1 Introduction and results

In this paper, we consider the Schrödinger differential operator

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

where \(V(x)\) is a non-negative potential belonging to the reverse Hölder class \(\mathit{RH}_{q}\) for \(q\geq n/2\).

A non-negative locally \(L_{q}\) integrable function \(V(x)\) on \({\mathbb{R}^{n}}\) is said to belong to \(\mathit{RH}_{q}\), \(1< q\le \infty \), if there exists \(C>0\) such that the reverse Hölder inequality

$$\begin{aligned} \biggl(\frac{1}{ \vert B(x,r) \vert } \int_{B(x,r)}V^{q}(y)\,dy \biggr)^{1/q}\leq \biggl(\frac{C}{ \vert B(x,r) \vert } \int_{B(x,r)}V(y)\,dy \biggr) \end{aligned}$$
(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 non-negative polynomial, then \(V \in \mathit{RH}_{\infty }\). Obviously, \(\mathit{RH}_{q_{2}} \subset \mathit{RH}_{q_{1}}\), if \(q_{1}< q_{2}\). It is worth pointing out that the \(\mathit{RH}_{q}\) class is such that, if \(V \in \mathit{RH}_{q}\) for some \(q > 1\), then there exists an \(\epsilon> 0\), which depends only n and the constant C in (1.1), such that \(V \in \mathit{RH}_{q+\epsilon}\). Throughout this paper, we always assume that \(0 \neq V \in \mathit{RH}_{n/2}\).

For \(x\in {\mathbb{R}^{n}}\), the function \(\rho(x)\) is defined by

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

Obviously, \(0< m_{V}(x)<\infty \) if \(V \neq0\). In particular, \(m_{V}(x)=1\) when \(V =1\) and \(m_{V}(x) \sim1+ \vert x \vert \) when \(V(x) = \vert x \vert ^{2}\).

According to [1], the new BMO space \(\mathit{BMO}_{\theta}(\rho)\) 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 \(b_{B}=\frac{1}{ \vert B \vert }\int_{B} b(y)\,dy\). A norm for \(b \in \mathit{BMO}_{\theta}(\rho)\), denoted by \([b]_{\theta}\), is given by the infimum of the constants in the inequality above. Clearly, \(\mathit{BMO}\subset \mathit{BMO}_{\theta}(\rho)\).

The classical Morrey spaces were originally introduced by Morrey in [2] 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 [2–4]. The classical version of Morrey spaces is equipped with the norm

$$\begin{aligned} \Vert f \Vert _{M_{p,\lambda}}:=\sup_{x\in {\mathbb{R}}^{n}}\sup _{r>0} r^{-\frac {\lambda}{p}} \Vert f \Vert _{L_{p}(B(x,r))}, \end{aligned}$$

where \(0\le\lambda< n\) and \(1\le p<\infty\). The generalized Morrey spaces are defined with \(r^{\lambda}\) replaced by a general non-negative function \(\varphi(x,r)\) satisfying some assumptions (see, for example, [5–8]).

The vanishing Morrey space \(\mathit{VM}_{p,\lambda}\) in its classical version was introduced in [9], where applications to PDE were considered. We also refer to [10] and [11] for some properties of such spaces. This is a subspace of functions in \(M_{p,\lambda}({\mathbb{R}}^{n})\), which satisfy the condition

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in {\mathbb{R}}^{n}, 0< t< r} t^{-\frac{\lambda }{p}} \Vert f \Vert _{L_{p}(B(x,t))}=0. \end{aligned}$$

We now present the definition of generalized Morrey spaces (including weak version) associated with Schrödinger operator, which introduced by second author in [12].

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 \(V\in \mathit{RH}_{q}\), \(q\ge1\). We denote by \(M_{p,\varphi }^{\alpha ,V}=M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) the generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in L_{\mathrm{loc}}^{p}({\mathbb{R}}^{n})\) with finite norm

$$\begin{aligned} \Vert f \Vert _{M_{p,\varphi}^{\alpha ,V}}=\sup_{x\in {\mathbb{R}}^{n}, r>0} \biggl(1+ \frac {r}{\rho(x)} \biggr)^{\alpha}\varphi(x,r)^{-1} r^{-n/p} \Vert f \Vert _{L_{p}(B(x,r))}. \end{aligned}$$

Also \(WM_{p,\varphi}^{\alpha ,V}=WM_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) we denote the weak generalized Morrey space associated with Schrödinger operator, the space of all functions \(f\in WL_{\mathrm{loc}}^{p} ({\mathbb{R}}^{n})\) with

$$\begin{aligned} \Vert f \Vert _{WM_{p,\varphi}^{\alpha ,V}}=\sup_{x\in {\mathbb{R}}^{n}, r>0} \biggl(1+ \frac {r}{\rho(x)} \biggr)^{\alpha}\varphi(x,r)^{-1} r^{-n/p} \Vert f \Vert _{WL_{p}(B(x,r))}< \infty. \end{aligned}$$

Remark 1.1

  1. (i)

    When \(\alpha=0\), and \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) is the classical Morrey space \(L_{p,\lambda}({\mathbb{R}}^{n})\) introduced by Morrey in [2].

  2. (ii)

    When \(\varphi(x,r)=r^{(\lambda-n)/p}\), \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) is the Morrey space associated with Schrödinger operator \(L_{p,\lambda}^{\alpha ,V}({\mathbb{R}}^{n})\) studied by Tang and Dong in [13].

  3. (iii)

    When \(\alpha=0\), \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) is the generalized Morrey space \(M_{p,\varphi}({\mathbb{R}}^{n})\) introduced by Mizuhara and Nakai in [7, 8].

  4. (iv)

    The generalized Morrey space associated with Schrödinger operator \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) was introduced by the second author in [12].

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 1.2

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

$$\begin{aligned} \lim_{r\rightarrow0} \sup_{x \in {\mathbb{R}^{n}}} \mathfrak{A}_{p,\varphi }^{\alpha ,V}(f;x,r) = 0. \end{aligned}$$
(1.2)

The vanishing weak generalized Morrey space \(VWM_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) associated with Schrödinger operator 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 \(\mathit{VM}_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) and \(\mathit{VWM}_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) are Banach spaces with respect to the norm

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

respectively.

We define the Marcinkiewicz integral associated with the Schrödinger operator L by

$$\begin{aligned} \mu_{j}^{L} f(x)= \biggl( \int_{0}^{\infty}\biggl\vert \int_{ \vert x-y \vert \le t} K_{j}^{L} (x,y) f(y) \,dy \biggr\vert ^{2} \,\frac{dt}{t^{3}} \biggr)^{1/2}, \end{aligned}$$

where \(K_{j}^{L}(x,y)=\widetilde{K_{j}^{L}}(x,y) \vert x-y \vert \) and \(\widetilde {K_{j}^{L}}(x,y)\) is the kernel of \(R_{j}^{L}=\frac{\partial}{\partial x_{j}} L^{-1/2}\), \(j=1,\ldots,n\).

Let b be a locally integrable function, the commutator generalized by \(\mu_{j}^{L}\) and b be defined by

$$\begin{aligned} \bigl[b,\mu_{j}^{L}\bigr] f(x)= \biggl( \int_{0}^{\infty}\biggl\vert \int_{ \vert x-y \vert \le t} K_{j}^{L} (x,y) \bigl(b(x)-b(y) \bigr)f(y)\,dy \biggr\vert ^{2} \,\frac{dt}{t^{3}} \biggr)^{1/2}. \end{aligned}$$

Let \(\widetilde{K_{j}^{\triangle}}(x,y)\) denote the kernel of the classical Riesz transform \(R_{j}=\frac{\partial}{\partial x_{j}} \triangle^{-1/2}\). When \(V=0\), then \(K_{j}^{\triangle}(x,y) =\widetilde{K_{j}^{\triangle}}(x,y) \vert x-y \vert =\frac {(x_{j}-y_{j})/ \vert x-y \vert }{ \vert x-y \vert ^{n-1}}\). Obviously, \(\mu_{j}^{\triangle}f(x)\) is the classical Marcinkiewicz integral. Therefore, it will be an interesting thing to study the property of \(\mu_{j}^{L}\).

The area of Marcinkiewicz integral associated with the Schrödinger operator has been under intensive research recently. Gao and Tang in [14] showed that \(\mu_{j}^{L}\) is bounded on \(L_{p}({\mathbb{R}}^{n})\) for \(1< p<\infty\), and bounded from \(L_{1}({\mathbb{R}}^{n})\) to weak \(WL_{1}({\mathbb{R}}^{n})\). Chen and Zou in [15] proved that the commutator \([b,\mu_{j}^{L}]\) is bounded on \(L_{p}({\mathbb{R}}^{n})\) for \(1< p<\infty\), where b belongs to \(\mathit{BMO}_{\theta}(\rho)\). In [16–18], Akbulut et al. obtained the boundedness of \(\mu_{j}^{L}\) and \([b,\mu _{j}^{L}]\) on the generalized Morrey space \(M_{p,\varphi}\), Chen and Jin in [19] showed the boundedness of \(\mu_{j}^{L}\) and \([b,\mu_{j}^{L}]\) on the Morrey spaces \(L_{p,\lambda}^{\alpha ,V}\) associated with Schrödinger operator.

In this paper, we study the boundedness of the Marcinkiewicz integral operators \(\mu_{j}^{L}\) on generalized Morrey space \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) associated with Schrödinger operator and vanishing generalized Morrey space \(\mathit{VM}_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) associated with Schrödinger operator. When b belongs to the new BMO function spaces \(\mathit{BMO}_{\theta}(\rho)\), we also show that \([b,\mu_{j}^{L}]\) is bounded on \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\).

Definition 1.3

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

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

respectively.

For the non-triviality of the space \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in\Omega_{p}^{\alpha ,V}\). Our main results are as follows.

Theorem 1.1

Let \(V\in \mathit{RH}_{n/2}\), \(\alpha\ge0\), \(1\le p<\infty\) and \(\varphi_{1},\varphi_{2} \in\Omega_{p}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\frac{\operatorname {ess\,sup}_{t< s< \infty}\varphi_{1}(x,s)s^{\frac {n}{p}}}{t^{\frac{n}{p}}} \,\frac{dt}{t} \le c_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.3)

where \(c_{0}\) does not depend on x and r. Then the operator \(\mu _{j}^{L}\) is bounded from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{p,\varphi_{2}}^{\alpha ,V}\) for \(p>1\) and from \(M_{1,\varphi_{1}}^{\alpha ,V}\) to \(WM_{1,\varphi_{2}}^{\alpha ,V}\). Moreover, for \(p>1\)

$$\begin{aligned} \bigl\Vert \mu_{j}^{L} f \bigr\Vert _{M_{p,\varphi_{2}}^{\alpha ,V}}\le C \Vert f \Vert _{M_{p,\varphi _{1}}^{\alpha ,V}} \end{aligned}$$

and for \(p=1\)

$$\begin{aligned} \bigl\Vert \mu_{j}^{L} f \bigr\Vert _{WM_{1,\varphi_{2}}^{\alpha ,V}}\le C \Vert f \Vert _{M_{1,\varphi _{1}}^{\alpha ,V}}. \end{aligned}$$

Theorem 1.2

Let \(V\in \mathit{RH}_{n/2}\), \(b \in \mathit{BMO}_{\theta}(\rho)\), \(1< p<\infty\), and \(\varphi_{1},\varphi_{2} \in\Omega_{p}^{\alpha ,V}\) satisfy the condition

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

where \(c_{0}\) does not depend on x and r. Then the operator \([b,\mu _{j}^{L}]\) is bounded from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{p,\varphi_{2}}^{\alpha ,V}\) and

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

Definition 1.4

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

$$\begin{aligned} \inf_{x\in {\mathbb{R}^{n}}} \inf_{r>\delta} \biggl(1+\frac{r}{\rho(x)} \biggr)^{-\alpha } \varphi(x,r)>0, \quad \mbox{for some } \delta>0, \end{aligned}$$
(1.5)

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 \(\mathit{VM}_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}})\) we always assume that \(\varphi\in\Omega_{p,1}^{\alpha ,V}\).

Theorem 1.3

Let \(V\in \mathit{RH}_{n/2}\), \(\alpha\ge0\), \(1\le p<\infty\) and \(\varphi_{1},\varphi_{2} \in\Omega_{p,1}^{\alpha ,V}\) satisfy the condition

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

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

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

where \(C_{0}\) does not depend on \(x\in {\mathbb{R}^{n}}\) and \(r>0\). Then the operator \(\mu_{j}^{L}\) is bounded from \(\mathit{VM}_{p,\varphi_{1}}^{\alpha ,V}\) to \(\mathit{VM}_{p,\varphi_{2}}^{\alpha ,V}\) for \(p>1\) and from \(\mathit{VM}_{1,\varphi_{1}}^{\alpha ,V}\) to \(VWM_{1,\varphi_{2}}^{\alpha ,V}\).

Theorem 1.4

Let \(V\in \mathit{RH}_{n/2}\), \(b \in \mathit{BMO}_{\theta}(\rho)\), \(1< p<\infty\), and \(\varphi_{1},\varphi_{2} \in\Omega_{p,1}^{\alpha ,V}\) satisfy the condition

$$\begin{aligned} \int_{r}^{\infty}\biggl(1+\ln\frac{t}{r} \biggr) \varphi_{1}(x,t) \,\frac{dt}{t}\le c_{0} \varphi_{2}(x,r), \end{aligned}$$
(1.7)

where \(c_{0}\) does not depend on x and r,

$$\begin{aligned} \lim_{r \to0} \frac{\ln\frac{1}{r}}{\inf_{x \in {\mathbb{R}^{n}}}\varphi_{2}(x,r)}=0 \end{aligned}$$
(1.8)

and

$$\begin{aligned} c_{\delta}:= \int_{\delta}^{\infty} \bigl(1+ \vert \ln t \vert \bigr) \sup_{x\in {\mathbb{R}^{n}}}\varphi_{1}(x,t) \,\frac{dt}{t} < \infty \end{aligned}$$
(1.9)

for every \(\delta>0\).

Then the operator \([b,\mu_{j}^{L}]\) is bounded from \(\mathit{VM}_{p,\varphi_{1}}^{\alpha ,V}\) to \(\mathit{VM}_{p,\varphi_{2}}^{\alpha ,V}\).

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 preliminaries

We would like to recall the important properties concerning the function \(\rho(x)\).

Lemma 2.1

[20]

Let \(V\in \mathit{RH}_{n/2}\). 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

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

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

Proof

By (2.1) we get

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

 □

We give some inequalities about the new BMO space \(\mathit{BMO}_{\theta}(\rho)\).

Lemma 2.3

[1]

Let \(1\le s <\infty\). If \(b\in \mathit{BMO}_{\theta}(\rho)\), then

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

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).

Lemma 2.4

[1]

Let \(1\le s<\infty\), \(b\in \mathit{BMO}_{\theta}(\rho)\), and \(B=B(x,r)\). Then

$$\begin{aligned} \biggl(\frac{1}{ \vert 2^{k} B \vert } \int_{2^{k} B} \bigl\vert b(y)-b_{B} \bigr\vert ^{s} \,dy \biggr)^{1/s}\le [b]_{\theta}k \biggl(1+ \frac{2^{k} r}{\rho(x)} \biggr)^{\theta'} \end{aligned}$$

for all \(k\in {\mathbb{N}}\), with \(\theta'\) as in Lemma 2.3.

The following results give the estimates of the kernel of \(\mu_{j}^{L}\) the boundedness of \(\mu_{j}^{L}\) and their commutators on \(L_{p}\).

Lemma 2.5

[20]

If \(V\in \mathit{RH}_{n/2}\), then, for every N, there exists a constant C such that

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

Lemma 2.6

[16]

Let \(V\in \mathit{RH}_{n/2}\). Then

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{L_{p}({\mathbb{R}}^{n})} \le C \Vert f \Vert _{L_{p}({\mathbb{R}}^{n})} \end{aligned}$$

holds for \(1< p<\infty\), and

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{WL_{1}({\mathbb{R}}^{n})} \le C \Vert f \Vert _{L_{1}({\mathbb{R}}^{n})}. \end{aligned}$$

Lemma 2.7

[15]

Let \(V\in \mathit{RH}_{n/2}\), \(1< p<\infty\) and \(b\in \mathit{BMO}_{\theta}(\rho)\). Then

$$\begin{aligned} \bigl\Vert \bigl[b,\mu_{j}^{L}\bigr](f) \bigr\Vert _{L_{p}({\mathbb{R}}^{n})}\le C [b]_{\theta} \Vert f \Vert _{L_{p}({\mathbb{R}}^{n})}. \end{aligned}$$

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

Lemma 2.8

[21]

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

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

Lemma 2.9

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

  1. (i)

    If

    $$\begin{aligned} \sup_{t< r< \infty } \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \frac {r^{-\frac{n}{p}}}{\varphi(x,r)}=\infty\quad\textit{for some } t>0 \textit{ and for all } x\in {\mathbb{R}^{n}}, \end{aligned}$$
    (2.3)

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

  2. (ii)

    If

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

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

Proof

(i) Let (2.3) be satisfied and f be not equivalent to zero. Then \(\sup_{x\in {\mathbb{R}^{n}}} \Vert f \Vert _{L_{p}(B(x,t))}>0 \), hence

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

Therefore \(\Vert f \Vert _{M_{p,\varphi}^{\alpha ,V}}=\infty\).

(ii) Let \(f\in M_{p,\varphi}^{\alpha ,V}({\mathbb{R}^{n}}) \) and (2.4) be satisfied. Then there are two possibilities:

  • Case 1: \(\sup_{ 0< r< t} (1+\frac{r}{\rho(x)} )^{\alpha } \varphi(x,r)^{-1}=\infty\) for all \(t>0\).

  • Case 2: \(\sup_{ 0< r< t} (1+\frac{r}{\rho(x)} )^{\alpha } \varphi(x,r)^{-1}<\infty\) for some \(t\in(0,\tau)\).

For Case 1, by the Lebesgue differentiation theorem, for almost all \(x\in {\mathbb{R}^{n}}\),

$$\begin{aligned} \lim_{r\to0+}\frac{ \Vert f\chi_{B(x,r)} \Vert _{L_{p}}}{ \Vert \chi_{B(x,r)} \Vert _{L_{p}}}= \bigl\vert f(x) \bigr\vert . \end{aligned}$$
(2.5)

We claim that \(f(x)=0 \) for all those x. Indeed, fix x and assume \(\vert f(x) \vert >0\). Then by (2.5) there exists \(t_{0}>0 \) such that

$$\begin{aligned} r^{-\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x,r))}\geq2^{-1} v_{n}^{\frac{1}{p}} \bigl\vert f(x) \bigr\vert \end{aligned}$$

for all \(0< r\leq t_{0}\), where \(v_{n}\) is the volume of the unit ball in \({\mathbb{R}^{n}}\). Consequently,

$$\begin{aligned} \Vert f \Vert _{M_{p,\varphi}^{\alpha ,V}} & \geq\sup_{0< r< t_{0}} \biggl(1+ \frac{r}{\rho(x)} \biggr)^{\alpha } \varphi(x,r)^{-1} r^{-\frac {n}{p}} ~ \Vert f \Vert _{L_{p}(B(x,r))} \\ & \geq2^{-1} v_{n}^{\frac{1}{p}} \bigl\vert f(x) \bigr\vert \sup_{0< r< t_{0}} \biggl(1+\frac {r}{\rho(x)} \biggr)^{\alpha } \varphi(x,r)^{-1}. \end{aligned}$$

Hence \(\Vert f \Vert _{M_{p,\varphi}^{\alpha ,V}}=\infty\), so \(f\notin M_{p,\varphi}({\mathbb{R}^{n}}) \) and we arrive at a contradiction.

Note that Case 2 implies that \(\sup_{t< r<\tau} (1+\frac{r}{\rho (x)} )^{\alpha } \varphi(x,r)^{-1}=\infty\), hence

$$\begin{aligned} \sup_{s< r< \infty} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi (x,r)^{-1} r^{-\frac{n}{p}} & \geq\sup_{t< r< \tau} \biggl(1+\frac {r}{\rho(x)} \biggr)^{\alpha } \varphi(x,r)^{-1} r^{-\frac{n}{p}} \\ & \geq\tau^{-\frac{n}{p}} \sup_{t< r< \tau} \biggl(1+ \frac{r}{\rho (x)} \biggr)^{\alpha } \varphi(x,r)^{-1}=\infty, \end{aligned}$$

which is the case in (i). □

3 Proof of Theorem 1.1

We first prove the following conclusions.

Theorem 3.1

Let \(V\in \mathit{RH}_{n/2}\). If \(1< p<\infty \), then the inequality

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{L_{p}(B(x_{0},r))} \lesssim r^{\frac{n}{p}} \int _{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \end{aligned}$$

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

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{WL_{1}(B(x_{0},r))} \lesssim r^{n} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n}}\,\frac{dt}{t} \end{aligned}$$

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

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

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{L_{p}(B(x_{0},r))} \le \bigl\Vert \mu_{j}^{L}(f_{1}) \bigr\Vert _{L_{p}(B(x_{0},r))}+ \bigl\Vert \mu_{j}^{L}(f_{2}) \bigr\Vert _{L_{p}(B(x_{0},r))}. \end{aligned}$$

Since \(f_{1}\in L_{p}({\mathbb{R}^{n}})\) and from the boundedness of \(\mu_{j}^{L}\) on \(L_{p}({\mathbb{R}}^{n})\), \(p>1\), it follows that

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f_{1}) \bigr\Vert _{L_{p}(B(x_{0},r))} &\lesssim \Vert f \Vert _{L_{p}(B(x_{0},r))} \\ & \lesssim r^{\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x_{0},2r))} \int_{2r}^{\infty}\frac{dt}{t^{\frac{n}{p}+1}} \\ & \lesssim r^{\frac{n}{p}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}} \,\frac{dt}{t}. \end{aligned}$$
(3.1)

To estimate \(\Vert \mu_{j}^{L}(f_{2}) \Vert _{L_{p}(B(x_{0},2r))}\) obverse that \(x\in B\), \(y\in(2B)^{c}\) implies \(\frac{1}{2} \vert x_{0}-y \vert \le \vert x-y \vert \le\frac {3}{2} \vert x_{0}-y \vert \). Then by (2.2) and Minkowski’s inequality

$$\begin{aligned} \sup_{x\in B(x_{0},r)}\mu_{j}^{L}(f_{2}) (x) & \lesssim \int_{(2B)^{c}}\frac { \vert f(y) \vert }{ \vert x_{0}-y \vert ^{n-1}} \biggl( \int_{ \vert x_{0}-y \vert }^{\infty}\,\frac{dt}{t^{3}} \biggr)^{1/2} \,dy \\ & \lesssim\sum_{k=1}^{\infty}\bigl(2^{k+1} r\bigr)^{-n} \int_{2^{k+1}B} \bigl\vert f(y) \bigr\vert \,dy. \end{aligned}$$

By Hölder’s inequality we get

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

Then

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f_{2}) \bigr\Vert _{L_{p}(B(x_{0},r))}\lesssim r^{\frac{n}{p}} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \end{aligned}$$
(3.3)

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

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{L_{p}(B(x_{0},r))}\lesssim r^{\frac{n}{p}} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \end{aligned}$$
(3.4)

holds for \(1< p<\infty\).

When \(p=1\), from the boundedness of \(\mu_{j}^{L}\) from \(L_{1}({\mathbb{R}}^{n})\) to \(WL_{1}({\mathbb{R}}^{n})\), we get

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f_{1}) \bigr\Vert _{WL_{1}(B(x_{0},r))} & \lesssim \Vert f \Vert _{L_{1}(B(x_{0},r))} \lesssim r^{n} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n}}\,\frac{dt}{t}. \end{aligned}$$

From (3.3) we have

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f_{2}) \bigr\Vert _{WL_{1}(B(x_{0},r))} & \le \bigl\Vert \mu_{j}^{L}(f_{2}) \bigr\Vert _{L_{1}(B(x_{0},r))} \lesssim r^{n} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n}}\,\frac{dt}{t}. \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{WL_{1}(B(x_{0},r))} \lesssim r^{n} \int_{2 r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{n}}\,\frac{dt}{t}. \end{aligned}$$

 □

Remark 3.1

Note that another proof of Theorem 3.1 is given in [16].

Proof of Theorem 1.1

From Lemma 2.8, 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 \(\Vert f \Vert _{L_{p}(B(x_{0},t))}\) is a nondecreasing function of t, and \(f\in M_{p,\varphi_{1}}^{\alpha ,V}\), then

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

Since \(\alpha\ge0\), and \((\varphi_{1},\varphi_{2})\) satisfies the condition (1.3), then

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

Then by Theorem 3.1 we have

$$\begin{aligned} & \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{M_{p,\varphi_{2}}^{\alpha ,V}} \\ & \quad \lesssim\sup_{x_{0}\in {\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{-n/p} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{L_{p}(B(x_{0},r))} \\ & \quad \lesssim\sup_{x_{0}\in {\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \\ & \quad \lesssim \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha ,V}}. \end{aligned}$$

Let \(p=1\). Similar to (3.5) we get

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

From Theorem 3.1 we have

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{WM_{1,\varphi_{2}}^{\alpha ,V}} &\lesssim\sup_{x_{0}\in {\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} r^{-n/p} \bigl\Vert \mu_{j}^{L}(f) \bigr\Vert _{WL_{1}(B(x_{0},r))} \\ &\lesssim\sup_{x_{0}\in {\mathbb{R}}^{n}, r>0} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha}\varphi_{2}(x_{0},r)^{-1} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{1}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \\ &\lesssim \Vert f \Vert _{M_{1,\varphi_{1}}^{\alpha ,V}}. \end{aligned}$$

 □

4 Proof of Theorem 1.2

Similar to the proof of Theorem 1.1, it suffices to prove the following result.

Theorem 4.1

Let \(V\in \mathit{RH}_{n/2}\), \(b\in \mathit{BMO}_{\theta}(\rho)\). If \(1< p<\infty\), then the inequality

$$\begin{aligned} & \bigl\Vert \bigl[b,\mu_{j}^{L}(f)\bigr] \bigr\Vert _{L_{p}(B(x_{0},r))}\lesssim[b]_{\theta}r^{\frac {n}{p}} \int_{2r}^{\infty}\biggl(1+\ln\frac{t}{r} \biggr)\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac {n}{p}}}\,\frac{dt}{t} \end{aligned}$$
(4.1)

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

Proof

We write f as \(f=f_{1}+f_{2}\), where \(f_{1}(y)=f(y)\chi _{B(x_{0},2r)}(y)\). Then

$$\begin{aligned} \bigl\Vert \bigl[b,\mu_{j}^{L}\bigr](f) \bigr\Vert _{L_{p}(B(x_{0},r))}\le \bigl\Vert \bigl[b,\mu_{j}^{L} \bigr](f_{1}) \bigr\Vert _{L_{p}(B(x_{0},r))}+ \bigl\Vert \bigl[b, \mu_{j}^{L}\bigr](f_{2}) \bigr\Vert _{L_{p}(B(x_{0},r))}. \end{aligned}$$

From the boundedness of \([b,\mu_{j}^{L}]\) on \(L_{p}({\mathbb{R}}^{n})\) and (3.1) we get

$$\begin{aligned} \bigl\Vert \bigl[b,\mu_{j}^{L} \bigr](f_{1}) \bigr\Vert _{L_{p}(B(x_{0},r))} & \lesssim[b]_{\theta} \Vert f \Vert _{L_{p}(B(x_{0},2r))} \\ & \lesssim[b]_{\theta}r^{\frac{n}{p}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \\ & \lesssim[b]_{\theta}r^{\frac{n}{p}} \int_{2r}^{\infty}\biggl(1+\ln\frac{t}{r} \biggr)\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac {n}{p}}}\,\frac{dt}{t}. \end{aligned}$$
(4.2)

We now turn to deal with the term \(\Vert [b,\mu_{j}^{L}](f_{2}) \Vert _{L_{p}(B(x_{0},r))}\). For any given \(x\in B(x_{0},2r)\) we have

$$\begin{aligned} \bigl[b,\mu_{j}^{L}\bigr](f_{2}) (x) & = \biggl( \int_{0}^{\infty}\biggl\vert \int_{ \vert x-y \vert \le t}K_{j}^{L}(x,y) \bigl(b(x)-b(y) \bigr)f(y) \,dy \biggr\vert ^{2} \,\frac{dt}{t^{3}} \biggr)^{1/2} \\ & \le \bigl\vert b(x)-b_{2B} \bigr\vert \mu_{j}^{L}(f_{2}) (x)+\mu_{j}^{L}\bigl((b-b_{2B})f_{2} \bigr) (x). \end{aligned}$$

By (2.2), Lemma 2.2 and (3.2) we have

$$\begin{aligned} \sup_{x\in B(x_{0},r)}\mu_{j}^{L}(f_{2}) (x) & \lesssim \int_{(2B)^{c}}\frac {1}{ (1 +\frac{ \vert x-y \vert }{\rho(x)} )^{N}}\frac{ \vert f(y) \vert }{ \vert x_{0}-y \vert ^{n-1}} \biggl( \int_{ \vert x_{0}-y \vert }^{\infty}\frac{dt}{t^{3}} \biggr)^{1/2} \,dy \\ & \lesssim\frac{1}{ (1+\frac{2r}{\rho(x)} )^{N}}\sum_{k=1}^{\infty}\bigl(2^{k+1} r\bigr)^{-n} \int_{2^{k+1}B} \bigl\vert f(y) \bigr\vert \,dy \\ & \lesssim\frac{1}{ (1+\frac{2r}{\rho(x_{0})} )^{N/(k_{0}+1)}} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. \end{aligned}$$

Then by Lemma 2.3, and taking \(N\ge(k_{0}+1)\theta\) we get

$$\begin{aligned} \bigl\Vert \bigl(b(x)-b_{2B}\bigr) \mu_{j}^{L}(f_{2}) \bigr\Vert _{L_{p}(B(x_{0},r))} &\lesssim[b]_{\theta}r^{\frac{n}{p}} \biggl(1+\frac{2r}{\rho (x_{0})} \biggr)^{\theta-N/(k_{0}+1)} \int_{2r}^{\infty}\frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac {dt}{t} \\ &\lesssim[b]_{\theta}r^{\frac{n}{p}} \int_{2r}^{\infty}\biggl(1+\ln \frac{t}{r} \biggr) \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. \end{aligned}$$
(4.3)

Finally, let us estimate \(\Vert \mu_{j}^{L}((b-b_{2B})f_{2}) \Vert _{L_{p}(B(x_{0},r))}\). By (2.2), Lemma 2.2 and (3.2) we have

$$\begin{aligned} \mu_{j}^{L}\bigl((b-b_{2B})f_{2} \bigr) (x) & \lesssim \int_{(2B)^{c}}\frac{1}{ (1+\frac{ \vert x-y \vert }{\rho(x)} )^{N}} \frac{ \vert b(y)-b_{2B} \vert \vert f(y) \vert }{ \vert x_{0}-y \vert ^{n-1}} \biggl( \int_{ \vert x_{0}-y \vert }^{\infty}\frac{dt}{t^{3}} \biggr)^{1/2} \,dy \\ &\lesssim\sum_{k=1}^{\infty}\frac{1}{(2^{k} r)^{n} (1+\frac{2^{k} r}{\rho(x)} )^{N}} \int_{2^{k+1}B} \bigl\vert b(y)-b_{2B} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ & \lesssim\sum_{k=1}^{\infty}\frac{1}{(2^{k} r)^{n} (1+\frac{2^{k} r}{\rho(x_{0})} )^{N/(k_{0}+1)}} \int_{2^{k+1}B} \bigl\vert b(y)-b_{2B} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy. \end{aligned}$$

Note that

$$\begin{aligned} \int_{2^{k+1}B} \bigl\vert b(y)-b_{2B} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy & \lesssim \biggl( \int_{2^{k+1}B} \bigl\vert b(y)-b_{2B} \bigr\vert ^{p'} \biggr)^{1/p'} \Vert f \Vert _{L_{p}(B(x_{0},2^{k+1}r))} \\ &\lesssim[b]_{\theta}k \biggl(1+\frac{2^{k} r}{\rho(x_{0})} \biggr)^{\theta'} \bigl(2^{k} r\bigr)^{\frac{n}{p'}} \Vert f \Vert _{L_{p}(B(x_{0},2^{k+1}r))}. \end{aligned}$$

Then

$$\begin{aligned} \sup_{x\in B(x_{0},r)}\mu_{j}^{L} \bigl((b-b_{B})f_{2}\bigr) (x) &\lesssim[b]_{\theta}\sum_{k=1}^{\infty}\frac{k}{ (1+\frac{2^{k} r}{\rho(x_{0})} )^{N/(k_{0}+1)-\theta'}} \bigl(2^{k} r\bigr)^{-\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x_{0},2^{k+1}r))} \\ &\lesssim[b]_{\theta}\sum_{k=1}^{\infty}k\bigl(2^{k} r\bigr)^{-\frac{n}{p}} \Vert f \Vert _{L_{p}(B(x_{0},2^{k+1}r))} \\ &\lesssim[b]_{\theta}\sum_{k=1}^{\infty}k \int_{2^{k} r}^{2^{k+1}r} \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. \end{aligned}$$

Since \(2^{k} r\le t \le2^{k+1}r\), \(k\approx\ln\frac{t}{r}\). Thus

$$\begin{aligned} \sup_{x\in B(x_{0},r)}\mu_{j}^{L} \bigl((b-b_{B})f_{2}\bigr) (x) &\lesssim[b]_{\theta}\sum_{k=1}^{\infty}k \int_{2^{k} r}^{2^{k+1}r} \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \\ &\lesssim[b]_{\theta}\sum_{k=1}^{\infty}\int_{2^{k} r}^{2^{k+1}r} \ln \frac{t}{r} \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t} \\ &\lesssim[b]_{\theta}\int_{2r}^{\infty}\biggl(1+\ln\frac{t}{r} \biggr) \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert \mu_{j}^{L}\bigl((b-b_{2B})f_{2} \bigr) \bigr\Vert _{L_{p}(B(x_{0},r))}\lesssim [b]_{\theta}r^{\frac{n}{p}} \int_{2r}^{\infty}\biggl(1+\ln\frac {t}{r} \biggr) \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. \end{aligned}$$
(4.4)

Combining (4.2), (4.3) and (4.4), the proof of Theorem 4.1 is completed. □

Remark 4.1

Note that, in the case \(b \in \mathit{BMO}\), Theorem 4.1 was proved in [18].

Proof of Theorem 1.2

Since \(f\in M_{p,\varphi _{1}}^{\alpha ,V}\) and \((\varphi_{1},\varphi_{2})\) satisfies the condition (1.4), by (3.5) we have

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

Then from Theorem 4.1 we get

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

 □

5 Proof of Theorem 1.3

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 from Theorem 1.1. So we only have to prove that

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in {\mathbb{R}^{n}}} \mathfrak{A}_{p,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad\Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in {\mathbb{R}^{n}}} \mathfrak{A}_{p,\varphi_{2}}^{\alpha ,V} \bigl(\mu_{j}^{L} (f);x,r\bigr)=0 \end{aligned}$$
(5.1)

and

$$\begin{aligned} \lim_{r\rightarrow0}\sup_{x\in {\mathbb{R}^{n}}} \mathfrak{A}_{1,\varphi_{1}}^{\alpha ,V}(f;x,r)=0\quad\Rightarrow\quad \lim _{r\rightarrow0}\sup_{x\in {\mathbb{R}^{n}}} \mathfrak{A}_{1,\varphi_{2}}^{W,\alpha ,V} \bigl(\mu_{j}^{L} (f);x,r\bigr)=0. \end{aligned}$$
(5.2)

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

$$\begin{aligned} \biggl(1+\frac{r}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,r)^{-1} r^{-n/p} \bigl\Vert \mu_{j}^{L} (f) \bigr\Vert _{L_{p}(B(x,r))}\leq C \bigl[I_{\delta_{0}}(x,r)+J_{\delta_{0}}(x,r)\bigr], \end{aligned}$$
(5.3)

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

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

and

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

and it is supposed that \(r<\delta_{0}\). We use the fact that \(f \in \mathit{VM}_{p,\varphi_{1}}^{\alpha ,V}({\mathbb{R}^{n}})\) 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 (5.3). This allows us to estimate the first term uniformly in \(r\in (0,\delta_{0})\):

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

The estimation of the second term now can be made by the choice of sufficiently small \(r>0\). Indeed, thanks to the condition (1.5) we have

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

where \(c_{\sigma_{0}}\) is the constant from (1.2). Then, by (1.5) it suffices to choose r small enough so that

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

which completes the proof of (5.1).

The proof of (5.2) is similar to the proof of (5.1).

6 Proof of Theorem 1.4

The norm inequality is provided by Theorem 1.2, therefore, we only have to prove the implication

$$\begin{aligned} & \lim_{r\rightarrow0}\sup_{x\in {\mathbb{R}^{n}}} \biggl(1+ \frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{1}(x,t)^{-1} t^{-n/p} \Vert f \Vert _{L_{p}(B(x,t))} = 0 \\ & \quad \Longrightarrow\quad \lim_{r\rightarrow0} \sup_{x\in {\mathbb{R}^{n}}} \biggl(1+\frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,t)^{-1} t^{-n/p} \bigl\Vert \bigl[b,\mu_{j}^{L} (f)\bigr] \bigr\Vert _{L_{p}(B(x,t))} =0. \end{aligned}$$

To check that

$$\sup_{x\in {\mathbb{R}^{n}}} \biggl(1+\frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,t)^{-1} t^{-n/p} \bigl\Vert \bigl[b, \mu_{j}^{L} (f)\bigr] \bigr\Vert _{L_{p}(B(x,t))}< \varepsilon\quad \mbox{for small } r, $$

we use the estimate (4.1):

$$\varphi_{2}(x,t)^{-1} t^{-n/p} \bigl\Vert \bigl[b,\mu_{j}^{L} (f)\bigr] \bigr\Vert _{L_{p}(B(x,t))} \lesssim \frac{[b]_{\theta}}{\varphi_{2}(x,r)} \int_{r}^{\infty} \biggl(1+\ln\frac{t}{r} \biggr) \frac{ \Vert f \Vert _{L_{p}(B(x_{0},t))}}{t^{\frac{n}{p}}}\,\frac{dt}{t}. $$

We take \(r<\delta_{0}\) where \(\delta_{0}\) will be chosen small enough and split the integration:

$$\begin{aligned} \biggl(1+\frac{t}{\rho(x)} \biggr)^{\alpha } \varphi_{2}(x,t)^{-1} t^{-n/p} \bigl\Vert \bigl[b, \mu_{j}^{L} (f)\bigr] \bigr\Vert _{L_{p}(B(x,t))} \leq C \bigl[I_{\delta_{0}}(x,r)+J_{\delta_{0}}(x,r)\bigr], \end{aligned}$$
(6.1)

where

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

and

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

We choose a fixed \(\delta_{0}>0\) such that

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

where C and \(C_{0}\) are constants from (6.1) and (1.7), which yields the estimate of the first term uniform in \(r\in(0,\delta_{0})\): \(\sup_{x\in {\mathbb{R}^{n}}}\mathit{CI}_{\delta_{0}}(x,r)<\frac{\varepsilon}{2}\), \(0< r<\delta_{0}\).

For the second term, writing \(1+\ln\frac{t}{r}\le1+ \vert \ln t \vert +\ln \frac{1}{r}\), we obtain

$$J_{\delta_{0}}(x,r)\leq\frac{c_{\delta_{0}}+ \widetilde{c_{\delta_{0}}} \ln\frac{1}{r}}{\varphi_{2}(x,r)} \Vert f \Vert _{M_{p,\varphi_{1}}^{\alpha ,V}}, $$

where \(c_{\delta_{0}}\) is the constant from (1.9) with \(\delta =\delta_{0}\) and \(\widetilde{c_{\delta_{0}}}\) is a similar constant with omitted logarithmic factor in the integrand. Then, by (1.8) we can choose small r such that \(\sup_{x\in {\mathbb{R}^{n}}}J_{\delta_{0}}(x,r)<\frac{\varepsilon}{2}\), which completes the proof.

7 Conclusions

In this paper, we study the boundedness of the Marcinkiewicz integral operators \(\mu_{j}^{L}\) and their commutators \([b,\mu_{j}^{L}]\) with \(b \in \mathit{BMO}_{\theta}(\rho)\) on generalized Morrey spaces \(M_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) associated with Schrödinger operator and vanishing generalized Morrey spaces \(\mathit{VM}_{p,\varphi}^{\alpha ,V}({\mathbb{R}}^{n})\) associated with Schrödinger operator. We find the sufficient conditions on the pair \((\varphi_{1},\varphi_{2})\) which ensure the boundedness of the operators \(\mu_{j}^{L}\) from one vanishing generalized Morrey space \(\mathit{VM}_{p,\varphi_{1}}^{\alpha ,V}\) to another \(\mathit{VM}_{p,\varphi_{2}}^{\alpha ,V}\), \(1< p<\infty\) and from the space \(\mathit{VM}_{1,\varphi_{1}}^{\alpha ,V}\) to the weak space \(VWM_{1,\varphi_{2}}^{\alpha ,V}\). When b belongs to \(\mathit{BMO}_{\theta}(\rho)\) and \((\varphi_{1},\varphi _{2})\) satisfies some conditions, we also show that \([b,\mu_{j}^{L}]\) is bounded from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{p,\varphi_{2}}^{\alpha ,V}\) and from \(\mathit{VM}_{p,\varphi_{1}}^{\alpha ,V}\) to \(\mathit{VM}_{p,\varphi_{2}}^{\alpha ,V}\), \(1< p<\infty\).

Our results about the boundedness of \(\mu_{j}^{L}\) and \([b,\mu_{j}^{L}]\) from \(M_{p,\varphi_{1}}^{\alpha ,V}\) to \(M_{p,\varphi_{2}}^{\alpha ,V}\) (Theorems 1.1 and 1.2) are based on the local estimates for the Marcinkiewicz integrals (Theorem 3.1) and their commutators (Theorem 4.1).