Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces

Abstract

Let $L=-\Delta+V$ be a Schrödinger operator, where Δ is the Laplacian on $\mathbb{R}^{n}$ and the non-negative potential V belongs to the reverse Hölder class $\mathit{RH}_{q}$ for $q \ge n/2$. 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 $\mathit{VM}_{p,\varphi_{2}}^{\alpha,V}$, $1< p<\infty$.

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

• (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].

• (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].

• (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].

• (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 $V\in \mathit{RH}_{q}$, $q\ge1$.

• (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 ${\mathbb{R}^{n}}$.

• (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).