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Schrödinger type operators on generalized Morrey spaces

Journal of Inequalities and Applications20152015:229

https://doi.org/10.1186/s13660-015-0747-8

Received: 25 October 2014

Accepted: 1 July 2015

Published: 23 July 2015

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.

Keywords

generalized Morrey spacesSchrödinger operatorcommutatorreverse Hölder class

MSC

42B3542B20

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 \(\pmb{L^{p,q,\lambda}_{\alpha,\beta, V}}\)

θ = 0, α = 0, p = q, 0<λ<1

Morrey space \(L^{p,\lambda}(\mathbb{R}^{n})\) [10]

θ = 0, p = q, 0<λ<1

Morrey type space \(L_{\alpha,V}^{p,\lambda}(\mathbb{R}^{n})\) [6]

α = λ = 0, \(\theta\in\mathbb{R}\), 0<p,q<∞,

Herz spaces \(K_{p}^{\theta,q}\) [11]

α = 0, λ ≥ 0, \(\theta\in\mathbb{R}\), 0<p,q<∞

Morrey-Herz spaces \(MK_{p,q}^{\theta,\lambda}\) [12, 13]

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.
  1. (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.
     
  2. (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
  1. (a)

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

     
  2. (b)

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

     
  3. (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

  1. (i)

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

     
  2. (ii)

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

     
  3. (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
  1. (a)

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

     
  2. (b)
    there exists a kernel K such that for every \(f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\),
    $$Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)\, dy \quad \mbox{a.e. on }\{ \operatorname{supp}f\}^{c}; $$
     
  3. (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. (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. (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 \(b\in \mathit{BMO}(\mathbb{R}^{n})\).
  1. (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
    $$\bigl\Vert \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb {R}^{n})}\|b\|_{\mathit{BMO}}. $$
     
  2. (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
$$\bigl\Vert \bigl[b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f\bigr\Vert _{L^{p_{2},q,\lambda }_{\alpha,\theta,V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta,V}}\|b\|_{\mathit{BMO}}. $$
 □

Declarations

Acknowledgements

Project was supported by NSFC No. 11171203; New Teacher’s Fund for Doctor Stations, Ministry of Education No. 20114402120003; Guangdong Natural Science Foundation S2011040004131; Foundation for Distinguished Young Talents in Higher Education of Guangdong, China, LYM11063.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics, Qingdao University, Qingdao, China
(2)
Department of Mathematics, Shantou University, Shantou, China

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