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Schrödinger type operators on generalized Morrey spaces
Journal of Inequalities and Applications volume 2015, Article number: 229 (2015)
Abstract
In this paper we introduce a class of generalized Morrey spaces associated with the Schrödinger operator \(L=-\Delta+V\). Via a pointwise estimate, we obtain the boundedness of the operators \(V^{\beta_{2}}(-\Delta +V)^{-\beta_{1}}\) and their dual operators on these Morrey spaces.
1 Introduction
The investigation of Schrödinger operators on the Euclidean space \(\mathbb{R}^{n}\) with nonnegative potentials which belong to the reverse Hölder class has attracted attention of many authors. Shen [1] studied the Schrödinger operator \(L=-\Delta+ V\), assuming the nonnegative potential V belongs to the reverse Hölder class \(B_{q}\), \(q\geq\frac{n}{2}\). In [1], Shen proved the \(L^{p}\)-boundedness of the operators \((-\Delta+V)^{i\gamma}\), \(\nabla^{2}(-\Delta+V)^{-1}\), \(\nabla (-\Delta+V)^{-1/2}\) and \(\nabla(-\Delta+V)^{-1}\nabla\). For further information, we refer the reader to Guo et al. [2], Liu [3], Liu et al. [4, 5], Tang and Dong [6], Yang et al. [7, 8] and the references therein.
The purpose of this paper is to generalize the results of Shen [1] and Sugano [9] to a class of Morrey spaces associated with L, denoted by \(L_{\alpha,\theta,V}^{p,q,\lambda}(\mathbb {R}^{n})\). See Definition 2.8 below. The significance of these spaces is that for particular choices of the parameters p, q, λ, θ and α, one obtains many classical function spaces (see Table 1).
In Section 3, let T be one of the Schrödinger type operators \(\nabla(-\Delta+V)^{-1}\nabla\), \(\nabla(-\Delta +V)^{-1/2}\) and \((-\Delta+V)^{-1/2}\nabla\). With the help of the \(L^{p}\)-boundedness of T, it is easy to verify that T is bounded on \(L_{\alpha,\theta, V}^{p,q, \lambda}(\mathbb{R}^{n})\). For \(b\in \mathit{BMO}(\mathbb{R}^{n})\), we can also obtain the boundedness of the commutator \([b, T]\) on \(L_{\alpha,\theta, V}^{p,q, \lambda}(\mathbb {R}^{n})\). See Theorems 3.2 and 3.3. For \(\theta=0\), \(p=q\) and \(0<\lambda<1\), \(L_{\alpha,0, V}^{p,p, \lambda }(\mathbb{R}^{n})\) becomes the spaces \(L_{\alpha,V}^{p,\lambda }(\mathbb{R}^{n})\) introduced by Tang and Dong [6]. Hence, the results are generalizations of Theorems 1 and 2 in [6].
In recent years, the fractional integral operator \(I_{\alpha}=(-\Delta +V)^{-\alpha}\) has been studied extensively. We refer to Duong and Yan [14], Jiang [15], Tang and Dong [6] and Yang et al. [7] for details. Suppose that \(V\in B_{s}\), \(s\geq\frac {n}{2}\). For \(0\leq\beta_{2}\leq\beta_{1}<\frac{n}{2}\), let
Sugano [9] obtained the weighted estimates for \(T_{\beta_{1}, \beta_{2}}\), \(T^{\ast}_{\beta_{1},\beta_{2}}\), \(0<\beta_{2}\leq\beta_{1}<1\). If \(\beta_{2}=0\), we can see that \(T_{\beta_{1},0}=I_{\beta_{1}}\). So \(T_{\beta_{1},\beta_{2}}\) and \(T_{\beta_{1},\beta_{2}}^{\ast}\) can be seen as generalizations of \(I_{\alpha}\). Moreover, for \((\beta _{1}, \beta_{2})=(1,1)\) and \((1/2,1/2)\), \(T_{1,1}^{\ast}=(-\Delta +V)^{-1}V\) and \(T_{1/2,1/2}^{\ast}=(-\Delta+V)^{-1/2}V^{1/2}\), respectively, which are studied by Shen [1] thoroughly. In Section 4, assume that \(1< p_{1}<\infty\), \(1< p_{2}<{s}/{\beta_{2}}\) and \(1< q<\infty\). If the index \((q, \beta _{1},\beta_{2},\lambda,\alpha,\theta)\) satisfies
we prove that \(T_{\beta_{1},\beta_{2}}\) is bounded from \(L^{p_{1},q,\lambda}_{\alpha,\theta, V}(\mathbb{R}^{n})\) to \(L^{p_{2},q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})\). Specially, we know that \((-\Delta+V)^{-1}V\) and \((-\Delta +V)^{-1/2}V^{1/2}\) are bounded on \(L_{\alpha,\theta, V}^{p,q, \lambda }(\mathbb{R}^{n})\). See Theorems 4.7 and 4.8 for details.
In the research of harmonic analysis and partial differential equations, the commutators play an important role. If T is a Calderón-Zygmund operator, \(b\in \mathit{BMO}(\mathbb{R}^{n})\), the \(L^{p}\)-boundedness of \([b,T ]\) was first discovered by Coifman et al. [16]. Later, Strömberg [14] gave a simple proof, adopting the idea of relating commutators with the sharp maximal operator of Fefferman and Stein. In 2008, Guo et al. [2] introduced a condition \(H(m)\) and obtained \(L^{p}\)-boundedness of the commutator of Riesz transforms associated with L, where \(b\in \mathit{BMO}(\mathbb{R}^{n})\). For further information, we refer to Liu [17], Liu et al. [4, 5], Yang et al. [8] and the references therein.
In Section 5, by the boundedness of \(I_{\alpha}\) and \((-\Delta+V)^{-\beta}V^{\beta}\), we can deduce that the commutators \([b, T_{\beta_{1},\beta_{2}}]\) and \([b, T^{\ast}_{\beta_{1},\beta_{2}}]\) are bounded from \(L^{p_{1}}(\mathbb{R}^{n})\) to \(L^{p_{2}}(\mathbb {R}^{n})\) (see Theorem 5.1). Theorem 5.1 together with Lemmas 4.1 and 2.7 can be used to prove that the commutators \([b, T_{\beta_{1},\beta_{2}}]\) and \([b, T^{\ast}_{\beta_{1},\beta_{2}}]\) are bounded from \(L_{\alpha,\theta,V}^{p_{1},q,\lambda}(\mathbb{R}^{n})\) to \(L_{\alpha,\theta,V}^{p_{2},q,\lambda}(\mathbb{R}^{n})\), respectively (see Theorems 5.2 and 5.3).
Remark 1.1
Unlike the setting of the Lebesgue spaces, it is well known that the dual of \(L^{p,\lambda}(\mathbb{R}^{n})\) is not \(L^{p',-\lambda }(\mathbb{R}^{n})\). Hence, after obtaining Theorem 4.7, we cannot deduce Theorem 4.8 via the method of duality used by Guo et al. [2].
2 Preliminaries
2.1 Schrödinger operator and the auxiliary function
In this paper, we consider the Schrödinger differential operator \(L=-\Delta +V\) on \(\mathbb{R}^{n}\), \(n\geq3\), where V is a nonnegative potential belonging to the reverse Hölder class \(B_{s}\), \(s\geq\frac {n}{2}\), which is defined as follows.
Definition 2.1
Let V be a nonnegative function.
-
(i)
We say \(V\in B_{s}\), \(s>1\), if there exists \(C>0\) such that for every ball \(B\subset\mathbb{R}^{n}\), the reverse Hölder inequality
$$\biggl(\frac{1}{|B|}\int_{B}V^{s}(x)\, dx \biggr)^{\frac{1}{s}}\lesssim \biggl(\frac{1}{|B|}\int_{B}V(x) \, dx \biggr) $$holds.
-
(ii)
We say \(V\in B_{\infty}\) if there exists a constant C such that for every ball \(B\subset\mathbb{R}^{n}\),
$$\|V\|_{L^{\infty}(B)}=\frac{1}{|B|}\int_{B}V(x)\, dx. $$
Remark 2.2
Assume \(V\in B_{s}\), \(1< s<\infty\). Then \(V(y)\, dy\) is a doubling measure. Namely, there exists a constant \(C_{0}\) such that for any \(r>0\) and \(y\in\mathbb{R}^{n}\),
Definition 2.3
(Shen [1])
For \(x\in\mathbb{R}^{n}\), the function \(m_{V}(x)\) is defined as
Remark 2.4
The function \(m_{V}\) reflects the scale of V essentially, but behaves better. It is deeply studied in Shen [1] and plays a crucial role in our proof. We list a property of \(m_{V}\) which will be used in the sequel and refer the reader to Guo et al. [2] for the details.
We state some notations and properties of \(m_{V}\).
Lemma 2.5
(Lemma 1.4 in [1])
Suppose that \(V \in B_{s}\) with \(s\geq\frac {n}{2}\). Then there exist positive constants C and \(k_{0}\) such that
-
(a)
if \(|x-y|\leq\frac{C}{m_{V}(x)}\), \(m_{V}(x)\sim m_{V}(y)\);
-
(b)
\(m_{V}(y)\lesssim(1+|x-y| m_{V}(x))^{k_{0}}m_{V}(x)\);
-
(c)
\(m_{V}(y)\geq{Cm_{V}(x)}/\{1+|x-y|m_{V}(x)\}^{k_{0}/(k_{0}+1)}\).
Lemma 2.6
(Lemma 1.2 in [1])
Suppose that \(V\in B_{s}\), \(s>\frac{n}{2}\). There exists a constant C such that for \(0< r< R<\infty\),
Lemma 2.7
(Lemma 2.3 in [2])
Suppose \(V\in B_{s}\), \(s>\frac{n}{2}\). Then, for any \(N>\log_{2}C_{0}+1\), there exists a constant \(C_{N}\) such that for any \(x\in\mathbb{R}^{n}\) and \(r>0\),
2.2 Generalized Morrey spaces associated with L
Suppose that \(V\in B_{s}\), \(s>1\). Let \(L=-\Delta+V\) be the Schrödinger operator. Now we introduce a class of generalized Morrey spaces associated with L. For \(k\in\mathbb{Z}\), let \(E_{k}=B(x_{0},2^{k}r)\backslash B(x_{0},2^{k-1}r)\) and \(\chi_{k}\) be the characteristic function of \(E_{k}\).
Definition 2.8
Suppose that \(V\in B_{s}\), \(s>1\). Let \(p\in[1,+\infty)\), \(q\in [1,+\infty)\), \(\alpha\in(-\infty,+\infty)\) and \(\lambda\in (0,n)\), \(\theta\in(-\infty,+\infty)\). For \(f\in L_{\mathrm{loc}}^{q}(\mathbb {R}^{n})\), we say \(f\in L_{\alpha,\theta,V}^{p,q,\lambda}(\mathbb {R}^{n})\) provided that
where \(B(x_{0},r)\) denotes a ball centered at \(x_{0}\) and with radius r.
Proposition 2.9
-
(i)
For \(\alpha_{1}>\alpha_{2}\), \(L_{\alpha_{1},\theta, V}^{p,q,\lambda}(\mathbb{R}^{n})\subseteq L_{\alpha_{2},\theta, V}^{p,\lambda,q}(\mathbb{R}^{n})\).
-
(ii)
If \(\theta=0\), \(p=q\) and \(\alpha<0\), \(L^{p,\lambda }(\mathbb{R}^{n}) \subset L_{\alpha,\theta,V}^{p,q, \lambda }(\mathbb{R}^{n})\).
-
(iii)
If \(\theta=0\), \(p=q\) and \(\alpha>0\), \(L_{\alpha,\theta ,V}^{p,q, \lambda}(\mathbb{R}^{n}) \subset L^{p,\lambda}(\mathbb{R}^{n})\).
2.3 Calderón-Zygmund operators
We say that an operator T taking \(C_{c}^{\infty}(\mathbb{R}^{n})\) into \(L_{\mathrm{loc}}^{1}(\mathbb {R}^{n})\) is called a Calderón-Zygmund operator if
-
(a)
T extends to a bounded linear operator on \(L^{2}(\mathbb {R}^{n})\);
-
(b)
there exists a kernel K such that for every \(f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\),
$$Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)\, dy \quad \mbox{a.e. on }\{ \operatorname{supp}f\}^{c}; $$ -
(c)
the kernel \(K(x,y)\) satisfies the Calderón-Zygmund estimate
$$\begin{aligned}& \bigl\vert K(x,y)\bigr\vert \leq\frac{C}{|x-y|^{n}}; \\& \bigl\vert K(x+h,y)-K(x,y)\bigr\vert +\bigl\vert K(x,y+h)-K(x,y)\bigr\vert \leq\frac{C|h|^{\delta }}{|x-y|^{n+\delta}} \end{aligned}$$
for \(x,y\in\mathbb{R}^{n}\), \(|h|<\frac{|x-y|}{2}\) and for some \(\delta>0\).
Shen [1] obtained the following result.
Theorem 2.10
(Theorem 0.8 in [1])
Suppose \(V\in B_{n}\). Then
are Calderón-Zygmund operators.
Corollary 2.11
Suppose that \(V\in B_{n}\) and \(b\in \mathit{BMO}(\mathbb{R}^{n})\). The commutator \([b, T]\) is bounded on \(L^{p}(\mathbb{R}^{n})\).
In particular, let K denote the kernel of one of the above operators. Then K satisfies the following estimate:
for any \(N\in\mathbb{N}\). See (6.5) of Shen [1] for details.
Suppose \(V\in B_{s}\) for \(s\geq\frac{n}{2}\). Let \(L=-\Delta+V\). The semigroup generated by L is defined as
where \(K_{t}\) is the kernel of \(e^{-tL}\).
Lemma 2.12
([18])
Let \(K_{t}(x,y)\) be as in (2.3). For every nonnegative integer k, there is a constant \(C_{k}\) such that
Some notations
Throughout the paper, c and C will denote unspecified positive constants, possibly different at each occurrence. The constants are independent of the functions. \(\mathsf{U}\approx \mathsf{V}\) represents that there is a constant \(c>0\) such that \(c^{-1}\mathsf{V}\le\mathsf {U}\le c\mathsf{V}\) whose right inequality is also written as \(\mathsf{U}\lesssim\mathsf{V}\). Similarly, if \(\mathsf{V}\ge c\mathsf{U}\), we denote \(\mathsf{V}\gtrsim\mathsf{U}\).
3 Riesz transforms and the commutators on \(L^{p,q,\lambda }_{\alpha,\theta,V}(\mathbb{R}^{n})\)
Throughout this paper, for \(p\in(1, \infty)\), denote by \(p'\) the conjugate of p, that is, \(\frac{1}{p}+\frac{1}{p'}=1\). Let \(V\in B_{n}\). In this section, we assume that T is one of the Schrödinger type operators \(\nabla(-\Delta+V)^{-1}\nabla\), \(\nabla(-\Delta +V)^{-1/2}\) and \((-\Delta+V)^{-1/2}\nabla\). We study the boundedness on \(L^{p,q,\lambda}_{\alpha,\theta ,V}(\mathbb{R}^{n})\) of T and its commutator \([b, T]\) with \(b\in \mathit{BMO}(\mathbb{R}^{n})\). The bounded mean oscillation space \(\mathit{BMO}(\mathbb {R}^{n})\) is defined as follows.
Definition 3.1
A locally integrable function b is said to belong to \(\mathit{BMO}(\mathbb {R}^{n})\) if
where the supremum is taken over all balls B in \(\mathbb{R}^{n}\). Here \(b_{B}=\frac{1}{|B|}\int_{B}b(x)\, dx\) stands for the mean value of b over the ball B and \(|B|\) means the measure of B.
We first prove that T is bounded on \(L^{p,q,\lambda}_{\alpha,\theta ,V}(\mathbb{R}^{n})\).
Theorem 3.2
Suppose that \(\alpha\in(-\infty,0]\), \(\lambda\in(0,n)\) and \(1< q<\infty\). If \(1< p<\infty\), \(\frac{\lambda}{q}-\frac{1}{p}<\theta<\frac{\lambda }{q}+1-\frac{1}{p}\), then the operators T are bounded on \(L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})\).
Proof
For any ball \(B(x_{0},r)\), write
where \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). Hence, we have
For \(A_{2}\), by Theorem 2.10, we have
We first estimate the term \(E_{1}\). Note that if \(x\in E_{k}\), \(y\in E_{j}\) and \(j\leq k-2\), then \(|x-y|\sim2^{k}r\). By Lemma 2.5 and (2.2), we can get
where \(\frac{1}{p}+\frac{1}{p'}=1\). Since \(- \frac{1}{p}+ \frac{\lambda}{q} <\theta<(1- \frac{1}{p})+ \frac{\lambda}{q}\), we obtain
For \(A_{3}\), we can see that when \(x\in E_{k}\), \(y\in E_{j}\), then \(|x-y|\sim2^{j}r\) for \(j\geq k+2\). Similar to \(E_{1}\), we have
Since \(-\frac{1}{p}+\frac{\lambda}{q}<\theta<(1-\frac{1}{p})+\frac {\lambda}{q}\), choosing N large enough, we obtain
Let \(N=[-\frac{\alpha}{q}+1](k_{0}+1)\). Finally, \(\|Tf\|_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p,q,\lambda}_{\alpha,\theta, V}(\mathbb {R}^{n})}\). This completes the proof of Theorem 3.2. □
Suppose that \(b\in \mathit{BMO}(\mathbb{R}^{n})\) and \(V\in B_{n}\). Let T be one of the Schrödinger type operators \(\nabla(-\Delta+V)^{-1}\nabla \), \(\nabla(-\Delta+V)^{-1/2}\) and \((-\Delta+V)^{-1/2}\nabla\). The commutator \([b, T]\) is defined as
Theorem 3.3
Suppose that \(V\in B_{n}\) and \(b\in \mathit{BMO}(\mathbb{R}^{n})\). Let \(1< p<\infty\), \(1< q<\infty\), \(\alpha\in(-\infty, 0]\), \(\lambda\in(0,n)\). If the index \((p,q,\theta,\lambda)\) satisfies \(\frac{\lambda }{q}-\frac{1}{p}<\theta<\frac{\lambda}{q}+1-\frac{1}{p}\), then
Proof
For any ball \(B=B(x_{0},r)\), we can get
where \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). Hence, we have
For \(B_{2}\), by Corollary 2.11, we have
Denote by \(b_{2^{k}r}\) the mean value of b on the ball \(B(x_{0}, 2^{k}r)\). For \(B_{1}\), by Lemma 2.5 and (2.2), we have
where in the third inequality, we have used John-Nirenberg’s inequality [19]. Since \(- \frac{1}{p}+ \frac{\lambda}{q} <\theta<(1- \frac{1}{p})+ \frac{\lambda}{q}\), we obtain
For \(B_{3}\), similar to \(B_{1}\), we have
Since \(-\frac{1}{p}+\frac{\lambda}{q}<\theta<(1-\frac{1}{p})+\frac {\lambda}{q}\), choosing N large enough, we obtain
Let \(N=[-\frac{\alpha}{q}+1](k_{0}+1)\). We finally get
□
4 Schrödinger type operators on \(L^{p,q,\lambda}_{\alpha ,\theta,V}(\mathbb{R}^{n})\)
Let \(L=-\Delta+V\) be the Schrödinger operator, where \(V\in B_{s}\), \(s>n/2\). For \(0<\beta<\frac{n}{2}\), the fractional integral operator associated with L is defined by
Denote by \(K_{\beta}(x,y)\) the kernel of \(L^{-\beta}\). By Lemma 2.12, Bui [20] obtained the following pointwise estimate.
Lemma 4.1
(Proposition 3.3 in [20])
Let \(0<\beta<\frac{n}{2}\). For \(N\in\mathbb{N}\), there is a constant \(C_{N}\) such that
where \(K_{t}(\cdot, \cdot)\) is the kernel of the semigroup \(e^{-tL}\).
Definition 4.2
Let \(f\in L_{\mathrm{loc}}^{q}(\mathbb{R}^{n})\). Denote by \(|B|\) the Lebesgue measure of the ball \(B\subset\mathbb{R}^{n}\). The fractional Hardy-Littlewood maximal function \(M_{\sigma,\gamma}\) is defined by
Lemma 4.3
([16])
Suppose \(1<\gamma<p_{1}<\frac{n}{\sigma}\) and \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{\sigma}{n}\). Then
As a generalization of the fractional integral associated with L, the operators \(V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}\), \(0\leq\beta _{2}\leq\beta_{1}\leq1\), have been studied by Sugano [9] systematically. Applying the method of Sugano [9] together with Lemma 4.1, we can obtain the following result for \(V^{\beta _{2}}(-\Delta+V)^{-\beta_{1}}\), \(0\leq\beta_{2}\leq\beta_{1}\leq n/2\). We omit the proof.
Theorem 4.4
Suppose that \(V\in B_{\infty}\). Let \(1<\beta_{2}\leq\beta_{1}<\frac {n}{2}\). Then
In a similar way, by (4.1), we can get the following estimate for the operators \((-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\), \(0\leq\beta_{2}\leq\beta_{1}<\frac{n}{2}\).
Theorem 4.5
Suppose that \(V\in B_{s}\) for \(s>\frac{n}{2}\). Let \(0\leq\beta _{2}\leq\beta_{1}<\frac{n}{2}\). Then
where \((\frac{s}{\beta_{2}})'\) is the conjugate of \((\frac{s}{\beta_{2}})\).
Proof
Let \(r={1}/{m_{V}(x)}\). By Lemma 4.1 and Hölder’s inequality, we have
For \(k\geq1\), because \(V(y)\,dy\) is a doubling measure, we have
where \(k_{0}=2-n+\log_{2}C_{0}\). For \(k\leq0\), Lemma 2.6 implies that
Taking N large enough, we get
□
By Theorem 4.5 and the duality, we can obtain the following.
Corollary 4.6
Suppose \(V\in B_{s}\) for \(s>\frac{n}{2}\).
-
(1)
If \(1<(\frac{s}{\beta_{2}})'<p_{1}<\frac{n}{2\beta _{1}-2\beta_{2}}\) and \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta _{1}-2\beta_{2}}{n}\), then
$$\bigl\Vert (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}, $$where \(\frac{s}{\beta_{2}}+(\frac{s}{\beta_{2}})'=1\).
-
(2)
If \(1< p_{2}<\frac{s}{\beta_{2}}\) and \(\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}\), then
$$\bigl\Vert V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}. $$
Theorem 4.7
Suppose that \(V\in B_{s}\), \(s\geq\frac{n}{2}\), \(\alpha\in(-\infty ,0]\), \(\lambda\in(0,n)\). Let \(1< q<\infty\), \(1<\beta_{2}\leq\beta_{1}<\frac{n}{2}\) and \(1< p_{2}<\frac{s}{\beta _{2}}\) with \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}\). If \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac {2\beta_{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), then
Proof
For any ball \(B(x_{0},r)\), write
where \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). Hence, we have
We first estimate \(M_{2}\). For \(1< p_{2}<\frac{s}{\beta_{2}}\), by (2) of Corollary 4.6, we can get
Now we deal with the terms \(M_{1}\) and \(M_{3}\). We choose N large enough such that
and take positive \(N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). For \(M_{1}\), note that if \(x\in E_{k}\), \(y\in E_{j}\) and \(j\leq k-2\), then \(|x-y|\sim2^{k}r\). By Lemmas 4.1 and 2.7, we use Hölder’s inequality to obtain
where \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), we obtain
For \(M_{3}\), note that when \(x\in E_{k}\), \(y\in E_{j}\) and \(j\geq k+2\), then \(|x-y|\sim2^{j}r\). Similar to \(E_{1}\), we have
where \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), we obtain
Choosing N large enough, we obtain
□
Theorem 4.8
Suppose that \(V\in B_{s}\), \(s\geq\frac{n}{2}\), \(\alpha\in(-\infty ,0]\), \(\lambda\in(0,n)\) and \(1< q<\infty\). Let \(0<\beta_{2}\leq\beta_{1}<\frac{n}{2}\), \(\frac{s}{s-\beta _{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}\) with \(\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}\). If \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), then
Proof
For any ball \(B(x_{0}, r)\), let \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). We can decompose f as follows:
Similar to the proof of Theorem 4.7, we have
For \(L_{2}\), because \(1<\frac{s}{s-\beta_{2}}<p_{1}<\frac{n}{2\beta _{1}-\beta_{2}}\), we use Corollary 4.6 to obtain
For \(L_{1}\), we can see that if \(x\in E_{k}\) and \(y\in E_{j}\), then \(|x-y|\sim2^{k}r\) for \(j\leq k-2\). By Hölder’s inequality and the fact that \(V\in B_{s}\), we deduce from Lemmas 4.1 and 2.7 that
where \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}\) and \(N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), we obtain
For \(L_{3}\), note that when \(x\in E_{k}\), \(y\in E_{j}\) and \(j\geq k+2\), then \(|x-y|\sim2^{j}r\). Similar to \(E_{1}\), we have
where \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}\) and \(N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda }{q}-\frac{1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), we obtain
Let N be large enough. We finally get \(\|(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}f\|_{L^{p_{2},q,\lambda }_{\alpha,\theta,V}}\lesssim\|f\|_{L^{p_{1},q,\lambda}_{\alpha ,\theta,V}}\). □
5 Boundedness of the commutators on \(L^{p,q,\lambda}_{\alpha ,\theta,V}(\mathbb{R}^{n})\)
In this section, let \(b\in \mathit{BMO}(\mathbb{R}^{n})\). We consider the boundedness of commutators \([b, (-\Delta+V)^{-\beta_{1}}V^{\beta _{2}}]\) and its duality on the generalized Morrey spaces \(L^{p,q,\lambda}_{\alpha,\theta,V}(\mathbb{R}^{n})\). For this purpose, we prove the commutator \([b, (-\Delta+V)^{-\beta_{1}}V^{\beta _{2}}]\) is bounded from \(L^{p_{1}}(\mathbb{R}^{n})\) to \(L^{p_{2}}(\mathbb{R}^{n})\). For the sake of simplicity, we denote by \(b_{2^{k}r}\) the mean value of b on the ball \(B(x_{0}, 2^{k}r)\).
Theorem 5.1
Suppose that \(V\in B_{s}\), \(s\geq\frac{n}{2}\) and \(b\in \mathit{BMO}(\mathbb{R}^{n})\).
-
(i)
If \(0<\beta_{2}\leq\beta_{1}<\frac{n}{2}\), \(\frac {s}{s-\beta_{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}\), \(\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}\), then
$$\bigl\Vert \bigl[b, (-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}\bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb {R}^{n})}\|b\|_{\mathit{BMO}}. $$ -
(ii)
If \(1< p_{2}<\frac{s}{\beta_{2}}\) and \(\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}\), then
$$\bigl\Vert \bigl[b, V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}\bigr]f\bigr\Vert _{L^{p_{2}}(\mathbb {R}^{n})}\lesssim\|f\|_{L^{p_{1}}(\mathbb{R}^{n})}\|b\|_{\mathit{BMO}}. $$
Proof
We only prove (i). (ii) can be obtained by duality. Because \(\beta _{2}\leq\beta_{1}\), we can decompose the operator \((-\Delta +V)^{-\beta_{1}}V^{\beta_{2}}\) as
Denote by \(L^{\beta_{2}-\beta_{1}}\) and \(T_{\beta_{2}}\) the operators \((-\Delta+V)^{\beta_{2}-\beta_{1}}\) and \((-\Delta +V)^{-\beta_{2}}V^{\beta_{2}}\), respectively. Then we can get
By (1) of Corollary 4.6, we can get
This completes the proof. □
In the rest of this section, we prove the boundedness of the commutators \([b,V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}]\) and \([b,(-\Delta+V)^{-\beta_{1}}V^{\beta_{2}}]\) on \(L^{p_{2},q,\lambda }_{\alpha,\theta,V}(\mathbb{R}^{n})\), respectively.
Theorem 5.2
Suppose that \(V\in B_{s}\), \(s\geq\frac{n}{2}\), \(\alpha\in(-\infty,0]\) and \(\lambda\in(0,n)\). Let \(1< q<\infty\), \(1<\beta_{2}\leq\beta_{1}<\frac{n}{2}\) and \(1< p_{2}<\frac{s}{\beta _{2}}\) with \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}\). If \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta_{1}}{n}<\theta <\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), then for \(b\in \mathit{BMO}(\mathbb{R}^{n})\),
Proof
For any ball \(B(x_{0},r)\), we have
where \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). Hence, we have
For \(D_{2}\), by (ii) of Theorem 5.1, we have
For \(D_{1}\), by Lemmas 2.7 and 4.1, we obtain
where \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta_{1}-2\beta _{2}}{n}\) and \(N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta_{1}}{n}<\theta <\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), we obtain
For \(D_{3}\), because \(\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{2\beta _{1}-2\beta_{2}}{n}\) and \(N_{1}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta _{2}\), we have
where we have used the fact that \(|x-y|\sim2^{j}r\) for \(x\in E_{k}\), \(y\in E_{j}\) and \(j\geq k+2\). Since \(\frac{\lambda}{q}-\frac{1}{p_{1}}+\frac{2\beta _{1}}{n}<\theta<\frac{\lambda}{q}+1-\frac{1}{p_{1}}\), we obtain
Let N be large enough. Finally, we get
□
Theorem 5.3
Suppose that \(V\in B_{s}\), \(s\geq\frac{n}{2}\) and \(b\in \mathit{BMO}(\mathbb {R}^{n})\). Let \(\alpha\in(-\infty,0]\), \(\lambda\in(0,n)\) and \(1< q<\infty\). If \(0<\beta_{2}\leq\beta_{1}<\frac{n}{2}\), \(\frac{s}{s-\beta _{2}}< p_{1}<\frac{n}{2\beta_{1}-2\beta_{2}}\), \(\frac {1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta_{2}}{n}\), \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), then
Proof
Similarly, we can decompose f based on an arbitrary ball \(B(x_{0},r)\) as follows:
where \(E_{j}=B(x_{0},2^{j}r)\backslash B(x_{0},2^{j-1}r)\). Hence, we have
Applying Theorem 5.1, we can get
For \(F_{1}\), by Hölder’s inequality and the fact that \(V\in B_{s}\), we apply Lemmas 4.1 and 2.7 to deduce that
where \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}\) and \(N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda}{q}-\frac {1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), we obtain
For \(F_{3}\), note that when \(x\in E_{k}\), \(y\in E_{j}\) and \(j\geq k+2\), then \(|x-y|\sim2^{j}r\). Similar to \(F_{1}\), we have
where \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{2\beta_{1}-2\beta _{2}}{n}\) and \(N_{2}<(N/k_{0}+1)-(\log_{2}C_{0}+1)\beta_{2}\). Since \(\frac{\lambda}{q}-\frac{1}{p_{2}}<\theta<\frac{\lambda }{q}-\frac{1}{p_{2}}+1-\frac{2\beta_{1}}{n}\), we obtain
Let N be large enough. We finally get
□
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Acknowledgements
Project was supported by NSFC No. 11171203; New Teacher’s Fund for Doctor Stations, Ministry of Education No. 20114402120003; Guangdong Natural Science Foundation S2011040004131; Foundation for Distinguished Young Talents in Higher Education of Guangdong, China, LYM11063.
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Li, P., Wan, X. & Zhang, C. Schrödinger type operators on generalized Morrey spaces. J Inequal Appl 2015, 229 (2015). https://doi.org/10.1186/s13660-015-0747-8
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DOI: https://doi.org/10.1186/s13660-015-0747-8