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 \(\mathit{VM}_{p,\varphi_{2}}^{\alpha,V}\), \(1< p<\infty\).
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1 Introduction and results
In this paper, we consider the Schrödinger differential operator
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
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
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
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
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
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
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
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
and
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
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
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
respectively.
We define the Marcinkiewicz integral associated with the Schrödinger operator L by
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
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 \(t>0\),
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
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 \(p>1\)
and for \(p=1\)
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
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
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
and
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
for every \(\delta>0\), and
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
where \(c_{0}\) does not depend on x and r,
and
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
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
Proof
By (2.1) we get
□
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
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
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
Lemma 2.6
[16]
Let \(V\in \mathit{RH}_{n/2}\). Then
holds for \(1< p<\infty\), and
Lemma 2.7
[15]
Let \(V\in \mathit{RH}_{n/2}\), \(1< p<\infty\) and \(b\in \mathit{BMO}_{\theta}(\rho)\). Then
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
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
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 \(x\in {\mathbb{R}^{n}}\),
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
for all \(0< r\leq t_{0}\), where \(v_{n}\) is the volume of the unit ball in \({\mathbb{R}^{n}}\). Consequently,
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
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
holds for any \(f\in L_{\mathrm{loc}}^{p}({\mathbb{R}}^{n})\). Moreover, for \(p=1\) the inequality
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
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
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
By Hölder’s inequality we get
Then
holds for \(1\le p<\infty\). Therefore, by (3.1) and (3.3) we get
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
From (3.3) we have
Then
□
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
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
Since \(\alpha\ge0\), and \((\varphi_{1},\varphi_{2})\) satisfies the condition (1.3), then
Then by Theorem 3.1 we have
Let \(p=1\). Similar to (3.5) we get
From Theorem 3.1 we have
□
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
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
From the boundedness of \([b,\mu_{j}^{L}]\) on \(L_{p}({\mathbb{R}}^{n})\) and (3.1) we get
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
By (2.2), Lemma 2.2 and (3.2) we have
Then by Lemma 2.3, and taking \(N\ge(k_{0}+1)\theta\) we get
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
Note that
Then
Since \(2^{k} r\le t \le2^{k+1}r\), \(k\approx\ln\frac{t}{r}\). Thus
Then
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
Then from Theorem 4.1 we get
□
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
and
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):
where \(\delta_{0}>0\) (we may take \(\delta_{0}>1\)), and
and
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
where C and \(C_{0}\) are constants from (1.6) and (5.3). This allows us to estimate the first term uniformly in \(r\in (0,\delta_{0})\):
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
where \(c_{\sigma_{0}}\) is the constant from (1.2). Then, by (1.5) it suffices to choose r small enough so that
which completes 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
To check that
we use the estimate (4.1):
We take \(r<\delta_{0}\) where \(\delta_{0}\) will be chosen small enough and split the integration:
where
and
We choose a fixed \(\delta_{0}>0\) such that
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
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).
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The authors thank the referees for careful reading the paper and useful comments.
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The research of A Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.007). The research of VS Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.008) and by the Ministry of Education and Science of the Russian Federation (Agreement number: 02.a03.21.0008).
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This work was carried out in collaboration between all authors. VSG raised these interesting problems in the research. VSG, AA and MNO proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, and they read and approved the manuscript.
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Akbulut, A., Guliyev, V.S. & Omarova, M.N. Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces. Bound Value Probl 2017, 121 (2017). https://doi.org/10.1186/s13661-017-0851-4
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DOI: https://doi.org/10.1186/s13661-017-0851-4