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

Boundary Value Problems20172017:121

https://doi.org/10.1186/s13661-017-0851-4

Received: 12 May 2017

Accepted: 4 August 2017

Published: 21 August 2017

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 \(\mathit{VM}_{p,\varphi_{2}}^{\alpha,V}\), \(1< p<\infty\).

Keywords

Schrödinger operatorMarcinkiewicz integralvanishing generalized Morrey spacecommutatorBMO

MSC

42B3535J10

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 [24]. 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, [58]).
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 [1618], 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).

Declarations

Acknowledgements

The authors thank the referees for careful reading the paper and useful comments.

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

Authors’ Affiliations

(1)
Department of Mathematics, Ahi Evran University, Kirsehir, Turkey
(2)
S.M. Nikolskii Institute of Mathematics at RUDN University, Moscow, Russia
(3)
Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan
(4)
Baku State University, Baku, Azerbaijan

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