Abstract
In this paper, we obtain the boundedness of the multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent.
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1 Introduction
The study of multilinear integral operators was motivated not only as the generalization of the theory of linear ones but also their natural appearance in analysis. It has increasing attention and much development in recent years, such as the study of the bilinear Hilbert transform by Lacey and Thiele [21, 22] and the systematic treatment of multilinear Calderón–Zygmund operators by Grafokas and Torres [13, 14], Grafokas and Kalton [12]. The importance of fractional integral operators is owing to the fact that they are smooth operators and have been extensively used in various areas such as potential analysis, harmonic analysis and partial differential equations. As one of the most important operators, the multilinear fractional integral operator (also known as the multilinear Riesz potential) has also attracted more attention, see for example [3, 11, 18, 24].
It is well known that function spaces with variable exponent arouse strong interest not only in harmonic analysis but also in applied mathematics. The theory of function spaces with variable exponent has made great progress since some elementary properties were given by Kováčik and Rákosník [19] in 1991. Lebesgue and Sobolev spaces with integrability exponent have been widely studied, see [5, 8] and the references therein. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids [26], in the study of image processing [4], and in differential equations with nonstandard growth [15]. On the other hand, the λ-central bounded mean oscillation spaces, Morrey type spaces and related function spaces have interesting applications in studying boundedness of operators including singular integral operators; see for example [1, 9, 19, 27,28,29]. In 2015, Mizuta, Ohno and Shimomura introduced the non-homogeneous central Morrey spaces of variable exponent in [23]. Recently, Fu et al. introduced the λ-central BMO spaces and the central Morrey spaces with variable exponent and gave the boundedness of some operators in [10]. In [2, 6, 7, 17] and [31,32,33,34], the authors proved the boundedness of some integral operators on variable function spaces, respectively. Meanwhile, some authors gave the boundedness of multilinear integral operators and their commutators on variable exponent function spaces, such as [16, 30, 35].
Motivated by [9, 10, 29], we will study the boundedness of the multilinear fractional integral operators and their commutators on the central Morrey spaces with variable exponent.
Let us explain the outline of this article. In Sect. 2, we first briefly recall some standard notations and lemmas in variable Lebesgue spaces. Then we will recall the definiton of the λ-central BMO spaces and central Morrey spaces with variable exponent. In Sect. 3, we will establish the boundedness for a class of multi-sublinear fractional integral operators on central Morrey spaces with variable exponent. Subsequently the boundedness of multilinear fractional integral commutators on central Morrey spaces with variable exponent will be obtained in Sect. 4. In Sect. 5, we will also consider the boundedness of another multilinear fractional integral commutators.
In addition, we denote the Lebesgue measure and the characteristic function of a measurable set \(A\subset\mathbb{R}^{n}\) by \(|A|\) and \(\chi_{A}\), respectively. The notation \(f\approx g\) means that there exist constants \(C_{1},C_{2}>0\) such that \(C_{1}g\leq f\leq C_{2}g\).
2 Variable exponent function spaces
Firstly we give some notation and basic definitions on variable exponent Lebesgue spaces.
Given an open set \(E\subset\mathbb{R}^{n}\), and a measurable function \(p(\cdot):E\rightarrow[1,\infty)\). \(p'(\cdot)\) is the conjugate exponent defined by \(p'(\cdot)=p(\cdot)/(p(\cdot)-1)\).
The set \(\mathcal{P}(E)\) consists of all \(p(\cdot):E\rightarrow[1,\infty)\) satisfying
By \(L^{p(\cdot)}(E)\) we denote the space of all measurable functions f on E such that, for some \(\lambda>0\),
This is a Banach function space with respect to the Luxemburg–Nakano norm,
The space \(L_{\mathrm{loc}}^{p(\cdot)}(\varOmega)\) is defined by \(L_{\mathrm{loc}}^{p(\cdot)}(\varOmega):=\{f: f\in L^{p(\cdot)}(E)\) for all compact subsets \({E\subset\varOmega\}}\).
Let \(f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\), the Hardy–Littlewood maximal operator is defined by
where \(B_{r}(x)=\{y\in\mathbb{R}^{n}:|x-y|< r\}\). The set \(\mathcal{B}(\mathbb{R}^{n})\) consists of \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) satisfying the condition that M is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\).
In variable \(L^{p}\) spaces there are some important lemmas as follows.
Lemma 2.1
([7])
If \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\)and satisfies
and
then \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n})\), that is, the Hardy–Littlewood maximal operatorMis bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\).
Lemma 2.2
([20] (Generalized Hölder inequality))
Let \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\). If \(f\in L^{p(\cdot)}(\mathbb {R}^{n})\)and \(g\in L^{p'(\cdot)}(\mathbb{R}^{n})\), thenfgis integrable on \(\mathbb {R}^{n}\)and
where
Lemma 2.3
([17])
Suppose \(p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\). Then there exists a positive constantCsuch that, for all ballsBin \(\mathbb{R}^{n}\),
Lemma 2.4
([17])
Let \(p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\). Then there exists a positive constantCsuch that, for all ballsBin \(\mathbb{R}^{n}\)and all measurable subsets \(S\subset B\),
and
where \(\delta_{1}\), \(\delta_{2}\)are constants with \(0<\delta_{1}, \delta_{2}<1\).
Lemma 2.5
([8])
Let \(p(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\)satisfies conditions (2.1) and (2.2) in Lemma 2.1. Then
for every cube (or ball) \(Q\subset\mathbb{R}^{n}\), where \(p(\infty)=\lim_{x\rightarrow\infty}p(x)\).
Lemma 2.6
([8])
Let \(p(\cdot), q(\cdot), s(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\)be such that
for almost every \(x\in\mathbb{R}^{n}\). Then
for all \(f\in L^{p(\cdot)}(\mathbb{R}^{n})\)and \(g\in L^{q(\cdot )}(\mathbb{R}^{n})\).
Now we recall that the central Morrey space with variable exponent and the λ-central bounded mean oscillation space with variable exponent in [10] are defined as follows.
Definition 2.1
([10])
Let \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(\lambda\in \mathbb{R}\). The central Morrey space with variable exponent \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb {R}^{n})\) is defined by
where
Definition 2.2
([10])
Let \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(\lambda<1/n\). The λ-central BMO space with variable exponent \(\mathrm {CBMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) is defined by
where
Remark 2.1
Denote by \(\mathcal{B}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) and \(\mathrm{CMO}^{q(\cdot),\lambda}(\mathbb{R}^{n})\) the inhomogeneous versions of the central Morrey space and the λ-central BMO space with variable exponent, which are defined, respectively, by taking the supremum over \(R\geq1\) in Definition 2.1 and Definition 2.2 instead of \(R>0\) there.
Remark 2.2
Our results in this paper remain true for the inhomogeneous versions of λ-central BMO spaces and central Morrey spaces with variable exponent.
3 Multilinear fractional integral operators
Let \(m\in\mathbb{N}\) and \(K(y_{0},y_{1},\ldots,y_{m})\) be a function defined away from the diagonal \(y_{0}=y_{1}=\cdots=y_{m}\) in \((\mathbb {R}^{n})^{m+1}\). We denote by f⃗ the m-tuple \((f_{1},\ldots ,f_{m})\). Now we consider that T is an m-linear operator defined on the product of test functions such that, for K, the integral representation below is valid:
whenever \(f_{j}\), \(j=1,\ldots,m\), are smooth functions with compact support and \(x\notin \bigcap_{j=1}^{m}\operatorname{supp}f_{j}\).
Particularly, there is a kind of multilinear operator \(T_{\alpha,m}\), which is called multilinear fractional integral operator, whose kernel is
In 1999, Kenig and Stein [18] gave the boundedness of the above multilinear fractional integral operator \(T_{\alpha,m}\) on the product of Lebesgue spaces.
Theorem A
([18])
Let \(0< \alpha<mn\)and \(T_{\alpha,m}\)be anm-linear fractional integral operator with kernelKsatisfying (3.1). Suppose \(1\leq p_{1},p_{2},\ldots,p_{m}\leq \infty\), \(1/q=1/p_{1}+\cdots+1/p_{m}- \alpha/n>0\).
- (1)
If each \(p_{j}>1\), \(j=1,\ldots,m\), then
$$\bigl\Vert T_{\alpha,m}(\vec{f}) \bigr\Vert _{L^{q}(\mathbb{R}^{n})}\leq C\prod _{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}}(\mathbb{R}^{n})}. $$ - (2)
If each \(p_{j}=1\)for somej, then
$$\bigl\Vert T_{\alpha,m}(\vec{f}) \bigr\Vert _{L^{q,\infty}(\mathbb{R}^{n})}\leq C\prod _{i=1}^{m} \Vert f_{i} \Vert _{L^{p_{i}}(\mathbb{R}^{n})}. $$
In the variable exponent case, Tan, Liu and Zhao [30] gave the following result.
Theorem B
([30])
Let \(m\in\mathbb{N}\), \(0< \alpha<mn\), \(q(\cdot),p_{1}(\cdot),\ldots,p_{m}(\cdot)\in\mathcal {P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=1/p_{1}(\cdot)+\cdots+1/p_{m}(\cdot)-\alpha/n\). Then
Next we will give the boundedness of a class of multi-sublinear fractional integral operators T on the product of central Morrey spaces with variable exponent.
Theorem 3.1
Let \(m\in\mathbb{N}\), \(0< \alpha<mn\)andTbe a multi-sublinear fractional integral operator such that
for any integrable functions \(f_{1},\ldots,f_{m}\)with compact support and \(x\notin\bigcap_{j=1}^{m}\operatorname{supp}f_{j}\). Suppose \(\lambda _{j}<-\frac{\alpha}{mn}\), \(\lambda=\sum_{j=1}^{m}\lambda_{j}+\alpha/n\), \(p_{j}(\cdot)\) (\(j=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=\sum_{j=1}^{m}1/p_{j}(\cdot)-\alpha/n>0\). IfTis bounded from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times\cdots \times L^{p_{m}(\cdot)}(\mathbb{R}^{n})\)into \(L^{q(\cdot)}(\mathbb {R}^{n})\), thenTis also bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\).
If \(0< \alpha<mn\) and \(T_{\alpha,m}\) is an m-linear fractional integral operator, then the condition (3.2) is obviously satisfied by (3.1). By Theorem B we can get the following corollary of Theorem 3.1.
Corollary 3.1
Let \(m\in\mathbb{N}\), \(0< \alpha<mn\)and \(T_{\alpha,m}\)be anm-linear fractional integral operator with kernelKsatisfying (3.1). Suppose \(\lambda_{j}<-\frac{\alpha}{mn}\), \(\lambda=\sum_{j=1}^{m}\lambda_{j}+\alpha/n\), \(p_{j}(\cdot)\) (\(j=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1and \(1/q(\cdot)=\sum_{j=1}^{m}1/p_{j}(\cdot)-\alpha/n>0\). Then \(T_{\alpha,m}\)is bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\).
Proof of Theorem 3.1
In order to simplify the proof, we consider only the situation when \(m=2\). Actually, a similar procedure works for all \(m\in\mathbb{N}\). Let \(f_{1}\), \(f_{2}\) be functions in \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda _{1}}(\mathbb{R}^{n})\) and \(\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda _{2}}(\mathbb{R}^{n})\), respectively. For fixed \(R>0\), denote \(B(0,R)\) by B. We need to prove
where C is a constant independent of R.
By the Minkowski inequality we write
We first estimate \(I_{1}\). Using Lemma 2.4 and the boundedness of T from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times L^{p_{2}(\cdot)}(\mathbb {R}^{n})\) into \(L^{q(\cdot)}(\mathbb{R}^{n})\), we have
where
Next we estimate \(I_{2}\). Noting that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq |x-y_{1}|^{2n-\alpha}\), by using (3.2), \({\lambda_{1}<-\frac{\alpha }{2n}}\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we have
Similarly, we estimate \(I_{3}\). Noticing that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq|x-y_{2}|^{2n-\alpha}\), by (3.2), \(\lambda _{2}<-\frac{\alpha}{2n}\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we obtain
For the estimate of \(I_{4}\). Noting that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\geq|x-y_{1}|^{n-\alpha/2}|x-y_{2}|^{n-\alpha/2}\), by (3.2), \(\lambda _{j}<-\frac{\alpha}{2n}\), \(j=1,2\), Lemma 2.3, the Minkowski inequality and the generalized Hölder inequality, we have
Combining the estimates of (3.3)–(3.7), we have
that is,
This completes the proof of Theorem 3.1. □
4 Multilinear fractional integral commutators
Let \(m\in\mathbb{N}\), \(\vec{b}=(b_{1},b_{2},\ldots,b_{m})\) and \(b_{i}\in \mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})\), \(i=1,\ldots,m\). Then the multilinear commutators of fractional integral operator are defined by
Theorem 4.1
Let \(0< \alpha<mn\), \(0< v_{i}<1/n\), \(\lambda_{i}<-\frac{\alpha}{mn}\), \(\lambda =\sum_{i=1}^{m} v_{i}+\sum_{i=1}^{m}\lambda_{i}+\alpha/n\), \(v_{i}+\lambda _{i}<-\alpha/n\), \(p_{i}(\cdot)\) (\(i=1,\ldots,m\)), \(q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1 \(1/p(\cdot)=\sum_{i=1}^{m}1/p_{i}(\cdot)-\alpha/n>0\)and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+\sum_{i=1}^{m}1/p_{i}(\cdot )-\alpha/n\). Then \([\vec{b},T_{\alpha,m}]\)is also bounded from \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda_{1}}(\mathbb{R}^{n})\times \cdots\times\dot{\mathcal{B}}^{p_{m}(\cdot),\lambda_{m}}(\mathbb {R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\)and the following inequality holds:
Proof
Without loss of generality, we still consider only the situation when \(m=2\). Let \(f_{1}\), \(f_{2}\) be functions in \(\dot{\mathcal{B}}^{p_{1}(\cdot),\lambda _{1}}(\mathbb{R}^{n})\) and \(\dot{\mathcal{B}}^{p_{2}(\cdot),\lambda _{2}}(\mathbb{R}^{n})\), respectively. For fixed \(R>0\), denote \(B(0,R)\) by B. We have the following decomposition:
Thus, by the Minkowski inequality we write
Because of the symmetry of \(f_{1}\) and \(f_{2}\), we can see that the estimate of \(L_{2}\) is analogous to that of \(L_{3}\). Then we will estimate \(L_{1}\), \(L_{2}\) and \(L_{4}\), respectively.
Next we will decompose \(f_{i}\) as \(f_{i}(y_{i})=f_{i}(y_{i})\chi _{2B}+f_{i}(y_{i})\chi_{(2B)^{c}}\), for \(i=1,2\).
(i) For \(L_{1}\), by using the Minkowski inequality we can write
Firstly we estimate \(L_{11}\). Noticing that \(\frac{1}{p(\cdot)}=\sum_{i=1}^{2}\frac{1}{p_{i}(\cdot)}-\frac{\alpha}{n}\), then \(\frac {1}{q(\cdot)}=\sum_{i=1}^{2}\frac{1}{u_{i}(\cdot)}+\frac{1}{p(\cdot )}\). By Lemma 2.6 and using the boundedness of \(T_{\alpha,2}\) from \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times L^{p_{2}(\cdot)}(\mathbb {R}^{n})\) into \(L^{q(\cdot)}(\mathbb{R}^{n})\) in Theorem B, we get
where
Now we estimate \(L_{12}\). Noticing that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\geq|x-y_{2}|^{2n-\alpha}\). By \(\lambda_{2}<-\frac{\alpha}{2n}\), Lemma 2.3 and the generalized Hölder inequality, we obtain
Let \(\frac{1}{u(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{u_{2}(\cdot )}\), by the fact \(1/p(\cdot)=\sum_{i=1}^{2}1/p_{i}(\cdot)-\alpha/n>0\) and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+\sum_{i=1}^{m}1/p_{i}(\cdot )-\alpha/n\), then \(u(\cdot)>q(\cdot)\). Thus by Lemma 2.5 and Lemma 2.6 we get
This yields
Similarly, we have
Now for the estimate of \(L_{14}\). Note that \(|(x-y_{1}, x-y_{2})|^{2n-\alpha}\geq|x-y_{1}|^{n-\alpha/2}|x-y_{2}|^{n-\alpha/2}\). Using \(\lambda_{j}<-\frac{\alpha}{2n}\), \(j=1,2\) and the generalized Hölder inequality, we have
Similar to the estimates for \(L_{12}\), we have
By the estimates of \(L_{1j}\), \(j=1,2,3,4\), we get
(ii) For \(L_{2}\), we obtain
Let \(\frac{1}{q(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{q_{1}(\cdot )}\), \(\frac{1}{q_{1}(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{g(\cdot )}-\frac{\alpha}{n}\), \(\frac{1}{g(\cdot)}=\frac{1}{p_{2}(\cdot )}+\frac{1}{u_{2}(\cdot)}\). By \(\lambda_{j}<-\frac{\alpha}{2n}\), \(j=1,2\) and boundedness of \(T_{\alpha,2}\) from \(L^{p_{1}(\cdot)}(\mathbb {R}^{n})\times L^{g(\cdot)}(\mathbb{R}^{n})\) into \(L^{q_{1}(\cdot )}(\mathbb{R}^{n})\), we get
where
For the estimate of \(L_{22}\), we have \(|(x-y_{1}, x-y_{2})|^{2n-\alpha }\approx|x-y_{1}|^{2n-\alpha}\). Using \(\frac{1}{p'_{2}(\cdot)}=\frac {1}{g'(\cdot)}+\frac{1}{u_{2}(\cdot)}\), \(\lambda_{1}<-\frac{\alpha }{2n}\), Lemmas 2.3, 2.6 and the generalized Hölder inequality, we have
Thus, from \(\frac{1}{q(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac {1}{q_{1}(\cdot)}\) and Lemmas 2.5, 2.6, we get
For \(L_{23}\), noticing that \(|(x-y_{1}, x-y_{2})|^{2n}\geq|x-y_{2}|^{2n}\) and \(\frac{1}{p'_{2}(\cdot)}=\frac{1}{g'(\cdot)}+\frac{1}{u_{2}(\cdot )}\). By Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\), the generalized Hölder inequality and the Minkowski inequality, we get
where
for \(v_{2}>0\).
Hence,
For \(L_{24}\), using Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\) and the generalized Hölder inequality, we obtain
Thus
Combining the estimates of \(L_{2j}\), \(j=1,2,3,4\), we can deduce that
(iii) For \(L_{4}\), we have
For \(L_{41}\), let \(\frac{1}{h_{i}(\cdot)}=\frac{1}{p_{i}(\cdot)}+\frac {1}{u_{i}(\cdot)}\), \(i=1,2\), then \(1/q(\cdot)=\sum_{i=1}^{2} 1/h_{i}(\cdot )-\alpha/n\). Using Lemmas 2.3, 2.5, 2.6 and the boundedness of \(T_{\alpha,2}\) from \(L^{h_{1}(\cdot)}(\mathbb{R}^{n})\times L^{h_{2}(\cdot)}(\mathbb{R}^{n})\) into \(L^{q(\cdot)}(\mathbb {R}^{n})\), we have
For \(L_{42}\), using Lemmas 2.3, 2.6, \(v_{2}+\lambda_{2}+\alpha/n<0\) and the generalized Hölder inequality, we get
This implies that
Similarly,
For \(L_{44}\), using Lemmas 2.3, 2.6, \(v_{i}+\lambda_{i}+\alpha/n<0\), \(i=1,2\) and the generalized Hölder inequality, we get
So
From the estimates of \(L_{4j}\), \(j=1,2,3,4\), we have
Furthermore, we obtain
Thus, we have
This completes the proof of Theorem 4.1. □
5 Multilinear fractional integral commutators of the second kind
There is another kind of multilinear commutators \([\vec{b}, T_{\alpha }]\), which was introduced by Pérez and Trujillo-González [25] in 2002, with the vector symbol \(\vec{b}=(b_{1},b_{2},\ldots,b_{m})\) defined by
where \(b_{i}\in\mathrm{CBMO}^{u_{i}(\cdot),v_{i}}(\mathbb{R}^{n})\), \(i=1,\ldots,m\). We have the following result.
Theorem 5.1
Let \(0< \alpha<n\), \(0< v_{i}<1/n\), \(\lambda=\sum_{i=1}^{m} v_{i}+\mu+\alpha /n<0\), \(p(\cdot), q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\)satisfy conditions (2.1) and (2.2) in Lemma 2.1 \(1/p(\cdot)>\alpha/n\)and \(1/q(\cdot)=\sum_{i=1}^{m} 1/u_{i}(\cdot)+1/p(\cdot)-\alpha/n\). Then \([\vec{b},T_{\alpha}]\)is bounded from \(\dot{\mathcal{B}}^{p(\cdot),\mu}(\mathbb{R}^{n})\)into \(\dot{\mathcal{B}}^{q(\cdot),\lambda}(\mathbb{R}^{n})\)and the following inequality holds:
Proof
Without loss of generality, we can assume that \(m=2\). For any fixed \(R>0\), denote \(B(0,R)\) by B and \(B(0,kR)\) by kB for \(k\in\mathbb{N}\). Let \(\{b\}_{E}\) denote the integral average of the function b over the set E. For \(f\in\dot{\mathcal{B}}^{p(\cdot ),\mu}(\mathbb{R}^{n})\) and any \(x\in\mathbb{R}^{n}\), we write
and then we may decompose \([\vec{b}, T_{\alpha}]\vec{f}(x)\) into four parts as follows:
Now we will give the estimates of four functions above, respectively.
(i) For \(M_{1}(x)\), using the Minkowski inequality we write
Firstly we estimate \(M_{11}\). Let \(\frac{1}{r(\cdot)}=\frac {1}{p(\cdot)}-\frac{\alpha}{n}\), then \(\frac{1}{q(\cdot)}=\sum_{i=1}^{2}\frac{1}{u_{i}(\cdot)}+\frac{1}{r(\cdot)}\). By Lemmas 2.3, 2.6 and the boundedness of \(T_{\alpha}\) from \(L^{p(\cdot)}(\mathbb {R}^{n})\) into \(L^{r(\cdot)}(\mathbb{R}^{n})\) in Theorem 1.8 of [2], we have
Since
using Lemma 2.6 and the generalized Hölder inequality, we obtain
(ii) For \(M_{2}(x)\), let \(\frac{1}{q_{1}(\cdot)}=\frac{1}{u_{2}(\cdot )}+\frac{1}{p(\cdot)}-\frac{\alpha}{n}\) and then \(\frac{1}{q(\cdot )}=\frac{1}{u_{1}(\cdot)}+\frac{1}{q_{1}(\cdot)}\). Using Lemma 2.5 and Lemma 2.6 we get
where
and
Let \(\frac{1}{l(\cdot)}=\frac{1}{u_{2}(\cdot)}+\frac{1}{p(\cdot)}\), then \(\frac{1}{q_{1}(\cdot)}=\frac{1}{l(\cdot)}-\frac{\alpha}{n}\). Using the \((L^{l(\cdot)},L^{q_{1}(\cdot)})\)-boundedness of \(T_{\alpha}\), Lemma 2.5 and Lemma 2.6 we have
For \(M_{22}\), by using \(\frac{1}{u'_{2}(\cdot)}=\frac{1}{l'(\cdot )}+\frac{1}{p(\cdot)}\), \(\mu+v_{2}+\frac{\alpha}{n}<0\), Lemmas 2.3, 2.5, 2.6 and the generalized Hölder inequality, we get
Thus
Combining the estimates for (5.6)–(5.8), we obtain
(iii) Observing that \(M_{3}\) is symmetric to \(M_{2}\), we have
(iv) Finally, we split \(M_{4}\) as follows:
Let \(\frac{1}{t(\cdot)}=\frac{1}{u_{1}(\cdot)}+\frac{1}{u_{2}(\cdot )}+\frac{1}{p(\cdot)}\), then by the \((L^{t(\cdot)}(\mathbb{R}^{n}), L^{q(\cdot)}(\mathbb{R}^{n}))\)-boundedness of \(T_{\alpha}\) we have
For \(M_{42}\), using \(\mu+v_{1}+v_{2}+\alpha/n<0\), Lemmas 2.3, 2.5, 2.6 and the generalized Hölder inequality, we get
Thus
In combination with the estimates of \(M_{41}\) and \(M_{42}\), we have
To sum up, combining the estimates of (i)–(iv),
□
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The authors are very grateful to the referees for their valuable comments.
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This work was supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2017MA041), Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J18KA225) and National Natural Science Foundation of China (Grant No. 11761026).
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Wang, H., Xu, J. Multilinear fractional integral operators on central Morrey spaces with variable exponent. J Inequal Appl 2019, 311 (2019). https://doi.org/10.1186/s13660-019-2264-7
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DOI: https://doi.org/10.1186/s13660-019-2264-7