1 Introduction and main results

Let us consider the Schrödinger operator

$$\begin{aligned} L=-\Delta+V\quad \mbox{on } {\mathbb{R}^{n}}, n \geq3, \end{aligned}$$

where V is a nonnegative, \(V \neq0\), and belongs to the reverse Hölder class \(RH_{q}\) for some \(q\geq n/2\), i.e., there exists a constant \(C>0\) such that the reverse Hölder inequality

$$\begin{aligned} \biggl(\frac{1}{ \vert B(x,r) \vert } \int_{B(x,r)}V^{q}(y)\,dy \biggr)^{1/q}\leq \frac{C}{ \vert B(x,r) \vert } \int_{B(x,r)}V(y)\,dy \end{aligned}$$
(1.1)

holds for every \(x \in{\mathbb{R}^{n}}\) and \(0< r<\infty \), where \(B(x,r)\) denotes the ball centered at x with radius r. In particular, if V is a nonnegative polynomial, then \(V \in RH_{\infty }\).

As in [29], for a given potential \(V \in RH_{q}\) with \(q\geq n/2\), we define the auxiliary function

$$\begin{aligned} \rho(x):= \frac{1}{m_{V}(x)} = \sup_{r>0} \biggl\{ r : \frac {1}{r^{n-2}} \int_{B(x,r)} V(y)\,dy \le1 \biggr\} . \end{aligned}$$

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

Let \(\theta>0\) and \(0<\nu<1\), in view of [22], the Campanato class, associated with the Schrödinger operator \(\Lambda_{\nu }^{\theta}(\rho)\) consists of the locally integrable functions b such that

$$\begin{aligned} \frac{1}{ \vert B(x,r) \vert ^{1+\nu/n}} \int_{B(x,r)} \bigl\vert b(y)-b_{B} \bigr\vert \,dy \le C \biggl(1+\frac{r}{\rho(x)} \biggr)^{\theta} \end{aligned}$$
(1.2)

for all \(x\in{\mathbb{R}}^{n}\) and \(r>0\). A seminorm of \(b \in\Lambda _{\nu }^{\theta}(\rho)\), denoted by \([b]_{\beta}^{\theta}\), is given by the infimum of the constants in the inequality above.

Note that if \(\theta=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is the classical Campanato space; if \(\nu=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is exactly the space \(BMO_{\theta}(\rho)\) introduced in [5].

We now present the definition of generalized Morrey spaces \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) (including the weak version) associated with a Schrödinger operator, which was introduced by the first author in [18].

The classical Morrey spaces \(L_{p,\lambda}({\mathbb{R}}^{n})\) was introduced by Morrey in [24] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the reader to [9–12, 24, 35]. The generalized Morrey spaces are defined with \(r^{\lambda}\) replaced by a general nonnegative function \(\varphi(x,r)\) satisfying some assumptions (see, for example, [15, 23, 25, 30]).

For brevity, in the sequel we use the notations

$$\mathfrak{A}_{p,\varphi}^{\alpha ,V}(f;x,r):= \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } r^{-n/p} \varphi(x,r)^{-1} \Vert f \Vert _{L_{p}(B(x,r))} $$

and

$$\mathfrak{A}_{\Phi,\varphi}^{W,\alpha ,V}(f;x,r):= \biggl(1+\frac {r}{\rho(x)} \biggr)^{\alpha } r^{-n/p} \varphi(x,r)^{-1} \Vert f \Vert _{WL_{p}(B(x,r))}. $$

Definition 1.1

Let \(\varphi(x,r)\) be a positive measurable function on \(\mathbb{R}^{n}\times(0,\infty)\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\). For any fixed \(x_{0} \in{\mathbb{R}^{n}}\) we denote by \(LM_{p,\varphi}^{\alpha,V,\{ x_{0}\} }=LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) the local generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in L^{loc}_{p}(\mathbb{R}^{n})\) with finite norm

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}= \sup_{r>0} \mathfrak {A}_{p,\varphi}^{\alpha,V}(f;x_{0},r). \end{aligned}$$

Also \(WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}=WLM_{p,\varphi}^{\alpha ,V,\{x_{0}\}}(\mathbb{R}^{n})\) we denote the weak local generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in WL^{loc}_{p} (\mathbb{R}^{n})\) with

$$\begin{aligned} \Vert f \Vert _{WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}= \sup_{r>0} \mathfrak {A}_{p,\varphi}^{W,\alpha,V}(f;x_{0},r)< \infty . \end{aligned}$$

The local spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) and \(WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) are Banach spaces with respect to the norm

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}} & = \sup_{r>0} \mathfrak {A}_{p,\varphi}^{\alpha,V}(f;x_{0},r),\qquad \Vert f \Vert _{WLM_{p,\varphi}^{\alpha,V,\{x_{0}\}}} = \sup_{r>0} \mathfrak {A}_{p,\varphi}^{W,\alpha,V}(f;x_{0},r), \end{aligned}$$

respectively.

Remark 1.1

  1. (i)

    When \(\alpha=0\), and \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}(\mathbb{R}^{n})\) is the local (central) Morrey space \(LM_{p,\lambda}^{\{0\}}(\mathbb {R}^{n})\) studied in [4].

  2. (ii)

    When \(\alpha=0\), \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\} }(\mathbb{R}^{n})\) is the local generalized Morrey space \(VM_{p,\varphi}^{\{x_{0}\}}(\mathbb{R}^{n})\) were introduced by the first author in [13]; see also [14, 16, 21] etc.

Definition 1.2

The vanishing generalized Morrey space associated with the Schrödinger operator \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) is defined as the spaces of functions \(f\in M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) such that

$$\begin{aligned} \lim_{r\rightarrow0} \sup_{x \in{\mathbb{R}^{n}}} \mathfrak {A}_{p,\varphi }^{\alpha ,V}(f;x,r) = 0. \end{aligned}$$
(1.3)

The vanishing weak generalized Morrey space associated with the Schrödinger operator \(VWM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) is defined as the spaces of functions \(f\in WM_{p,\varphi}^{\alpha ,V}({\mathbb {R}}^{n})\) such that

$$\begin{aligned} \lim_{r\rightarrow0} \sup_{x \in{\mathbb{R}^{n}}} \mathfrak {A}_{p,\varphi }^{W,\alpha ,V}(f;x,r) = 0. \end{aligned}$$

The vanishing spaces \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) and \(VWM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) are Banach spaces with respect to the norm

$$\begin{aligned} &\Vert f \Vert _{VM_{p,\varphi}^{\alpha ,V}} \equiv \Vert f \Vert _{M_{p,\varphi }^{\alpha ,V}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \mathfrak{A}_{p,\varphi}^{\alpha ,V}(f;x,r), \\ &\Vert f \Vert _{VWM_{p,\varphi}^{\alpha ,V}} \equiv \Vert f \Vert _{WM_{p,\varphi }^{\alpha ,V}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \mathfrak{A}_{W,p,\varphi }^{\alpha ,V}(f;x,r), \end{aligned}$$

respectively.

In the case \(\alpha=0\), and \(\varphi(x,r)=r^{(\lambda-n)/p}\) \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) is the vanishing Morrey space \(VM_{p,\lambda}\) introduced in [33], where applications to PDE were considered.

We refer to [3, 20, 27, 28] for some properties of vanishing generalized Morrey spaces.

Definition 1.3

Let \(L=-\Delta+V\) with \(V\in RH_{q_{1}}\), \(q_{1}>n/2\). The fractional integral associated with L is defined by

$$\begin{aligned} \mathcal{I}_{\beta}^{L} f(x)=L^{-\beta/2} f(x)= \int_{0}^{\infty}e^{-tL}(f) (x) t^{\beta/2-1} \,dt \end{aligned}$$

for \(0<\beta<n\). The commutator of \(\mathcal{I}_{\beta}^{L}\) is defined by

$$\begin{aligned} \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x)=b(x) \mathcal{I}_{\beta}^{L} f(x)-\mathcal{I}_{\beta}^{L}(bf) (x). \end{aligned}$$

Note that, if \(L=-\Delta\) is the Laplacian on \({\mathbb{R}}^{n}\), then \(\mathcal {I}_{\beta}^{L}\) and \([b,\mathcal{I}_{\beta}^{L}]\) are the Riesz potential \(I_{\beta}\) and the commutator of the Riesz potential \([b,I_{\beta}]\), respectively, that is,

$$\begin{aligned} I_{\beta}f(x)= \int_{{\mathbb{R}}^{n}} \frac{f(y)}{ \vert x-y \vert ^{n}} \,dy, \qquad [b,I_{\beta}]f(x)= \int_{{\mathbb{R}}^{n}} \frac{b(x)-b(y)}{ \vert x-y \vert ^{n}} f(y) \,dy. \end{aligned}$$

When \(b\in BMO\), Chanillo proved in [8] that \([b,I_{\beta }]\) is bounded from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-\beta /n\), \(1< p< n/\beta\). When b belongs to the Campanato space \(\Lambda _{\nu}\), \(0<\nu<1\), Paluszynski in [26] showed that \([b,I_{\beta}]\) is bounded from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-(\beta+\nu)/n\), \(1< p< n/(\beta+\nu)\). When \(b\in BMO_{\theta}(\rho)\), Bui in [6] obtained the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) from \(L_{p}({\mathbb {R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) with \(1/q=1/p-\beta/n\), \(1< p< n/\beta\).

Inspired by the above results, we are interested in the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) on generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) and the vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\), when b belongs to the new Campanato class \(\Lambda_{\nu}^{\theta}(\rho)\).

In this paper, we consider the boundedness of the commutator of \(\mathcal{I}_{\beta}^{L}\) on the local generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\), the generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}({\mathbb {R}}^{n})\) and the vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}} ^{n})\). When b belongs to the new Campanato space \(\Lambda_{\nu }^{\theta}(\rho)\), \(0<\nu<1\), we show that \([b,\mathcal{I}_{\beta }^{L}]\) are bounded from \(LM_{p,\varphi_{1}}^{\alpha ,V,\{x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\), from \(M_{p,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) to \(M_{q,\varphi }^{\alpha ,V}({\mathbb{R}}^{n})\) and from \(VM_{p,\varphi}^{\alpha,V}({\mathbb {R}}^{n})\) to \(VM_{q,\varphi}^{\alpha,V}({\mathbb{R}}^{n})\) with \(1/q=1/p-(\beta +\nu)/n\), \(1< p< n/(\beta+\nu)\).

Our main results are as follows.

Theorem 1.1

Let \(x_{0} \in{\mathbb{R}^{n}}\), \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(1\le p< n/(\beta+\nu )\), \(1/q=1/p-(\beta+\nu)/n\) and let \(\varphi_{1}, \varphi_{2}\in\Omega _{p,loc}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x_{0},s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \le c_{0} \varphi_{2}(x_{0},r), \end{aligned}$$
(1.4)

where \(c_{0}\) does not depend on r. Then the operator \([b,\mathcal {I}_{\beta}^{L}]\) is bounded from \(LM_{p,\varphi_{1}}^{\alpha,V,\{x_{0}\} }\) to \(LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\}}\) for \(p>1\) and from \(LM_{1,\varphi_{1}}^{\alpha,V,\{x_{0}\}}\) to \(WLM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V,\{x_{0}\}}\). Moreover, for \(p>1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{LM_{q,\varphi_{2}}^{\alpha,V,\{x_{0}\} }}\le C [b]_{\nu}^{\theta} \Vert f \Vert _{LM_{p,\varphi_{1}}^{\alpha,V,\{ x_{0}\}}}, \end{aligned}$$

and for \(p=1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WLM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V,\{x_{0}\}}}\le C [b]_{\nu}^{\theta} \Vert f \Vert _{LM_{1,\varphi_{1}}^{\alpha,V,\{x_{0}\}}}, \end{aligned}$$

where C does not depend on f.

Corollary 1.1

Let \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(1\le p< n/(\beta+\nu)\), \(1/q=1/p-(\beta+\nu)/n\) and let \(\varphi_{1}\in\Omega_{p}^{\alpha ,V}\), \(\varphi_{2} \in\Omega_{q}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x,s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \le c_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.5)

where \(c_{0}\) does not depend on x and r. Then the operator \([b,\mathcal{I}_{\beta}^{L}]\) is bounded from \(M_{p,\varphi_{1}}^{\alpha,V}\) to \(M_{q,\varphi_{2}}^{\alpha,V}\) for \(p>1\) and from \(M_{1,\varphi_{1}}^{\alpha,V}\) to \(WM_{\frac{n}{n-\beta-\nu},\varphi _{2}}^{\alpha,V}\). Moreover, for \(p>1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}}\le C [b]_{\theta} \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}, \end{aligned}$$

and for \(p=1\)

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V}}\le C \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}}, \end{aligned}$$

where C does not depend on f.

Theorem 1.2

Let \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(V\in RH_{q_{1}}\), \(q_{1}>n/2\), \(0<\nu<1\), \(\alpha\ge0\), \(b \in\Lambda_{\nu}^{\theta }(\rho)\), \(1< p<n/(\beta+\nu)\), \(1/q=1/p-(\beta+\nu)/n\), and let \(\varphi_{1}\in\Omega_{p,1}^{\alpha ,V}\), \(\varphi_{2} \in\Omega _{q,1}^{\alpha ,V}\) satisfy the conditions

$$\begin{aligned} c_{\delta}:= \int_{\delta}^{\infty}\sup_{x\in{\mathbb {R}^{n}}} \varphi_{1}(x,t) \,\frac{dt}{t} < \infty \end{aligned}$$

for every \(\delta>0\), and

$$\begin{aligned} \int_{r}^{\infty} \varphi_{1}(x,t)\, \frac{dt}{t^{1-\beta-\nu}} \leq C_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.6)

where \(C_{0}\) does not depend on \(x\in{\mathbb{R}^{n}}\) and \(r>0\). Then the operator \([b,\mathcal{I}_{\beta}^{L}]\) is bounded from \(VM_{p,\varphi _{1}}^{\alpha ,V}\) to \(VM_{q,\varphi_{2}}^{\alpha ,V}\) for \(p>1\) and from \(VM_{1,\varphi_{1}}^{\alpha ,V}\) to \(VWM_{\frac{n}{n-\beta-\nu },\varphi _{2}}^{\alpha ,V}\).

Remark 1.2

Note that, in the case of \(V \equiv0\), \(\nu=0\) Corollary 1.1 and Theorem 1.2 were proved in [19, Corollary 5.5 and 7.5] and in the case of \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(\nu=0\) in [32, Theorems 1.3 and 1.4].

In this paper, we shall use the symbol \(A\lesssim B\) to indicate that there exists a universal positive constant C, independent of all important parameters, such that \(A\le CB\). \(A \approx B\) means that \(A\lesssim B\) and \(B\lesssim A\).

2 Some technical lemmas and propositions

We would like to recall the important properties concerning the critical function.

Lemma 2.1

([29])

Let \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\). For the associated function ρ there exist C and \(k_{0}\ge1\) such that

$$\begin{aligned} C^{-1}\rho(x) \biggl(1+\frac{ \vert x-y \vert }{\rho(x)} \biggr)^{-k_{0}}\le\rho (y)\le C\rho(x) \biggl(1+\frac{ \vert x-y \vert }{\rho(x)} \biggr)^{\frac{k_{0}}{1+k_{0}}} \end{aligned}$$
(2.1)

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

Lemma 2.2

([2])

Suppose \(x\in B(x_{0},r)\). Then for \(k\in N\) we have

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

According to [5], the new BMO space \(BMO_{\theta}(\rho)\) with \(\theta\ge0\) is defined as a set of all locally integrable functions b such that

$$\begin{aligned} \frac{1}{ \vert B(x,r) \vert } \int_{B(x,r)} \bigl\vert b(y)-b_{B} \bigr\vert \,dy \le C \biggl(1+\frac {r}{\rho(x)} \biggr)^{\theta} \end{aligned}$$

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

Let \(\theta>0\) and \(0<\nu<1\), a seminorm on the Campanato class \(\Lambda_{\nu}^{\theta}(\rho)\) is denoted by \([b]_{\nu}^{\theta}\),

$$\begin{aligned}{} [b]_{\nu}^{\theta}:=\sup_{x\in{\mathbb{R}^{n}}, r>0} \frac{\frac {1}{ \vert B(x,r) \vert ^{1+\nu/n}}\int_{B(x,r)} \vert b(y)-b_{B} \vert \,dy}{(1+\frac {r}{\rho(x)})^{\theta}}< \infty. \end{aligned}$$

The Lipschitz space, associated with the Schrödinger operator (see [22]), consists of the functions f satisfying

$$\begin{aligned} \Vert f \Vert _{\mathrm{Lip}_{\nu}^{\theta}(\rho)}:=\sup_{x\in{\mathbb{R}^{n}}, r>0} \frac{ \vert f(x)-f(y) \vert }{ \vert x-y \vert ^{\nu} (1+\frac{ \vert x-y \vert }{\rho(x)}+\frac { \vert x-y \vert }{\rho(y)} )^{\theta}}< \infty. \end{aligned}$$

It is easy to see that this space is exactly the Lipschitz space when \(\theta=0\).

Note that if \(\theta=0\) in (1.2), \(\Lambda_{\nu}^{\theta }(\rho)\) is exactly the classical Campanato space; if \(\nu=0\), \(\Lambda_{\nu}^{\theta}(\rho)\) is exactly the space \(BMO_{\theta}(\rho)\); if \(\theta=0\) and \(\nu=0\), it is exactly the John–Nirenberg space \(BMO\).

The following relations between \(\mathrm{Lip}_{\nu}^{\theta}(\rho)\) and \(\Lambda_{\nu}^{\theta}(\rho)\) were proved in [22, Theorem 5].

Lemma 2.3

([22])

Let \(\theta>0\) and \(0<\nu<1\). Then following embedding is valid:

$$\begin{aligned} \Lambda_{\nu}^{\theta}(\rho) \subseteq\mathrm{Lip}_{\nu}^{\theta }( \rho) \subseteq\Lambda_{\nu}^{(k_{0}+1)\theta}(\rho), \end{aligned}$$

where \(k_{0}\) is the constant appearing in Lemma 2.1.

We give some inequalities about the Campanato space, associated with the Schrödinger operator \(\Lambda_{\nu}^{\theta}(\rho)\).

Lemma 2.4

([22])

Let \(\theta>0\) and \(1\le s <\infty\). If \(b\in\Lambda_{\nu}^{\theta}(\rho)\), then there exists a positive constant C such that

$$\begin{aligned} \biggl(\frac{1}{ \vert B \vert } \int_{B} \bigl\vert b(y)-b_{B} \bigr\vert ^{s} \,dy \biggr)^{1/s} \le C [b]_{\nu}^{\theta} r^{\nu} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\theta'} \end{aligned}$$

for all \(B=B(x,r)\), with \(x\in{\mathbb{R}}^{n}\) and \(r>0\), where \(\theta '=(k_{0}+1)\theta\) and \(k_{0}\) is the constant appearing in (2.1).

Let \(K_{\beta}\) be the kernel of \(\mathcal{I}_{\beta}^{L}\). The following result gives the estimate on the kernel \(K_{\beta}(x,y)\).

Lemma 2.5

([6])

If \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\), then, for every N, there exists a constant C such that

$$\begin{aligned} \bigl\vert K_{\beta}(x,y) \bigr\vert \le \frac{C}{ (1+\frac{ \vert x-y \vert }{\rho(x)} )^{N}}\frac{1}{ \vert x-y \vert ^{n-\beta}}. \end{aligned}$$
(2.2)

Finally, we recall a relationship between an essential supremum and an essential infimum.

Lemma 2.6

([34])

Let f be a real-valued nonnegative function and measurable on E. Then

$$\begin{aligned} \Bigl(\mathop{\operatorname{ess\inf}}\limits _{x\in E} f(x) \Bigr)^{-1}= \mathop{ \operatorname {ess\sup}}\limits _{x\in E}\frac{1}{f(x)}. \end{aligned}$$

It is natural, first of all, to find conditions ensuring that the spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\) and \(M_{p,\varphi }^{\alpha,V}\) are nontrivial, that is, consist not only of functions equivalent to 0 on \({\mathbb{R}^{n}}\).

Lemma 2.7

Let \(x_{0} \in{\mathbb{R}^{n}}\), \(\varphi(x,r)\) be a positive measurable function on \(\mathbb{R}^{n}\times(0,\infty)\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\). If

$$\begin{aligned} \sup_{t< r< \infty} \biggl(1+\frac{r}{\rho(x_{0})} \biggr)^{\alpha} \frac{r^{-\frac{n}{p}}}{\varphi(x_{0},r)}=\infty\quad\textrm{for some } t>0, \end{aligned}$$
(2.3)

then \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}({\mathbb{R}^{n}})=\Theta\), where Θ is the set of all functions equivalent to 0 on \({\mathbb{R}^{n}}\).

Proof

Let (2.4) be satisfied and f be not equivalent to zero. Then \(\Vert f \Vert _{L_{p}(B(x_{0},t))}>0 \), hence

$$\begin{aligned} \Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}}&\geq\sup_{ t< r< \infty} \biggl(1+ \frac{r}{\rho(x_{0})} \biggr)^{\alpha} \varphi (x_{0},r)^{-1} r^{-\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x_{0},r))} \\ & \geq \Vert f \Vert _{L_{p}(B(x_{0},t))}\sup_{ t< r< \infty} \biggl(1+ \frac {r}{\rho(x_{0})} \biggr)^{\alpha} \varphi(x_{0},r)^{-1} r^{-\frac{n}{p}}. \end{aligned}$$

Therefore \(\Vert f \Vert _{LM_{p,\varphi}^{\alpha,V,\{x_{0}\} }}=\infty\). □

Remark 2.1

We denote by \(\Omega_{p,loc}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty)\) such that, for all \(t>0\),

$$\sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac{r^{-\frac{n}{p}}}{\varphi(x,r)} \biggr\Vert _{L_{\infty }(t,\infty )} < \infty . $$

In what follows, keeping in mind Lemma 2.7, for the non-triviality of the space \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in\Omega_{p,loc}^{\alpha ,V}\).

Lemma 2.8

([2])

Let \(\varphi(x,r) \) be a positive measurable function on \({\mathbb{R}^{n}}\times(0,\infty )\), \(1\le p<\infty\), \(\alpha\ge0\), and \(V\in RH_{q}\), \(q\ge1\).

  1. (i)

    If

    $$\begin{aligned} \sup_{t< r< \infty } \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac {r^{-\frac{n}{p}}}{\varphi(x,r)}=\infty\quad\textit{for some } t>0 \textit{ and for all } x\in{\mathbb{R}^{n}}, \end{aligned}$$
    (2.4)

    then \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})=\Theta\).

  2. (ii)

    If

    $$\begin{aligned} \sup_{ 0< r< \tau} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi (x,r)^{-1} = \infty\quad\textit{ for some } \tau>0\textit{ and for all } x\in{\mathbb{R}^{n}}, \end{aligned}$$
    (2.5)

    then \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})=\Theta\).

Remark 2.2

We denote by \(\Omega_{p}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty)\) such that, for all \(t>0\),

$$\begin{aligned} &\sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac{r^{-\frac{n}{p}}}{\varphi(x,r)} \biggr\Vert _{L_{\infty }(t,\infty )} < \infty , \quad \mbox{and} \\ & \sup_{x\in{\mathbb{R}^{n}}} \biggl\Vert \biggl(1+ \frac{r}{\rho(x)} \biggr)^{\alpha } \varphi(x,r)^{-1} \biggr\Vert _{L_{\infty }(0, t)}< \infty , \end{aligned}$$

respectively. In what follows, keeping in mind Lemma 2.8, for the non-triviality of the space \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in \Omega_{p}^{\alpha ,V}\).

Remark 2.3

We denote by \(\Omega_{p,1}^{\alpha ,V}\) the sets of all positive measurable functions φ on \({\mathbb{R}^{n}}\times(0,\infty )\) such that

$$\begin{aligned} \inf_{x\in{\mathbb{R}^{n}}} \inf_{r>\delta} \biggl(1+\frac{r}{\rho (x)} \biggr)^{-\alpha } \varphi(x,r)>0, \quad \mbox{for some } \delta>0, \end{aligned}$$
(2.6)

and

$$\begin{aligned} \lim_{r \to0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac {r^{n/p}}{\varphi(x,r)} = 0. \end{aligned}$$

For the non-triviality of the space \(VM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in \Omega_{p,1}^{\alpha ,V}\).

3 Proof of Theorem 1.1

We first prove the following conclusions.

Lemma 3.1

Let \(0<\nu<1\), \(0< \beta+\nu<n\) and \(b\in\Lambda_{\nu}^{\theta }(\rho)\), then the following pointwise estimate holds:

$$\bigl\vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) \bigr\vert \lesssim[b]_{\nu }^{\theta} I_{\beta+\nu} \bigl( \vert f \vert \bigr) (x). $$

Proof

Note that

$$\begin{aligned} \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) &= b(x) \mathcal{I}_{\beta }^{L}(f) (x) - \mathcal{I}_{\beta}^{L}(bf) (x) \\ &= \int_{{\mathbb{R}^{n}}} \bigl[b(x)-b(y) \bigr] K_{\beta}(x,y) f(y) \,dy. \end{aligned}$$

If \(b\in\Lambda_{\nu}^{\theta}(\rho)\), then from Lemma 2.5 we have

$$\begin{aligned} \begin{aligned} \bigl\vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f(x) \bigr\vert & \le \int_{{\mathbb{R}^{n}}} \bigl\vert b(x)-b(y) \bigr\vert \bigl\vert K_{\beta}(x,y) \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ & \lesssim[b]_{\nu}^{\theta} \int_{{\mathbb{R}^{n}}} \vert x-y \vert ^{\nu} \bigl\vert K_{\beta }(x,y) \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ &= [b]_{\nu}^{\theta} I_{\beta+\nu} \bigl( \vert f \vert \bigr) (x). \end{aligned} \end{aligned}$$

 □

From Lemma 3.1 we get the following.

Corollary 3.1

Suppose \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\) and \(b\in\Lambda_{\nu }^{\theta}(\rho)\) with \(0<\nu<1\). Let \(0< \beta+\nu<n\) and let \(1\le p< q<\infty\) satisfy \(1/q=1/p-(\beta+\nu)/n\). Then for all f in \(L_{p}({\mathbb{R}^{n}})\) we have

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}({\mathbb{R}^{n}})} \lesssim \Vert f \Vert _{L_{p}({\mathbb{R}^{n}})} \end{aligned}$$

when \(p>1\), and also

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WL_{q}({\mathbb{R}^{n}})} \lesssim \Vert f \Vert _{L_{1}({\mathbb{R}^{n}})} \end{aligned}$$

when \(p=1\).

In order to prove Theorem 1.1, we need the following.

Theorem 3.1

Suppose \(V\in RH_{q_{1}}\) with \(q_{1} > n/2\), \(b\in\Lambda_{\nu}^{\theta}(\rho)\), \(\theta>0\), \(0<\nu<1\). Let \(0< \beta+\nu<n\) and let \(1\le p< q<\infty\) satisfy \(1/q=1/p-(\beta+\nu)/n\) then the inequality

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x_{0},r))} &\lesssim \bigl\Vert I_{\beta +\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))}\\ &\lesssim r^{\frac{n}{q}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$

holds for any \(f\in L_{loc}^{p}({\mathbb{R}}^{n})\). Moreover, for \(p=1\) the inequality

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L}f \bigr] \bigr\Vert _{WL_{\frac{n}{n-\beta-\nu }}(B(x_{0},r))} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac {n}{n-\beta-\nu}}(B(x_{0},r))} \\ & \lesssim r^{n-\beta} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t} \end{aligned}$$

holds for any \(f\in L_{loc}^{1}({\mathbb{R}}^{n})\).

Proof

For arbitrary \(x_{0}\in{\mathbb{R}}^{n}\), set \(B=B(x_{0},r)\) and \(\lambda B=B(x_{0},\lambda r)\) for any \(\lambda>0\). We write f as \(f=f_{1}+f_{2}\), where \(f_{1}(y)=f(y)\chi_{{B(x_{0},2r)}}(y)\), and \(\chi_{{B(x_{0},2r)}}\) denotes the characteristic function of \(B(x_{0},2r)\). Then

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x_{0},r))} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))} \\ & \le \Vert I_{\beta+\nu}f_{1} \Vert _{L_{q}(B(x_{0},r))}+ \Vert I_{\beta+\nu}f_{2} \Vert _{L_{q}(B(x_{0},r))}. \end{aligned}$$

Since \(f_{1}\in L_{p}({\mathbb{R}^{n}})\) and from the boundedness of \(I_{\beta+\nu}\) from \(L_{p}({\mathbb{R}}^{n})\) to \(L_{q}({\mathbb{R}}^{n})\) (see [31]) it follows that

$$\begin{aligned} \Vert I_{\beta+\nu}f_{1} \Vert _{L_{q}(B(x_{0},r))} &\lesssim \Vert f \Vert _{L_{p}(B(x_{0},2r))} \\ & \lesssim r^{\frac{n}{q}} \Vert f \Vert _{L_{p}(B(x_{0},2r))} \int_{2r}^{\infty}\frac{dt}{t^{\frac{n}{q}+1}} \lesssim r^{\frac{n}{q}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}} \, \frac{dt}{t}. \end{aligned}$$
(3.1)

To estimate \(\|I_{\beta+\nu}f_{2}\|_{L_{p}(B(x_{0},r))}\), the obverse of \(x\in B\), \(y\in(2B)^{c}\) implies \(|x-y|\approx|x_{0}-y|\). Then by (2.2) we have

$$\begin{aligned} \sup_{x\in B} \bigl\vert I_{\beta+\nu}f_{2}(x) \bigr\vert & \lesssim \int_{(2B)^{c}}\frac { \vert f(y) \vert }{ \vert x_{0}-y \vert ^{n-\beta-\nu}}\,dy \lesssim\sum _{k=1}^{\infty}\bigl(2^{k+1} r \bigr)^{-n+\beta} \int_{2^{k+1}B} \bigl\vert f(y) \bigr\vert \,dy. \end{aligned}$$

By Hölder’s inequality we get

$$\begin{aligned} \sup_{x\in B} \bigl\vert I_{\beta+\nu}f_{2}(x) \bigr\vert & \lesssim\sum_{k=1}^{\infty} \Vert f \Vert _{L_{p}(2^{k+1}B)} \bigl(2^{k+1} r \bigr)^{-1-\frac{n}{p}+\beta} \int_{2^{k} r}^{2^{k+1}r} \,dt \\ & \lesssim\sum_{k=1}^{\infty}\int_{2^{k} r}^{2^{k+1}r} \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \lesssim \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac {n}{q}}}\, \frac{dt}{t}. \end{aligned}$$
(3.2)

Then

$$\begin{aligned} \Vert I_{\beta+\nu}f_{2} \Vert _{L_{q}(B(x_{0},r))} \lesssim r^{\frac{n}{q}} \int _{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$
(3.3)

holds for \(1\le p< n/\beta\). Therefore, by (3.1) and (3.3) we get

$$\begin{aligned} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{q}(B(x_{0},r))}\lesssim r^{\frac{n}{q}} \int _{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \end{aligned}$$
(3.4)

for \(1< p<n/\beta\).

When \(p=1\), by the boundedness of \(I_{\beta+\nu}\) from \(L_{1}({\mathbb{R}}^{n})\) to \(WL_{\frac{n}{n-\beta-\nu}}({\mathbb{R}}^{n})\), we get

$$\begin{aligned} \Vert I_{\beta+\nu}f_{1} \Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} & \lesssim \Vert f \Vert _{L_{1}(B(x_{0},2r))} \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

By (3.3) we have

$$\begin{aligned} \Vert I_{\beta+\nu}f_{2} \Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} & \le \Vert I_{\beta+\nu}f_{2} \Vert _{L_{\frac{n}{n-\beta-\nu}} (B(x_{0},2r))} \\ & \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac{n}{n-\beta-\nu}}(B(x_{0},r))} \lesssim r^{n-\beta-\nu} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t}. \end{aligned}$$

 □

Proof of Theorem 1.1

From Lemma 2.6, we have

$$\begin{aligned} \frac{1}{\operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}} =\mathop{\operatorname{ess\sup}}_{t< s< \infty}\frac{1}{\varphi _{1}(x,s)s^{\frac{n}{p}}}. \end{aligned}$$

Note the fact that \(\|f\|_{L_{p}(B(x_{0},t))}\) is a nondecreasing function of t, and \(f\in M_{p,\varphi_{1}}^{\alpha,V}\), then

$$\begin{aligned} \frac{ (1+\frac{t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}& \lesssim\operatorname{ess\sup}_{t< s< \infty} \frac{ (1+\frac {t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \varphi_{1}(x,s)s^{\frac{n}{p}}} \\ & \lesssim\sup_{0< s< \infty} \frac{ (1+\frac{s}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},s))}}{ \varphi_{1}(x,s)s^{\frac{n}{p}}} \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

Since \(\alpha\ge0\), and \((\varphi_{1},\varphi_{2})\) satisfies the condition (1.5),

$$\begin{aligned} & \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}} \, \frac{dt}{t} \\ &\quad = \int_{2r}^{\infty}\frac{ (1+\frac{t}{\rho(x_{0})} )^{\alpha} \Vert f \Vert _{L_{p}(B(x_{0},t))}}{ \operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac {n}{p}}}\frac{\operatorname{ess\inf} _{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}{ (1+\frac{t}{\rho(x_{0})} )^{\alpha}t^{\frac{n}{q}}} \,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \int_{2r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac{n}{p}}}{ (1+\frac{t}{\rho(x_{0})} )^{\alpha}t^{\frac{n}{q}}}\,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{-\alpha} \int_{r}^{\infty}\frac{\operatorname{ess\inf}_{t< s< \infty}\varphi _{1}(x,s)s^{\frac{n}{p}}}{ t^{\frac{n}{q}}}\,\frac{dt}{t} \\ &\quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{-\alpha} \varphi_{2}(x_{0},r). \end{aligned}$$
(3.5)

Then by Theorem 3.1 we get

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}} & \lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{M_{q,\varphi_{2}}^{\alpha,V}} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{-n/q} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{L_{p}(B(x_{0},r))} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{q}}}\, \frac{dt}{t} \\ & \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

Let \(q=\frac{n}{n-\beta-\nu}\), similar to the estimates of (3.5) we have

$$\begin{aligned} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\,\frac{dt}{t} \lesssim \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}} \biggl(1+\frac{r}{\rho(x_{0})} \biggr)^{-\alpha} \varphi_{2}(x_{0},r). \end{aligned}$$

Thus by Theorem 3.1 we get

$$\begin{aligned} \bigl\Vert \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{WM_{\frac{n}{n-\beta-\nu },\varphi_{2}}^{\alpha,V}} &\lesssim \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WM_{\frac {n}{n-\beta-\nu},\varphi_{2}}^{\alpha,V}} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{\beta-n} \bigl\Vert I_{\beta+\nu} \bigl( \vert f \vert \bigr) \bigr\Vert _{WL_{\frac {n}{n-\beta-\nu}}(B(x_{0},r))} \\ & \lesssim\sup_{x_{0}\in{\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho (x_{0})} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n-\beta-\nu}}\, \frac{dt}{t} \\ & \lesssim \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha,V}}. \end{aligned}$$

 □

4 Proof of Theorem 1.2

The statement is derived from the estimate (3.4). The estimation of the norm of the operator, that is, the boundedness in the non-vanishing space, immediately follows by Theorem 1.1. So we only have to prove that

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{p,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad \Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{q,\varphi_{2}}^{\alpha ,V} \bigl( \bigl[b,\mathcal{I}_{\beta}^{L} \bigr]f;x,r \bigr)=0 \end{aligned}$$
(4.1)

and

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{1,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad \Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in{\mathbb{R}^{n}}} \mathfrak{A}_{n/(n-\beta),\varphi_{2}}^{W,\alpha ,V} \bigl( \bigl[b,\mathcal {I}_{\beta}^{L} \bigr]f;x,r \bigr)=0. \end{aligned}$$
(4.2)

To show that \(\sup_{x\in{\mathbb{R}^{n}}} (1+\frac{r}{\rho (x)} )^{\alpha } \varphi_{2}(x,r)^{-1} r^{-n/p} \|[b,\mathcal{I}_{\beta}^{L}]f\| _{L_{q}(B(x,r))}<\varepsilon\) for small r, we split the right-hand side of (3.4):

$$\begin{aligned} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,r)^{-1} r^{-n/p} \bigl\Vert \bigl[b, \mathcal{I}_{\beta}^{L} \bigr]f \bigr\Vert _{L_{q}(B(x,r))}\leq C \bigl[I_{\delta_{0}}(x,r)+J_{\delta_{0}}(x,r) \bigr], \end{aligned}$$
(4.3)

where \(\delta_{0}>0\) (we may take \(\delta_{0}>1\)), and

$$\begin{aligned} I_{\delta_{0}}(x,r):=\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \int_{r}^{\delta_{0}} t^{-\frac{n}{q}-1} \Vert f \Vert _{L_{p}(B(x,t))} \,dt \end{aligned}$$

and

$$\begin{aligned} J_{\delta_{0}}(x,r):=\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \int_{\delta_{0}}^{\infty } t^{-\frac{n}{q}-1} \Vert f \Vert _{L_{p}(B(x,t))} \,dt \end{aligned}$$

and it is supposed that \(r<\delta_{0}\). We use the fact that \(f \in VM_{p,\varphi_{1}}^{\alpha ,V}({\mathbb{R}^{n}})\) and choose any fixed \(\delta_{0}>0\) such that

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}} \biggl(1+\frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{1}(x,t)^{-1} t^{-n/p} \Vert f \Vert _{L_{p}(B(x,t))}< \frac{\varepsilon}{2CC_{0}}, \end{aligned}$$

where C and \(C_{0}\) are constants from (1.6) and (4.3). This allows one to estimate the first term uniformly in \(r\in (0,\delta_{0})\):

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}}CI_{\delta_{0}}(x,r)< \frac{\varepsilon}{2},\quad 0< r< \delta_{0}. \end{aligned}$$

The estimation of the second term now my be made already by the choice of r sufficiently small. Indeed, thanks to the condition (2.6) we have

$$\begin{aligned} J_{\delta_{0}}(x,r)\leq c_{\sigma_{0}} \frac{ (1+\frac{r}{\rho (x)} )^{\alpha }}{\varphi_{1}(x,r)} \Vert f \Vert _{VM_{p,\varphi _{1}}^{\alpha ,V}}, \end{aligned}$$

where \(c_{\sigma_{0}}\) is the constant from (1.3). Then by (2.6) it suffices to choose r small enough such that

$$\begin{aligned} \sup_{x\in{\mathbb{R}^{n}}}\frac{ (1+\frac{r}{\rho(x)} )^{\alpha }}{\varphi_{2}(x,r)} \leq\frac{\varepsilon}{2c_{\sigma_{0}} \Vert f \Vert _{VM_{p,\varphi_{1}}^{\alpha ,V}}}, \end{aligned}$$

which completes the proof of (4.1).

The proof of (4.2) is similar to the proof of (4.1).

5 Conclusions

In this paper, we study the boundedness of the commutators \([b,\mathcal {I}_{\beta}^{L}]\) with \(b \in\Lambda_{\nu}^{\theta}(\rho)\) on local generalized Morrey spaces \(LM_{p,\varphi}^{\alpha,V,\{x_{0}\}}\), generalized Morrey spaces \(M_{p,\varphi}^{\alpha,V}\) and vanishing generalized Morrey spaces \(VM_{p,\varphi}^{\alpha,V}\) associated with the Schrödinger operator, respectively. When b belongs to \(\Lambda_{\nu}^{\theta}(\rho)\) with \(\theta >0\), \(0<\nu<1\) and \((\varphi_{1},\varphi_{2})\) satisfies some conditions, we show that the commutator operator \([b,\mathcal {I}_{\beta}^{L}]\) are bounded from \(LM_{p,\varphi_{1}}^{\alpha ,V,\{ x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\), from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{q,\varphi_{2}}^{\alpha ,V}\) and from \(VM_{p,\varphi_{1}}^{\alpha ,V}\) to \(VM_{q,\varphi_{2}}^{\alpha ,V}\), \(1/p-1/q=(\beta+\nu)/n\).

Our results about the boundedness of \([b,\mathcal{I}_{\beta}^{L}]\) with \(b \in\Lambda_{\nu}^{\theta}(\rho)\) from \(LM_{p,\varphi _{1}}^{\alpha ,V,\{x_{0}\}}\) to \(LM_{q,\varphi_{2}}^{\alpha ,V,\{x_{0}\}}\) (Theorem 1.1) are based on the local estimate for the commutators \([b,\mathcal {I}_{\beta}^{L}]\) (Theorem 3.1).