such that for every $f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})$,

  $$Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)\, dy \quad \mbox{a.e. on }\{ \operatorname{supp}f\}^{c}; $$

• (c) the kernel $K(x,y)$ satisfies the Calderón-Zygmund estimate

  $$\begin{aligned}& \bigl\vert K(x,y)\bigr\vert \leq\frac{C}{|x-y|^{n}}; \\& \bigl\vert K(x+h,y)-K(x,y)\bigr\vert +\bigl\vert K(x,y+h)-K(x,y)\bigr\vert \leq\frac{C|h|^{\delta }}{|x-y|^{n+\delta}} \end{aligned}$$

for $x,y\in\mathbb{R}^{n}$, $|h|<\frac{|x-y|}{2}$ and for some $\delta>0$.

Shen [1] obtained the following result.

Theorem 2.10

(Theorem 0.8 in [1])

Suppose $V\in B_{n}$. Then

$$\nabla(-\Delta+V)^{-1}\nabla,\qquad \nabla(-\Delta+V)^{-\frac {1}{2}} \quad \textit{and}\quad (-\Delta+V)^{-\frac{1}{2}}\nabla $$

are Calderón-Zygmund operators.

Corollary 2.11

Suppose that $V\in B_{n}$ and $b\in \mathit{BMO}(\mathbb{R}^{n})$. The commutator $[b, T]$ is bounded on $L^{p}(\mathbb{R}^{n})$.

In particular, let K denote the kernel of one of the above operators. Then K satisfies the following estimate:

$$ \bigl\vert K(x,y)\bigr\vert \leq\frac{C_{N}}{(1+| x-y| m_{V}(x))^{N}} \frac{1}{|x-y|^{n}} $$

(2.2)

for any $N\in\mathbb{N}$. See (6.5) of Shen [1] for details.

Suppose $V\in B_{s}$ for $s\geq\frac{n}{2}$. Let $L=-\Delta+V$. The semigroup generated by L is defined as

$$ T_{t}f(x)=e^{-tL}f(x)=\int _{\mathbb{R}^{n}}K_{t}(x,y)f(y)\, dy,\quad f\in L^{2}\bigl(\mathbb{R}^{n}\bigr), t>0, $$

(2.3)

where $K_{t}$ is the kernel of $e^{-tL}$.

Lemma 2.12

([18])

Let $K_{t}(x,y)$ be as in (2.3). For every nonnegative integer k, there is a constant $C_{k}$ such that

$$0\leq K_{t}(x,y)\leq C_{k}t^{-\frac{n}{2}}\exp\bigl(-{| x-y | ^{2}}/{5t}\bigr) \bigl(1+\sqrt{t} m_{V}(x)+\sqrt{t} m_{V}(y)\bigr)^{-k}. $$

Some notations

Throughout the paper, c and C will denote unspecified positive constants, possibly different at each occurrence. The constants are independent of the functions. $\mathsf{U}\approx \mathsf{V}$ represents that there is a constant $c>0$ such that $c^{-1}\mathsf{V}\le\mathsf {U}\le c\mathsf{V}$ whose right inequality is also written as $\mathsf{U}\lesssim\mathsf{V}$. Similarly, if $\mathsf{V}\ge c\mathsf{U}$, we denote $\mathsf{V}\gtrsim\mathsf{U}$.

3 Riesz transforms and the commutators on $L^{p,q,\lambda }_{\alpha,\theta,V}(\mathbb{R}^{n})$

Throughout this paper, for $p\in(1, \infty)$, denote by $p'$ the conjugate of p, that is, $\frac{1}{p}+\frac{1}{p'}=1$. Let $V\in B_{n}$. In this section, we assume that T is one of the Schrödinger type operators $\nabla(-\Delta+V)^{-1}\nabla$, $\nabla(-\Delta +V)^{-1/2}$ and $(-\Delta+V)^{-1/2}\nabla$. We study the boundedness on $L^{p,q,\lambda}_{\alpha,\theta ,V}(\mathbb{R}^{n})$ of T and its commutator $[b, T]$ with $b\in \mathit{BMO}(\mathbb{R}^{n})$. The bounded mean oscillation space $\mathit{BMO}(\mathbb {R}^{n})$ is defined as follows.

Definition 3.1

A locally integrable function b is said to belong to $\mathit{BMO}(\mathbb {R}^{n})$ if

$$ \|b\|_{\mathit{BMO}}=:\sup_{B}\frac{1}{|B|}\int _{B}\bigl\vert b(x)-b_{B}\bigr\vert \, dx< \infty, $$

where the supremum is taken over all balls B in $\mathbb{R}^{n}$. Here $b_{B}=\frac{1}{|B|}\int_{B}b(x)\, dx$ stands for the mean value of b over the ball B and $|B|$ means the measure of B.

We first prove that T is bounded on $L^{p,q,\lambda}_{\alpha,\theta ,V}(\mathbb{R}^{n})$.

Theorem 3.2

Suppose that $\alpha\in(-\infty,0]$, $\lambda\in(0,n)$ and $1< q<\infty$. If $1< p<\infty$, $\frac{\lambda}{q}-\frac{1}{p}<\theta<\frac{\lambda }{q}+1-\frac{1}{p}$, then the operators T are bounded on $L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})$.

Proof

For any ball $B(x_{0},r)$, write

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{j}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y), $$

where $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. Hence, we have

$$\begin{aligned}& {\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}} {r^{\lambda n}}\sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \|\chi_{k}Tf\|^{q}_{L^{p}(\mathbb {R}^{n})} \\& \quad \lesssim{\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}} {r^{-\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\| \chi_{k}Tf_{j}\|_{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {}+ {\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}} {r^{-\lambda n}}\sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\| \chi_{k}Tf_{j}\| _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {}+ {\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}} {r^{-\lambda n}}\sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\| \chi_{k}Tf_{j}\| _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \quad = A_{1}+A_{2}+A_{3}. \end{aligned}$$

For $A_{2}$, by Theorem 2.10, we have

$$\begin{aligned} A_{2} \lesssim& \bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|Tf_{j}\| _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}}. \end{aligned}$$

We first estimate the term $E_{1}$. Note that if $x\in E_{k}$, $y\in E_{j}$ and $j\leq k-2$, then $|x-y|\sim2^{k}r$. By Lemma 2.5 and (2.2), we can get

$$\begin{aligned} \|\chi_{k}Tf_{j}\|_{L^{p}(\mathbb{R}^{n})} \lesssim& \biggl(\int _{E_{k}}\biggl\vert \int_{\mathbb{R}^{n}} \frac {1}{(1+| x-y| m_{V}(x))^{N}}\frac{1}{|x-y|^{n}}\bigl\vert f_{j}(y)\bigr\vert \, dy\biggr\vert ^{p}\, dx \biggr)^{\frac{1}{p}} \\ \lesssim&\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n}}|E_{k}|^{\frac{1}{p}} \int_{E_{j}}\bigl\vert f(y)\bigr\vert \, dy \\ \lesssim&\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}|E_{k}|^{\frac {1}{p}-1}|E_{j}|^{\frac{1}{p'}} \biggl(\int_{E_{j}}\bigl\vert f(y)\bigr\vert ^{p} \, dy \biggr)^{\frac{1}{p}}, \end{aligned}$$

where $\frac{1}{p}+\frac{1}{p'}=1$. Since $- \frac{1}{p}+ \frac{\lambda}{q} <\theta<(1- \frac{1}{p})+ \frac{\lambda}{q}$, we obtain

$$\begin{aligned} A_{1} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\frac {|E_{k}|^{\frac{1}{p}-1}|E_{j}|^{\frac{1}{p'}}\|\chi_{j}f\| _{L^{p}(\mathbb{R}^{n})}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\frac {2^{\frac{n(j-k)}{p'}}(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}} \\ &\Biggl.\Biggl.{}\times\bigl(2^{j}r\bigr)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta } \bigl(1+2^{j}rm_{V}(x_{0})\bigr)^{\frac{\alpha}{q}} \bigl(2^{j}r\bigr)^{-\frac{\lambda n}{q}}\bigl(|E_{j}|^{\theta q} \|\chi_{j}f\|^{q}_{L^{p}(\mathbb {R}^{n})}\bigr)^{\frac{1}{q}} \Biggr)\Biggr.^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{k-2}_{j=-\infty }2^{\frac{n(j-k)}{p'}}|E_{k}|^{\theta-\frac{\lambda }{q}}|E_{j}|^{\frac{\lambda}{q}-\theta} \Biggr)^{q}\|f\| ^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{k-2}_{j=-\infty }2^{(j-k)n(1-\frac{1}{p}+\frac{\lambda}{q}-\theta)} \Biggr)^{q}\|f\| ^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})} \\ \lesssim&\|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})}. \end{aligned}$$

For $A_{3}$, we can see that when $x\in E_{k}$, $y\in E_{j}$, then $|x-y|\sim2^{j}r$ for $j\geq k+2$. Similar to $E_{1}$, we have

$$\begin{aligned} \|\chi_{k}Tf_{j}\|_{L^{p}(\mathbb{R}^{n})} \lesssim& \frac {1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n}}|E_{k}|^{\frac{1}{p}}\int _{E_{j}}\bigl\vert f(y)\bigr\vert \,dy \\ \lesssim&\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n}}|E_{k}|^{\frac{1}{p}}|E_{j}|^{\frac{1}{p'}} \biggl(\int_{E_{j}}\bigl\vert f(y)\bigr\vert ^{p} \,dy \biggr)^{\frac{1}{p}} \\ \lesssim&\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}|E_{k}|^{\frac {1}{p}}|E_{j}|^{-\frac{1}{p}} \|\chi_{j}f\|_{L^{p}(\mathbb{R}^{n})}. \end{aligned}$$

Since $-\frac{1}{p}+\frac{\lambda}{q}<\theta<(1-\frac{1}{p})+\frac {\lambda}{q}$, choosing N large enough, we obtain

$$\begin{aligned} A_{3} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\frac {|E_{k}|^{\frac{1}{p}}|E_{j}|^{-\frac{1}{p}}\|\chi_{j}f\| _{L^{p}(\mathbb{R}^{n})}}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times\Biggl\{ \sum^{\infty}_{j=k+2}\frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha}{q}}(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\alpha}}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}} \\ &\Biggl.\Biggl.{}\times2^{(k-j)\frac{n}{p}}\bigl(1+2^{j}rm_{V}(x_{0}) \bigr)^{\frac{\alpha }{q}}\bigl(2^{j}r\bigr)^{-\frac{\lambda n}{q}} \bigl(|E_{j}|^{\theta q}\|\chi_{j}f\| ^{q}_{L^{p}(\mathbb{R}^{n})}\bigr)^{\frac{1}{q}} \Biggr\} \Biggr.^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{\infty }_{j=k+2}2^{(k-j)\frac{n}{p}}|E_{j}|^{\frac{\lambda}{q}-\theta } \Biggr)^{q}\|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb {R}^{n})} \\ \lesssim&\|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})}. \end{aligned}$$

Let $N=[-\frac{\alpha}{q}+1](k_{0}+1)$. Finally, $\|Tf\|_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb {R}^{n})}$. This completes the proof of Theorem 3.2. □

Suppose that $b\in \mathit{BMO}(\mathbb{R}^{n})$ and $V\in B_{n}$. Let T be one of the Schrödinger type operators $\nabla(-\Delta+V)^{-1}\nabla$, $\nabla(-\Delta+V)^{-1/2}$ and $(-\Delta+V)^{-1/2}\nabla$. The commutator $[b, T]$ is defined as

$$[b,T]f=bT(f)-T(bf). $$

Theorem 3.3

Suppose that $V\in B_{n}$ and $b\in \mathit{BMO}(\mathbb{R}^{n})$. Let $1< p<\infty$, $1< q<\infty$, $\alpha\in(-\infty, 0]$, $\lambda\in(0,n)$. If the index $(p,q,\theta,\lambda)$ satisfies $\frac{\lambda }{q}-\frac{1}{p}<\theta<\frac{\lambda}{q}+1-\frac{1}{p}$, then

$$\bigl\Vert [b,T]f\bigr\Vert _{L^{p,q,\lambda}_{\alpha,\theta,V}}\leq C\|f\| _{L^{p,q,\lambda}_{\alpha,\theta,V}}\|b \|_{\mathit{BMO}}. $$

Proof

For any ball $B=B(x_{0},r)$, we can get

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y), $$

where $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. Hence, we have

$$\begin{aligned}& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \bigl\Vert \chi_{k}[b,T]f\bigr\Vert ^{q}_{L^{p}(\mathbb{R}^{n})} \\& \quad \lesssim\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\bigl\Vert \chi_{k}[b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {} +\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi_{k}[b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {} +\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\bigl\Vert \chi_{k}[b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\& \quad =: B_{1}+B_{2}+B_{3}. \end{aligned}$$

For $B_{2}$, by Corollary 2.11, we have

$$\begin{aligned} B_{2} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert [b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p}(\mathbb{R}^{n})} \Biggr)^{q}\|b\|^{q}_{\mathit{BMO}} \\ \lesssim& \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

Denote by $b_{2^{k}r}$ the mean value of b on the ball $B(x_{0}, 2^{k}r)$. For $B_{1}$, by Lemma 2.5 and (2.2), we have

$$\begin{aligned}& \bigl\Vert \chi_{k}[b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n}} \\& \qquad {}\times\biggl[\int _{E_{k}} \biggl(\int_{E_{j}}\bigl\vert b(x)-b(y)\bigr\vert \bigl\vert f(y)\bigr\vert \,dy \biggr)^{p} \,dx \biggr]^{\frac{1}{p}} \\& \quad \lesssim \frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n}} \biggl[ \biggl(\int _{E_{k}}\bigl\vert b(x)-b_{2^{k}r}\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}}\int_{E_{j}}\bigl\vert f(y)\bigr\vert \,dy \\& \qquad {}+|E_{k}|^{\frac{1}{p}}\int_{E_{j}}\bigl\vert b(y)-b_{2^{k}r}\bigr\vert \bigl\vert f(y)\bigr\vert \,dy \biggr] \\& \quad \lesssim \frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n}} \biggl[|E_{k}|^{\frac{1}{p}}|E_{j}|^{1-\frac{1}{p}} \| b\|_{\mathit{BMO}}\|f_{j}\|_{L^{p}(\mathbb{R}^{n})} \\& \qquad {} +|E_{k}|^{\frac{1}{p}}\|f_{j}\|_{L^{p}(\mathbb{R}^{n})} \biggl(\int_{E_{j}}\bigl\vert b(y)-b_{2^{k}r}\bigr\vert ^{p'}\,dx \biggr)^{\frac{1}{p'}} \biggr] \\& \quad \lesssim \frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac{|E_{j}|^{1-\frac {1}{p}}}{|E_{k}|^{1-\frac{1}{p}}}(k-j)\|f_{j} \|_{L^{p}(\mathbb {R}^{n})}\|b\|_{\mathit{BMO}}, \end{aligned}$$

where in the third inequality, we have used John-Nirenberg’s inequality [19]. Since $- \frac{1}{p}+ \frac{\lambda}{q} <\theta<(1- \frac{1}{p})+ \frac{\lambda}{q}$, we obtain

$$\begin{aligned} B_{1} \lesssim& \frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\frac {(k-j)\|f_{j}\|_{L^{p}(\mathbb{R}^{n})}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}} \frac{|E_{j}|^{1-\frac{1}{p}}}{|E_{k}|^{1-\frac{1}{p}}} \Biggr)^{q}\| b\|^{q}_{\mathit{BMO}} \\ \lesssim& \frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl[\sum^{k-2}_{j=-\infty}\frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\|b \|^{q}_{\mathit{BMO}} \\ &\Biggl.\Biggl.{}\times(k-j) \bigl(2^{j}r\bigr)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta} \frac {|E_{j}|^{1-\frac{1}{p}}}{|E_{k}|^{1-\frac{1}{p}}} \Biggr]\Biggr.^{q}\|f\| ^{q}_{L_{\alpha,\theta,V}^{p,q,\lambda}} \\ \lesssim& \frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{k-2}_{j=-\infty }(k-j)2^{(k-j)n(\theta-\frac{\alpha}{q}+\frac{1}{p}-1)} \Biggr)^{q} \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}}\|b \|^{q}_{\mathit{BMO}} \\ \lesssim& \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

For $B_{3}$, similar to $B_{1}$, we have

$$\begin{aligned}& \bigl\Vert \chi_{k}[b,T]f_{j}\bigr\Vert _{L^{p}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n}} \biggl(\int _{E_{k}}\biggl\vert \int_{E_{j}}\bigl\vert \bigl(b(x)-b(y)\bigr)f(y)\bigr\vert \,dy\biggr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \\& \quad \lesssim\frac {j-k}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}|E_{k}|^{\frac {1}{p}}|E_{j}|^{-\frac{1}{p}} \|f_{j}\|_{L^{p}(\mathbb{R}^{n})}\|b\|_{\mathit{BMO}}. \end{aligned}$$

Since $-\frac{1}{p}+\frac{\lambda}{q}<\theta<(1-\frac{1}{p})+\frac {\lambda}{q}$, choosing N large enough, we obtain

$$\begin{aligned} B_{3} \lesssim& \bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\frac {|E_{k}|^{\frac{1}{p}}|E_{j}|^{-\frac{1}{p}}(j-k)\|f_{j}\|_{L^{p} (\mathbb{R}^{n})}}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}} \Biggr)^{q}\|b\| ^{q}_{\mathit{BMO}} \\ \lesssim& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl[\sum^{\infty}_{j=k+2}\frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}} \\ &\Biggl.\Biggl.{}\times(j-k) \bigl(2^{j}r\bigr)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta }|E_{k}|^{\frac{1}{p}}|E_{j}|^{-\frac{1}{p}} \Biggr]\Biggr.^{q}\|f\| ^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})}\|b\| ^{q}_{\mathit{BMO}} \\ \lesssim& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{\infty }_{j=k+2}2^{(k-j)n(\frac{1}{p}-\frac{\lambda}{q}+\theta)} \Biggr)^{q} \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})}\|b\| ^{q}_{\mathit{BMO}} \\ \lesssim& \|f\|^{q}_{L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb {R}^{n})}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

Let $N=[-\frac{\alpha}{q}+1](k_{0}+1)$. We finally get

$$\bigl\Vert [b,T]f\bigr\Vert _{L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb {R}^{n})}\|b \|_{\mathit{BMO}}. $$

 □

4 Schrödinger type operators on $L^{p,q,\lambda}_{\alpha ,\theta,V}(\mathbb{R}^{n})$

Let $L=-\Delta+V$ be the Schrödinger operator, where $V\in B_{s}$, $s>n/2$. For $0<\beta<\frac{n}{2}$, the fractional integral operator associated with L is defined by

$$L^{-\beta}(f) (x)=\int^{\infty}_{0}e^{-tL}(f) (x)t^{\beta-1}\, dt. $$

Denote by $K_{\beta}(x,y)$ the kernel of $L^{-\beta}$. By Lemma 2.12, Bui [20] obtained the following pointwise estimate.

Lemma 4.1

(Proposition 3.3 in [20])

Let $0<\beta<\frac{n}{2}$. For $N\in\mathbb{N}$, there is a constant $C_{N}$ such that

$$\begin{aligned} K_{\beta}(x,y) =&\int_{0}^{\infty}K_{t}(x,y)t^{\beta-1} \, dt \\ \leq&\frac {C_{N}}{(1+| x-y| m_{V}(x))^{N}}\frac{1}{|x-y|^{n-2\beta}}, \end{aligned}$$

(4.1)

where $K_{t}(\cdot, \cdot)$ is the kernel of the semigroup $e^{-tL}$.

Definition 4.2

Let $f\in L_{\mathrm{loc}}^{q}(\mathbb{R}^{n})$. Denote by $|B|$ the Lebesgue measure of the ball $B\subset\mathbb{R}^{n}$. The fractional Hardy-Littlewood maximal function $M_{\sigma,\gamma}$ is defined by

$$M_{\sigma,\gamma}f(x)=\sup_{x\in B} \biggl(\frac{1}{|B|^{1-\frac {\sigma\gamma}{n}}} \int_{B}\bigl\vert f(y)\bigr\vert ^{\gamma}\,dy \biggr)^{\frac {1}{\gamma}}. $$

Lemma 4.3

([16])

Suppose $1<\gamma<p_{1}<\frac{n}{\sigma}$ and $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{\sigma}{n}$. Then

$$\|M_{\sigma,\gamma}f\|_{L^{p_{2}}(\mathbb{R}^{n})}\lesssim\|f\| _{L^{p_{1}}(\mathbb{R}^{n})}. $$

As a generalization of the fractional integral associated with L, the operators $V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}$, $0\leq\beta _{2}\leq\beta_{1}\leq1$, have been studied by Sugano [9] systematically. Applying the method of Sugano [9] together with Lemma 4.1, we can obtain the following result for $V^{\beta _{2}}(-\Delta+V)^{-\beta_{1}}$, $0\leq\beta_{2}\leq\beta_{1}\leq n/2$. We omit the proof.

Theorem 4.4

Suppose that $V\in B_{\infty}$. Let $1<\beta_{2}\leq\beta_{1}<\frac {n}{2}$. Then

$$\bigl\vert V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f(x)\bigr\vert \lesssim M_{2(\beta _{1}-\beta_{2}),1}f(x). $$

In a similar way, by (4.1), we can get the following estimate for the operators $(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}$, $0\leq\beta_{2}\leq\beta_{1}<\frac{n}{2}$.

Theorem 4.5

Suppose that $V\in B_{s}$ for $s>\frac{n}{2}$. Let $0\leq\beta _{2}\leq\beta_{1}<\frac{n}{2}$. Then

$$\bigl\vert (-\Delta+V)^{-\beta_{1}}\bigl(V^{\beta_{2}}f\bigr) (x)\bigr\vert \lesssim M_{2(\beta_{1}-\beta_{2})}(f)(x), $$

where $(\frac{s}{\beta_{2}})'$ is the conjugate of $(\frac{s}{\beta_{2}})$.

Proof

Let $r={1}/{m_{V}(x)}$. By Lemma 4.1 and Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned} &\bigl\vert (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}(x)f(x)\bigr\vert \\ &\quad \lesssim\sum^{\infty}_{k=-\infty}\int _{2^{k-1}r \leq|x-y|\leq 2^{k}r}\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}}V(y)^{\beta_{2}} \bigl\vert f(y)\bigr\vert \,dy \\ &\quad \lesssim\sum^{\infty}_{k=-\infty} \frac{(2^{k}r)^{2\beta _{2}}}{(1+2^{k})^{N}} \biggl(\frac{1}{(2^{k}r)^{n}}\int_{B(x,2^{k}r)}V(y)\,dy \biggr)^{\beta _{2}}M_{2(\beta_{1}-\beta_{2}),(\frac{s}{\beta_{2}})'}(f) (x). \end{aligned} \end{aligned}$$

For $k\geq1$, because $V(y)\,dy$ is a doubling measure, we have

$$\begin{aligned} \frac{(2^{k}r)^{2}}{(2^{k}r)^{n}}\int_{B(x,2^{k}r)}V(y)\,dy \lesssim& C_{0}^{k}\cdot2^{(2-n)k} \frac{r^{2}}{r^{n}}\int _{B(x,r)}V(y)\,dy \\ \lesssim&\bigl(2^{k}\bigr)^{k_{0}}, \end{aligned}$$

where $k_{0}=2-n+\log_{2}C_{0}$. For $k\leq0$, Lemma 2.6 implies that

$$\begin{aligned} \frac{(2^{k}r)^{2}}{(2^{k}r)^{n}}\int_{B(x,2^{k}r)}V(y)\,dy \lesssim & \biggl( \frac{r}{2^{k}r} \biggr)^{\frac{n}{s}-2}\frac{r^{2}}{r^{n}}\int _{B(x,r)}V(y)\,dy \\ \lesssim&\bigl(2^{k}\bigr)^{2-\frac{n}{s}}. \end{aligned}$$

Taking N large enough, we get

$$\bigl\vert (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f(x)\bigr\vert \lesssim M_{2(\beta _{1}-\beta_{2}),(\frac{s}{\beta_{2}})'}f(x). $$

 □

By Theorem 4.5 and the duality, we can obtain the following.

Corollary 4.6

Suppose $V\in B_{s}$ for $s>\frac{n}{2}$.

• (1) If $1<(\frac{s}{\beta_{2}})'<p_{1}<\frac{n}{2\beta _{1}-2\beta_{2}}$ and $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta _{1}-2\beta_{2}}{n}$, then

  $$\bigl\Vert (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}, $$

  where $\frac{s}{\beta_{2}}+(\frac{s}{\beta_{2}})'=1$.

• (2) If $1< p_{2}<\frac{s}{\beta_{2}}$ and $\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}$, then

  $$\bigl\Vert V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}. $$

Theorem 4.7

Suppose that $V\in B_{s}$, $s\geq\frac{n}{2}$, $\alpha\in(-\infty ,0]$, $\lambda\in(0,n)$. Let $1< q<\infty$, $1<\beta_{2}\leq\beta_{1}<\frac{n}{2}$ and $1< p_{2}<\frac{s}{\beta _{2}}$ with $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}$. If $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac {2\beta_{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, then

$$\bigl\Vert V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta, V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta, V}}. $$

Proof

For any ball $B(x_{0},r)$, write

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y), $$

where $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. Hence, we have

$$\begin{aligned}& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \bigl\Vert \chi_{k}V^{\beta_{2}}(-\Delta+V)^{-\beta _{1}}f\bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\ & \quad \lesssim\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\bigl\Vert \chi_{k}V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\ & \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi_{k}V^{\beta _{2}}(-\Delta+V)^{-\beta_{1}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})} \Biggr)^{q} \\ & \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\bigl\Vert \chi_{k}V^{\beta _{2}}(-\Delta+V)^{-\beta_{1}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})} \Biggr)^{q} \\ & \quad =M_{1}+M_{2}+M_{3}. \end{aligned}$$

We first estimate $M_{2}$. For $1< p_{2}<\frac{s}{\beta_{2}}$, by (2) of Corollary 4.6, we can get

$$ M_{2}\lesssim\frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \lesssim\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. $$

Now we deal with the terms $M_{1}$ and $M_{3}$. We choose N large enough such that

$$(N/k_{0}+1)-(\log_{2}C_{0}+1) \beta_{2}+{\alpha}/{q}>0 $$

and take positive $N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. For $M_{1}$, note that if $x\in E_{k}$, $y\in E_{j}$ and $j\leq k-2$, then $|x-y|\sim2^{k}r$. By Lemmas 4.1 and 2.7, we use Hölder’s inequality to obtain

$$\begin{aligned}& \bigl\Vert \chi_{k}V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f_{j} \bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\ & \quad \lesssim \biggl(\int_{E_{k}}\biggl\vert V^{\beta_{2}}(x)\int_{E_{j}}\frac {1}{(1+| x-y| m_{v}(x))^{N}} \frac{1}{|x-y|^{n-2{\beta _{1}}}}f(y)\,dy\biggr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\ & \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}}\int_{E_{j}} \bigl\vert f(y)\bigr\vert \,dy \biggl(\int_{E_{k}}\bigl\vert V(x)\bigr\vert ^{\beta_{2} p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\ & \quad \lesssim\frac{|E_{j}|^{1-\frac{1}{p_{1}}}|E_{k}|^{\frac {1}{p_{2}}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}}\|f_{j} \|_{L^{p_{1}}(\mathbb{R}^{n})} \biggl(\frac{1}{|E_{k}|}\int_{E_{k}}V(x)^{s} \,dx \biggr)^{\frac{\beta _{2}}{s}} \\ & \quad \lesssim\frac{|E_{j}|^{1-\frac{1}{p_{1}}}|E_{k}|^{\frac {1}{p_{2}}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}}\|f_{j} \|_{L^{p_{1}}(\mathbb{R}^{n})} \biggl(\frac{1}{|B_{k}|}\int_{B_{k}}V(x)\,dx \biggr)^{\beta_{2}} \\ & \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac {1}{(2^{k}r)^{n-2\beta_{1}+2\beta_{2}}}|E_{k}|^{\frac {1}{p_{2}}}|E_{j}|^{1-\frac{1}{p_{1}}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}$. Since $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, we obtain

$$\begin{aligned} M_{1} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {1}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac{1}{(2^{k}r)^{n-2\beta _{1}+2\beta_{2}}}|E_{k}|^{\frac{1}{p_{2}}}|E_{j}|^{1-\frac {1}{p_{1}}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac{(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta}}{(2^{k}r)^{n-2\beta_{1}+2\beta _{2}}}|E_{k}|^{\frac{1}{p_{2}}}|E_{j}|^{1-\frac{1}{p_{1}}} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},\lambda,q}_{\alpha,v,\theta}} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{k-2}_{j=-\infty }2^{(j-k)n(\frac{\lambda}{q}-\theta-\frac{1}{p_{1}}+1)} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},\lambda,q}_{\alpha,v,\theta}} \\ \lesssim&\|f\|^{q}_{L^{p_{1},\lambda,q}_{\alpha,v,\theta}}. \end{aligned}$$

For $M_{3}$, note that when $x\in E_{k}$, $y\in E_{j}$ and $j\geq k+2$, then $|x-y|\sim2^{j}r$. Similar to $E_{1}$, we have

$$\begin{aligned}& \bigl\Vert \chi_{k}V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f_{j} \bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n-2{\beta_{1}}}}\int_{E_{j}} \bigl\vert f(y)\bigr\vert \,dy \biggl(\int_{E_{k}}\bigl\vert V(x)\bigr\vert ^{\beta_{2} p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}}|E_{j}|^{\frac {2\beta_{1}}{n}-\frac{1}{p_{1}}}|E_{k}|^{\frac{1}{p_{2}}-\frac {2\beta_{2}}{n}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}$. Since $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, we obtain

$$\begin{aligned} M_{3} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {1}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}}|E_{j}|^{\frac{2\beta _{1}}{n}-\frac{1}{p_{1}}}|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta _{2}}{n}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha}{q}}(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta}}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}} \frac{|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta _{2}}{n}}}{|E_{j}|^{\frac{2\beta_{1}}{n}-\frac{1}{p_{1}}}} \Biggr)^{q} \|f \|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{\infty }_{j=k+2}2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac{1}{p_{1}}+\frac {2\beta_{1}}{n})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim&\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. \end{aligned}$$

Choosing N large enough, we obtain

$$\bigl\Vert V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta,V}} \lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. $$

 □

Theorem 4.8

Suppose that $V\in B_{s}$, $s\geq\frac{n}{2}$, $\alpha\in(-\infty ,0]$, $\lambda\in(0,n)$ and $1< q<\infty$. Let $0<\beta_{2}\leq\beta_{1}<\frac{n}{2}$, $\frac{s}{s-\beta _{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}$ with $\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}$. If $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, then

$$\bigl\Vert (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta,V}} \lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. $$

Proof

For any ball $B(x_{0}, r)$, let $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. We can decompose f as follows:

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y). $$

Similar to the proof of Theorem 4.7, we have

$$\begin{aligned}& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \bigl\Vert \chi_{k}(-\Delta+V)^{-\beta_{1}}V^{\beta _{2}}f\bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty}\bigl\Vert \chi_{k}(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {}+C\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi_{k}(-\Delta +V)^{-\beta_{1}}V^{\beta_{2}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})} \Biggr)^{q} \\& \qquad {}+C\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\bigl\Vert \chi_{k}(-\Delta +V)^{-\beta_{1}}V^{\beta_{2}}f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})} \Biggr)^{q} \\& \quad =L_{1}+L_{2}+L_{3}. \end{aligned}$$

For $L_{2}$, because $1<\frac{s}{s-\beta_{2}}<p_{1}<\frac{n}{2\beta _{1}-\beta_{2}}$, we use Corollary 4.6 to obtain

$$ L_{2}\lesssim\frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \lesssim\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. $$

For $L_{1}$, we can see that if $x\in E_{k}$ and $y\in E_{j}$, then $|x-y|\sim2^{k}r$ for $j\leq k-2$. By Hölder’s inequality and the fact that $V\in B_{s}$, we deduce from Lemmas 4.1 and 2.7 that

$$\begin{aligned}& \bigl\Vert \chi_{k}(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f_{j} \bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {|E_{k}|^{\frac{1}{p_{2}}}}{(2^{k}r)^{n-2{\beta_{1}}}}\int_{E_{j}}V(x)^{\beta_{2}} \bigl\vert f(y)\bigr\vert \,dy \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {|E_{k}|^{\frac{1}{p_{2}}}}{(2^{k}r)^{n-2{\beta _{1}}}}|E_{j}|^{1-\frac{1}{p_{1}}} \biggl(\frac{1}{|B_{j}|}\int_{B_{j}}V(x)\,dx \biggr)^{\beta_{2}}\| f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}}\frac {|E_{k}|^{\frac{1}{p_{2}}}}{(2^{k}r)^{n-2{\beta_{1}}}} |E_{j}|^{1-\frac{1}{p_{1}}} \bigl(2^{j}r\bigr)^{-2\beta_{2}}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}$ and $N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. Since $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, we obtain

$$\begin{aligned} L_{1} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {1}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}} \frac{|E_{k}|^{\frac{1}{p_{2}}}}{(2^{k}r)^{n-2{\beta _{1}}}}|E_{j}|^{1-\frac{1}{p_{1}}} \bigl(2^{j}r\bigr)^{-2\beta_{2}}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac{(1+2^{j}rm_{V}(x_{0}))^{-\frac {\alpha}{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}} \frac{(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta }}{(2^{k}r)^{n-2{\beta_{1}}}}\frac{|E_{k}|^{\frac {1}{p_{2}}}|E_{j}|^{1-\frac{1}{p_{1}}}}{(2^{j}r)^{2\beta_{2}}} \Biggr)^{q}\|f \|^{q}_{L^{p_{1},\lambda,q}_{\alpha,V,\theta}} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{k-2}_{j=-\infty }2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac{1}{p_{2}}-1+\frac{2\beta _{1}}{n})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},\lambda,q}_{\alpha,\theta,V}} \\ \lesssim&\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha, V,\theta}}. \end{aligned}$$

For $L_{3}$, note that when $x\in E_{k}$, $y\in E_{j}$ and $j\geq k+2$, then $|x-y|\sim2^{j}r$. Similar to $E_{1}$, we have

$$\begin{aligned}& \bigl\Vert \chi_{k}(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f_{j} \bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {|E_{k}|^{\frac{1}{p_{2}}}}{(2^{j}r)^{n-2{\beta_{1}}}}\int_{E_{j}}V(x)^{\beta_{2}} \bigl\vert f(y)\bigr\vert \,dy \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N_{2}}}\frac {|E_{k}|^{\frac{1}{p_{2}}}}{(2^{j}r)^{n-2{\beta _{1}}}}|E_{j}|^{1-\frac{1}{p_{1}}} \bigl(2^{j}r\bigr)^{-2\beta_{2}} \|f_{j} \|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}$ and $N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. Since $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda }{q}-\frac{1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, we obtain

$$\begin{aligned} L_{3} \lesssim&\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {1}{(1+2^{j}rm_{V}(x_{0}))^{N_{2}}}\frac{|E_{k}|^{\frac {1}{p_{2}}}}{(2^{j}r)^{n-2{\beta_{1}}}}|E_{j}|^{1-\frac {1}{p_{1}}} \bigl(2^{j}r\bigr)^{-2\beta_{2}}\|f_{j}\|_{L^{p_{1}}(\mathbb {R}^{n})} \Biggr)^{q} \\ \lesssim&\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{\infty }_{j=k+2}2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac{1}{p_{2}})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim&\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}. \end{aligned}$$

Let N be large enough. We finally get $\|(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f\|_{L^{p_{2},q,\lambda }_{\alpha,\theta,V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta,V}}$. □

5 Boundedness of the commutators on $L^{p,q,\lambda}_{\alpha ,\theta,V}(\mathbb{R}^{n})$

In this section, let $b\in \mathit{BMO}(\mathbb{R}^{n})$. We consider the boundedness of commutators $[b, (-\Delta+V)^{-\beta_{1}}V^{\beta _{2}}]$ and its duality on the generalized Morrey spaces $L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})$. For this purpose, we prove the commutator $[b, (-\Delta+V)^{-\beta_{1}}V^{\beta _{2}}]$ is bounded from $L^{p_{1}}(\mathbb{R}^{n})$ to $L^{p_{2}}(\mathbb{R}^{n})$. For the sake of simplicity, we denote by $b_{2^{k}r}$ the mean value of b on the ball $B(x_{0}, 2^{k}r)$.

Theorem 5.1

Suppose that $V\in B_{s}$, $s\geq\frac{n}{2}$ and $b\in \mathit{BMO}(\mathbb{R}^{n})$.

• (i) If $0<\beta_{2}\leq\beta_{1}<\frac{n}{2}$, $\frac {s}{s-\beta_{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}$, $\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}$, then

  $$\bigl\Vert \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb {R}^{n})}\|b\|_{\mathit{BMO}}. $$

• (ii) If $1< p_{2}<\frac{s}{\beta_{2}}$ and $\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}$, then

  $$\bigl\Vert \bigl[b, V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}\bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}\|b\|_{\mathit{BMO}}. $$

Proof

We only prove (i). (ii) can be obtained by duality. Because $\beta _{2}\leq\beta_{1}$, we can decompose the operator $(-\Delta +V)^{-\beta_{1}}V^{\beta_{2}}$ as

$$(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}=(-\Delta+V)^{\beta_{2}-\beta _{1}}(- \Delta+V)^{-\beta_{2}}V^{\beta_{2}}. $$

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

$$\begin{aligned}& \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f(x) \\& \quad =\bigl[b, (-\Delta+V)^{\beta_{2}-\beta_{1}}(-\Delta+V)^{-\beta _{2}}V^{\beta_{2}} \bigr]f(x) \\& \quad = bL^{\beta_{2}-\beta_{1}}T_{\beta_{2}}f(x)-L^{\beta_{2}-\beta _{1}}T_{\beta_{2}}(b f) (x) \\& \quad =bL^{\beta_{2}-\beta_{1}}T_{\beta_{2}}f(x)-L^{\beta_{2}-\beta _{1}} \bigl(bT_{\beta_{2}}f(x)\bigr) \\& \qquad {}+L^{\beta_{2}-\beta_{1}}\bigl(bT_{\beta_{2}}f(x)\bigr) -L^{\beta_{2}-\beta_{1}}T_{\beta_{2}}(b f) (x) \\& \quad =\bigl[b, L^{\beta_{2}-\beta_{1}}\bigr]T_{\beta_{2}}f(x)+L^{\beta_{2}-\beta _{1}}[b, T_{\beta_{2}}]f(x). \end{aligned}$$

By (1) of Corollary 4.6, we can get

$$\begin{aligned}& \bigl\Vert \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim \bigl\Vert \bigl[b, L^{\beta_{2}-\beta_{1}} \bigr]T_{\beta _{2}}f\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})}+\bigl\Vert L^{\beta_{2}-\beta _{1}} [b, T_{\beta_{2}} ]f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})} \\& \quad \lesssim \Vert T_{\beta_{2}}f\Vert _{L^{p_{1}}(\mathbb {R}^{n})}+\bigl\Vert [b, T_{\beta_{2}} ]f\bigr\Vert _{L^{p_{1}}(\mathbb{R}^{n})} \\& \quad \lesssim \|f\|_{L^{p_{1}}(\mathbb{R}^{n})}. \end{aligned}$$

This completes the proof. □

In the rest of this section, we prove the boundedness of the commutators $[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}]$ and $[b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}]$ on $L^{p_{2},q,\lambda }_{\alpha,\theta,V}(\mathbb{R}^{n})$, respectively.

Theorem 5.2

Suppose that $V\in B_{s}$, $s\geq\frac{n}{2}$, $\alpha\in(-\infty,0]$ and $\lambda\in(0,n)$. Let $1< q<\infty$, $1<\beta_{2}\leq\beta_{1}<\frac{n}{2}$ and $1< p_{2}<\frac{s}{\beta _{2}}$ with $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}$. If $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta_{1}}{n}<\theta <\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, then for $b\in \mathit{BMO}(\mathbb{R}^{n})$,

$$\bigl\Vert \bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}\bigr]f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta,V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta,V}}\|b\|_{\mathit{BMO}}. $$

Proof

For any ball $B(x_{0},r)$, we have

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y), $$

where $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. Hence, we have

$$\begin{aligned}& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \bigl\Vert \chi_{k} \bigl[b,V^{\beta_{2}}(-\Delta +V)^{-\beta_{1}} \bigr]f\bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\ & \quad \lesssim\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty} \bigl\Vert \chi_{k} \bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\ & \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi_{k} \bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\ & \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\bigl\Vert \chi_{k} \bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\ & \quad =:D_{1}+D_{2}+D_{3}. \end{aligned}$$

For $D_{2}$, by (ii) of Theorem 5.1, we have

$$\begin{aligned} D_{2} \lesssim& \bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q}\|b\|^{q}_{\mathit{BMO}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

For $D_{1}$, by Lemmas 2.7 and 4.1, we obtain

$$\begin{aligned}& \bigl\Vert \chi_{k}\bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\ & \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}} \biggl(\int _{E_{k}}\biggl\vert \int_{E_{j}}V^{\beta _{2}}(x) \bigl(b(x)-b(y)\bigr)f(y)\,dy\biggr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\ & \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}} \biggl[ \biggl(\int _{E_{k}}V^{\beta _{2}p_{2}}(x)\bigl\vert b(x)-b_{2^{k}r} \bigr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}}\int_{E_{j}} \bigl\vert f(y)\bigr\vert \,dy \\ & \qquad {}+ \biggl(\int_{E_{k}}V^{\beta_{2}p_{2}}(x)\,dx \biggr)^{\frac {1}{p_{2}}}\int_{E_{j}}\bigl\vert b(y)-b_{2^{k}r}\bigr\vert \bigl\vert f(y)\bigr\vert \,dy \biggr] \\ & \quad \lesssim\frac{\|b\|_{\mathit{BMO}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}} \biggl[ \biggl(\int _{E_{k}}V(x)\,dx \biggr)^{\beta_{2}}|E_{k}|^{\frac {1}{p_{2}}-\beta_{2}} \int_{E_{j}}\bigl\vert f(y)\bigr\vert \,dy \\ & \qquad {}+ \biggl(\int_{E_{k}}V(x)\,dx \biggr)^{\beta_{2}}|E_{k}|^{\frac {1}{p_{2}}-\beta_{2}}|E_{j}|^{1-\frac{1}{p_{1}}}(k-j) \|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \biggr] \\ & \quad \lesssim\frac{\|b\|_{\mathit{BMO}}}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac {k-j}{(2^{k}r)^{n-2{\beta_{1}}}}|E_{k}|^{\frac{1}{p_{2}}-\frac {2\beta_{2}}{n}}|E_{j}|^{1-\frac{1}{p_{1}}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}$ and $N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. Since $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta_{1}}{n}<\theta <\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, we obtain

$$\begin{aligned} D_{1} \lesssim& \|b\|^{q}_{\mathit{BMO}} \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {1}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac{k-j}{(2^{k}r)^{n-2{\beta _{1}}}}|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta _{2}}{n}}|E_{j}|^{1-\frac{1}{p_{1}}} \|f_{j}\|_{L^{p_{1}}(\mathbb {R}^{n})} \Biggr)^{q} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N_{1}}}\frac{(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta}}{(2^{k}r)^{n-2\beta_{1}}}\frac {|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta_{2}}{n}}}{|E_{j}|^{\frac {1}{p_{1}}-1}}(k-j) \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\frac{(1+rm_{V}(x_{0}))^{\alpha }}{r^{\lambda n}}\sum ^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl( \sum^{k-2}_{j=-\infty}(k-j)2^{(j-k)n(\frac{\lambda}{q}-\theta-\frac {1}{p_{1}}+1)} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

For $D_{3}$, because $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}$ and $N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta _{2}$, we have

$$\begin{aligned}& \bigl\Vert \chi_{k}\bigl[b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n-2{\beta_{1}}}} \biggl(\int _{E_{k}}\biggl\vert \int_{E_{j}}V(x)^{\beta_{2} } \bigl(b(x)-b(y)\bigr)f(y)\,dy\biggr\vert ^{p_{2}}\,dx \biggr)^{\frac {1}{p_{2}}} \\& \quad \lesssim\frac{j-k}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}}|E_{j}|^{\frac {2\beta_{1}}{n}-\frac{1}{p_{1}}}|E_{k}|^{\frac{1}{p_{2}}-\frac {2\beta_{2}}{n}} \|b\|_{\mathit{BMO}}\|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where we have used the fact that $|x-y|\sim2^{j}r$ for $x\in E_{k}$, $y\in E_{j}$ and $j\geq k+2$. Since $\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}$, we obtain

$$\begin{aligned} D_{3} \lesssim& \bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {j-k}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}}|E_{j}|^{\frac{2\beta _{1}}{n}-\frac{1}{p_{1}}}|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta _{2}}{n}} \|b\|_{\mathit{BMO}}\|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha}{q}}(2^{j}r)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta}}{(1+2^{j}rm_{V}(x_{0}))^{N_{1}}}\frac {|E_{k}|^{\frac{1}{p_{2}}-\frac{2\beta_{2}}{n}}}{|E_{j}|^{\frac {2\beta_{1}}{n}-\frac{1}{p_{1}}}}(j-k) \Biggr)^{q} \|f \|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{\infty }_{j=k+2}(j-k)2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac {1}{p_{1}}+\frac{2\beta_{1}}{n})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

Let N be large enough. Finally, we get

$$\bigl\Vert \bigl[b, V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}} \bigr]f\bigr\Vert _{L^{p_{2},q,\lambda}_{\alpha,\theta,V}}\lesssim\|f\| _{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|_{\mathit{BMO}}. $$

 □

Theorem 5.3

Suppose that $V\in B_{s}$, $s\geq\frac{n}{2}$ and $b\in \mathit{BMO}(\mathbb {R}^{n})$. Let $\alpha\in(-\infty,0]$, $\lambda\in(0,n)$ and $1< q<\infty$. If $0<\beta_{2}\leq\beta_{1}<\frac{n}{2}$, $\frac{s}{s-\beta _{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}$, $\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}$, $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, then

$$\bigl\Vert \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f\bigr\Vert _{L^{p_{2},q,\lambda}_{\alpha,\theta,V}}\lesssim\|f\| _{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|_{\mathit{BMO}}. $$

Proof

Similarly, we can decompose f based on an arbitrary ball $B(x_{0},r)$ as follows:

$$f(y)=\sum^{\infty}_{j=-\infty}f(y) \chi_{E_{j}}(y)=\sum^{\infty }_{j=-\infty}f_{j}(y), $$

where $E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)$. Hence, we have

$$\begin{aligned}& \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \bigl\Vert \chi_{k} \bigl[b, (-\Delta+V)^{-\beta _{1}}V^{\beta_{2}} \bigr]f\bigr\Vert ^{q}_{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k-2}_{j=-\infty} \bigl\Vert \chi_{k} \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\bigl\Vert \chi_{k} \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\& \qquad {}+\bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty }|E_{k}|^{\theta q} \Biggl(\sum^{\infty}_{j=k+2}\bigl\Vert \chi_{k} \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \Biggr)^{q} \\& \quad =F_{1}+F_{2}+F_{3}. \end{aligned}$$

Applying Theorem 5.1, we can get

$$\begin{aligned} F_{2} \lesssim& \frac{(1+rm_{V}(x_{0}))^{\alpha}}{r^{\lambda n}}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \Biggl(\sum^{k+1}_{j=k-1}\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q}\|b\|^{q}_{\mathit{BMO}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

For $F_{1}$, by Hölder’s inequality and the fact that $V\in B_{s}$, we apply Lemmas 4.1 and 2.7 to deduce that

$$\begin{aligned}& \bigl\Vert \chi_{k}\bigl[b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}} \biggl(\int _{E_{k}}\biggl\vert \int_{E_{j}} \bigl(b(x)-b(y)\bigr)V^{\beta_{2}}(y)f(y)\,dy\biggr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\& \quad \lesssim\frac{1}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{k}r)^{n-2{\beta_{1}}}} \\& \qquad {}\times\biggl[ \biggl( \int _{E_{k}}\bigl\vert b(x)-b_{2^{k}r}\bigr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}}\int_{E_{j}}\bigl\vert V^{\beta_{2}}(y)f(y)\bigr\vert \,dy \\& \qquad {}+|E_{k}|^{\frac{1}{p_{2}}}\int_{E_{j}}\bigl\vert b(y)-b_{2^{k}r}\bigr\vert \bigl\vert V^{\beta _{2}}(y)f(y) \bigr\vert \,dy \biggr] \\& \quad \lesssim\frac{(\int_{E_{j}}V(y)\,dy)^{\beta _{2}}}{(1+2^{k}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {k-j}{(2^{k}r)^{n-2{\beta_{1}}}} |E_{k}|^{\frac{1}{p_{2}}}|E_{j}|^{1-\frac{1}{p_{1}}} \|b\|_{\mathit{BMO}}\| f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \\& \quad \lesssim\|b\|_{\mathit{BMO}}\frac {k-j}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}}|E_{k}|^{\frac{1}{p_{2}}+\frac {2\beta_{1}}{n}-1}|E_{j}|^{1-\frac{1}{p_{1}}-\frac{2\beta_{2}}{n}} \|f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})}, \end{aligned}$$

where $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}$ and $N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. Since $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, we obtain

$$\begin{aligned} F_{1} \lesssim& \|b\|^{q}_{\mathit{BMO}} \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac {k-j}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}}|E_{k}|^{\frac{1}{p_{2}}+\frac {2\beta_{1}}{n}-1}|E_{j}|^{1-\frac{1}{p_{1}}-\frac{2\beta_{2}}{n}} \| f_{j}\|_{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{k-2}_{j=-\infty} \frac{(1+2^{j}rm_{V}(x_{0}))^{-\frac {\alpha}{q}}}{(1+2^{k}rm_{V}(x_{0}))^{N_{2}}} \bigl(2^{j}r\bigr)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta} \frac {|E_{j}|^{1-\frac{1}{p_{2}}-\frac{2\beta_{1}}{n}}}{|E_{k}|^{1-\frac {1}{p_{2}}-\frac{2\beta_{1}}{n}}}(k-j) \Biggr)^{q}\|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\frac{(1+rm_{V}(x_{0}))^{\alpha }}{r^{\lambda n}}\sum ^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl( \sum^{k-2}_{j=-\infty}(k-j)2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac {1}{p_{2}}-1+\frac{2\beta_{1}}{n})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

For $F_{3}$, note that when $x\in E_{k}$, $y\in E_{j}$ and $j\geq k+2$, then $|x-y|\sim2^{j}r$. Similar to $F_{1}$, we have

$$\begin{aligned}& \bigl\Vert \chi_{k}\bigl[b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}} \bigr]f_{j}\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})} \\& \quad \lesssim\frac{1}{(1+2^{j}rm_{V}(x_{0}))^{N/k_{0}+1}}\frac {1}{(2^{j}r)^{n-2{\beta_{1}}}} \biggl(\int _{E_{k}}\biggl\vert \int_{E_{j}} \bigl(b(x)-b(y)\bigr)V(y)^{\beta_{2}}f(y)\,dy\biggr\vert ^{p_{2}}\,dx \biggr)^{\frac{1}{p_{2}}} \\& \quad \lesssim\frac{j-k}{(1+2^{j}rm_{V}(x_{0}))^{N_{2}}}|E_{k}|^{\frac {1}{p_{2}}}|E_{j}|^{-\frac{1}{p_{2}}} \|f_{j}\|_{L^{p_{1}}(\mathbb {R}^{n})}\|b\|_{\mathit{BMO}}, \end{aligned}$$

where $\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}$ and $N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}$. Since $\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda }{q}-\frac{1}{p_{2}}+1-\frac{2\beta_{1}}{n}$, we obtain

$$\begin{aligned} F_{3} \lesssim& \|b\|^{q}_{\mathit{BMO}} \bigl(1+rm_{V}(x_{0})\bigr)^{\alpha}r^{-\lambda n} \sum^{0}_{k=-\infty}|E_{k}|^{\theta q} \\ &{}\times \Biggl(\sum^{\infty}_{j=k+2} \frac {(1+2^{j}rm_{V}(x_{0}))^{-\frac{\alpha }{q}}}{(1+2^{j}rm_{V}(x_{0}))^{N_{2}}}\bigl(2^{j}r\bigr)^{\frac{\lambda n}{q}}|E_{j}|^{-\theta} \frac{|E_{k}|^{\frac {1}{p_{2}}}}{|E_{j}|^{\frac{1}{p_{2}}}}(j-k)\|f_{j}\| _{L^{p_{1}}(\mathbb{R}^{n})} \Biggr)^{q} \\ \lesssim& \|b\|^{q}_{\mathit{BMO}}\bigl(1+rm_{V}(x_{0}) \bigr)^{\alpha}r^{-\lambda n}\sum^{0}_{k=-\infty}|E_{k}|^{\lambda} \Biggl(\sum^{\infty }_{j=k+2}(j-k)2^{(k-j)n(\theta-\frac{\lambda}{q}+\frac {1}{p_{2}})} \Biggr)^{q} \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}} \\ \lesssim& \|f\|^{q}_{L^{p_{1},q,\lambda}_{\alpha,\theta,V}}\|b\|^{q}_{\mathit{BMO}}. \end{aligned}$$

Let N be large enough. We finally get

$$\bigl\Vert \bigl[b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta,V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta,V}}\|b\|_{\mathit{BMO}}. $$

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