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 \(B\lesssim A\).