In this article, we study necessary and sufficient conditions on the parameters of the boundedness on Morrey spaces and modified Morrey spaces for \(T_{\Omega,\alpha}\) and \(M_{\Omega,\alpha}\), which are a multilinear fractional integral and a multilinear fractional maximal operator with rough kernel, respectively. Our results extend some known results significantly.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
1 Introduction
Suppose that Ω is homogeneous of degree zero on \(\mathbb {R}^{n}\) and \(\Omega\in L^{s}(\mathbb{S}^{n-1})\) with \(1< s\leq\infty\), where \(\mathbb{S}^{n-1}\) denotes the unit sphere of \(\mathbb {R}^{n}\). Moreover, \(m\geq1\) will denote an integer, \(\theta_{j}\) (\(j=1,\ldots,m\)) will be fixed, distinct, and nonzero real numbers, and \(0<\alpha<n\). We denote \({\mathbf{f}}=(f_{1},\ldots,f_{m})\), then the multilinear fractional integral operator on \(\mathbb {R}^{n}\) is given by the formula
Spanne and Adams obtained two remarkable results on Morrey spaces (see Definition 2.1 in Section 2) for \(I_{\alpha}\). Their results can be summarized as follows.
If\(p>1\), then the condition\(1/p-1/q=\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(I_{\alpha}\)from\(L^{p,\lambda}(\mathbb{R}^{n})\)to\(L^{q,\lambda}(\mathbb{R}^{n})\).
(ii)
If\(p=1\), then the condition\(1-1/q=\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(I_{\alpha}\)from\(L^{1,\lambda}(\mathbb{R}^{n})\)to\(WL^{q,\lambda}(\mathbb{R}^{n})\).
If \(\lambda=0\), then the statement of Propositions 1.1 and 1.2 reduces to the well-known Hardy-Littlewood-Sobolev inequality. On the other hand, in 2011, Guliyev et al. [6] found this inequality in modified Morrey spaces (see Definition 2.2 in Section 2) was also valid and proved the following.
If\(p>1\), then the condition\(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(I_{\alpha}\)from\(\widetilde{L}^{p,\lambda}\)to\(\widetilde{L}^{q,\lambda}\).
(ii)
If\(p=1\), then the condition\(\alpha/n\leq1-1/q\leq\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(I_{\alpha}\)from\(\widetilde{L}^{1,\lambda}\)to\(W\widetilde{L}^{q,\lambda}\).
When \(m\geq2\) and \(\Omega\equiv1\), Grafakos [7] studied Lebesgue boundedness of \(T_{1,\alpha}\). Recently, Gunawan [8] extended Grafakos’ result to Morrey spaces and provided a multi-version for the sufficiency of conclusion (i) in Proposition 1.2.
Let\(0<\alpha<n\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\), \(1< p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\), \(1/p-1/q=\alpha/(n-\lambda)\), then the operator\(T_{1,\alpha}\)is bounded from\(L^{p_{1},\lambda}(\mathbb{R}^{n})\times\cdots\times L^{p_{m},\lambda}(\mathbb{R}^{n})\)to\(L^{q,\lambda}(\mathbb{R}^{n})\).
When \(m\geq2\) and \(\Omega\in L^{s}(\mathbb{S}^{n-1})\), Ding and Lu [9] studied the \(L^{p_{1}}\times\cdots\times L^{p_{m}}\) boundedness for \(T_{\Omega,\alpha}\). After this work above, a natural question is: what properties does the operator \(T_{\Omega,\alpha}\) have on Morrey and modified Morrey spaces? We give answers as follows.
Theorem 1.1
Let\(0<\alpha<n\), \(\Omega\in L^{s}(\mathbb{S}^{n-1})\)with\(1< s\leq\infty\), \(s'=s/(s-1)\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\), \(0\leq\lambda< n-\alpha p\), \(1\leq p<{n}/\alpha\)and satisfy
$$ \frac{\lambda}{p}=\sum_{j=1}^{m} \frac{\lambda_{j}}{p_{j}} \quad\textit{for } 0\leq\lambda_{j}< n. $$
(1.1)
(i)
If\(p>s'\), then the condition\(1/p-1/q=\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(T_{\Omega,\alpha}\)from\(L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(L^{q,\lambda}(\mathbb{R}^{n})\).
(ii)
If\(p=s'\), then the condition\(1/{s'}-1/q=\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(T_{\Omega,\alpha}\)from\(L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(WL^{q,\lambda}(\mathbb{R}^{n})\).
Moreover, similar conclusions hold for\(M_{\Omega,\alpha}\).
Theorem 1.2
Letα, Ω, s, \(p_{j}\), \(\lambda_{j}\), p, andλbe as in Theorem 1.1.
(i)
If\(p>s'\), then the condition\(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(T_{\Omega,\alpha}\)from\(\widetilde{L}^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(\widetilde{L}^{q,\lambda}(\mathbb{R}^{n})\).
(ii)
If\(p=s'\), then the condition\(\alpha/n\leq1/s'-1/q\leq\alpha/(n-\lambda)\)is necessary and sufficient for the boundedness of the operator\(T_{\Omega,\alpha}\)from\(\widetilde{L}^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(W\widetilde{L}^{q,\lambda}(\mathbb{R}^{n})\).
Moreover, similar estimates hold for\(M_{\Omega,\alpha}\).
Remark 1.1
Note that Theorems 1.1 and 1.2 covers Propositions 1.2 and 1.3, respectively. Also, the case \(\lambda=\lambda_{1}=\cdots=\lambda_{m}\) and \(\Omega\equiv1\) reduces to Proposition 1.4; the case \(\lambda=\lambda_{1}=\cdots=\lambda_{m}=0\) gives the result of Ding and Lu [9] on Lebesgue spaces.
We observe that, in Theorems 1.1 and 1.2, the boundedness in the limiting case \(p=(n-\lambda)/\alpha\) remains open. In fact, when \(p=n/\alpha\) (i.e.\(\lambda=0\)), Ding and Lu [9] found \(M_{\Omega,\alpha}\) is bounded from \(L^{p_{1}}\times\cdots\times L^{p_{m}}\) to \(L^{\infty}\), but this corresponding result for \(T_{\Omega,\alpha}\) in this case does not hold. Our next goal is to extend Ding and Lu’s result to the case \(0\leq\lambda< n-\alpha\), as the continuation of Theorems 1.1 and 1.2.
Theorem 1.3
Let\(0<\alpha<n\), \(0\leq\lambda< n-\alpha\), \(\Omega\in L^{s}(\mathbb{S}^{n-1})\)with\(1< s\leq\infty\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\)and satisfy (1.1).
(i)
If\(p=(n-\lambda)/\alpha\geq s'\), then the operator\(M_{\Omega,\alpha}\)is bounded from\(L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(L^{\infty}(\mathbb{R}^{n})\).
(ii)
If\(s'\leq(n-\lambda)/\alpha\leq p\leq n/\alpha\), then the operator\(M_{\Omega,\alpha}\)is bounded from\(\widetilde{L}^{p_{1},\lambda_{1}}(\mathbb{R}^{n})\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}(\mathbb{R}^{n})\)to\(L^{\infty}(\mathbb{R}^{n})\).
Finally we shall describe the organization of this paper. In the following section, we will study the boundedness of maximal operator \(M_{\Omega}\) on Morrey and modified Morrey spaces. The last section we will devote to the boundedness of \(T_{\Omega,\alpha}\) and \(M_{\Omega,\alpha}\) and to showing the proof of Theorems 1.1, 1.2 and 1.3.
Throughout this paper, we assume the letter C always remains to denote a positive constant that may vary at each occurrence but is independent of the essential variables.
2 Boundedness of maximal operator \(M_{\Omega}\)
In this part, we investigate the boundedness of maximal operator \(M_{\Omega}\) (see Section 1) on Morrey and modified Morrey spaces defined by the following definitions.
Let \(1\leq p<\infty\), \(0\leq\lambda\leq n\). We denote by \(L^{p,\lambda}=L^{p,\lambda}(\mathbb {R}^{n})\) the Morrey space, and by \(WL^{p,\lambda}=WL^{p,\lambda}(\mathbb {R}^{n})\) the weak Morrey space, as the set of locally integrable functions \(f(x)\), \(x\in \mathbb {R}^{n}\), with the finite norms
Let \(1\leq p<\infty\), \(0\leq\lambda\leq n\), \([t]_{1}=\min\{1,t\}\). We denote by \(\widetilde{L}^{p,\lambda}=\widetilde{L}^{p,\lambda}(\mathbb {R}^{n})\) the modified Morrey space, and by \(W\widetilde{L}^{p,\lambda}=W\widetilde{L}^{p,\lambda}(\mathbb {R}^{n})\) the weak modified Morrey space, as the set of locally integrable functions \(f(x)\), \(x\in \mathbb {R}^{n}\), with the finite norms
It is easy to see that \(L^{p,0}(\mathbb {R}^{n})=\widetilde{L}^{p,0}(\mathbb {R}^{n})=L^{p}(\mathbb {R}^{n})\), \(WL^{p,0}(\mathbb {R}^{n})=W\widetilde{L}^{p,0}(\mathbb {R}^{n})=WL^{p}(\mathbb {R}^{n})\). If \(\lambda<0\) or \(\lambda>n\), then \(\widetilde{L}^{p,\lambda}(\mathbb {R}^{n})=L^{p,\lambda}(\mathbb {R}^{n})=\Theta\) where Θ is the set of all functions equivalent to 0 on \(\mathbb {R}^{n}\). In addition, from [6], we know
Recall the definition of \(M_{\Omega}\), as a special case when \(m=1\), \(\Omega\equiv1\) and \(\theta_{1}=1\), \(M_{\Omega}\) is the Hardy-Littlewood maximal operator \(\mathcal{M}\). In 1994, Nakai [11] obtained the boundedness of \(\mathcal{M}\) on Morrey spaces, later Guliyev [6] studied the operator \(\mathcal{M}\) on modified Morrey spaces and get a result parallel to Nakai’s result.
When \(m\geq2\) and \(\Omega\in L^{s}(\mathbb{S}^{n-1})\), we find \(M_{\Omega}\) also has the same properties by providing the following multi-version of Lemmas 2.1 and 2.2.
Theorem 2.3
Let\(\Omega\in L^{s}(\mathbb{S}^{n-1})\)with\(1< s\leq\infty\), \(0\leq\lambda< n\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\), \(p\geq s'\)and satisfy (1.1).
(i)
If\(p>s'\), there exists a positive constantCsuch that
(ii) If \(p=s'\), for any \(\beta>0\), let \(\varepsilon_{0}=\beta\), \(\varepsilon_{m}=1\) and \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{m-1}>0\) be arbitrary which will be chosen later. From the pointwise estimate (2.1), we get
3 Boundedness of \(T_{\Omega,\alpha}\) and \(M_{\Omega,\alpha}\)
The present section consists of two parts which are about the bounded estimates on Morrey and modified spaces for the multilinear fractional integral operator \(T_{\Omega,\alpha}\) and the multilinear fractional maximal operator \(M_{\Omega,\alpha}\), respectively.
3.1 Boundedness on Morrey spaces
In this part, we will prove Theorem 1.1. Let us begin with a requisite Hedberg’s type estimates, which plays a key role in proving Theorem 1.1.
Lemma 3.1
Let\(0<\alpha<n\), \(\Omega\in L^{s}(\mathbb{S}^{n-1})\)with\(1< s\leq\infty\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\), \(0\leq\lambda< n-\alpha p\), \(s'\leq p<{n}/\alpha\)and satisfy (1.1), then there exists a positive constantCsuch that
Recalling the conditions of Lemma 3.1, we can see \(s'\leq p<(n-\lambda)/\alpha\), which implies \(\alpha<(n-\lambda)/p\leq(n-\lambda)/s'\), then we get
Taking the supremum for \(x\in \mathbb {R}^{n}\) and \(t>0\), we will get the desired conclusion.
Necessity. Suppose that \(T_{\Omega,\alpha}\) is bounded from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(L^{q,\lambda}\). Define \({\mathbf {f}}_{\epsilon}(x)=(f_{1}(\epsilon x),\ldots, f_{m}(\epsilon x))\) for \(\epsilon>0\). Then it is easy to show that
If \(1/p<1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{L^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
If \(1/p>1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{L^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Therefore we get \(1/p=1/q+\alpha/(n-\lambda)\).
We proceed to prove (ii). Sufficiency. For any \(\beta>0\), applying Lemma 3.1 and the conclusion (ii) of Theorem 2.3, we get
Necessity. Let \(T_{\Omega,\alpha}\) be bounded from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(WL^{q,\lambda}\). Because we have (3.3) for \({\mathbf {f}}_{\epsilon}(x)=(f_{1}(\epsilon x),\ldots, f_{m}(\epsilon x))\) with \(\epsilon>0\), then we obtain
If \(1/p<1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in L^{p_{1},\lambda}\times\cdots\times L^{p_{m},\lambda}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{WL^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
If \(1/p>1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in L^{p_{1},\lambda}\times\cdots\times L^{p_{m},\lambda}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{WL^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Consequently, we get \(1/p=1/q+\alpha/(n-\lambda)\).
Next, we prove conclusions (i) and (ii) hold for \(M_{\Omega,\alpha}\). By the same arguments as above we get the necessity part and the sufficiency part follows from the conclusion of \(T_{\Omega,\alpha}\) and the following lemma.
Then Hölder’s inequality implies \(L^{t,\lambda}(\mathbb{R}^{n})\subset L^{q,\mu}(\mathbb{R}^{n})\) and \(WL^{t,\lambda}(\mathbb{R}^{n})\subset WL^{q,\mu}(\mathbb{R}^{n})\). In fact, there exists a constant \(C>0\) such that
As an another application, by Hölder’s inequality, we obtain an Olsen’s inequality as in the following corollary, which is a multi-version of the results in considered by Olsen in [12] in the study of the Schrödinger equation with perturbed potentials W.
Corollary 3.2
Letα, Ω, s, \(p_{j}\), \(\lambda_{j}\), p, andλbe as in Theorem 1.1and let\(W\in L^{(n-\lambda)/\alpha,\lambda}\). If\(p>s'\)and\(1/p-1/q=\alpha/(n-\lambda)\), then there exists a positive constantCsuch that
$$ \|W\cdot T_{\Omega,\alpha}{\mathbf{f}}\|_{L^{p,\lambda}(\mathbb{R}^{n})} \leq C \|W\|_{L^{(n-\lambda)/\alpha,\lambda}(\mathbb{R}^{n})} \prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}. $$
Moreover, similar estimates hold for\(M_{\Omega,\alpha}\).
3.2 Boundedness on modified Morrey spaces
This part we will devote to the boundedness on modified Morrey spaces and show the proof Theorem 1.2 and 1.3. With the same arguments on Morrey spaces, we also begin with a requisite Hedberg’s type estimates on modified Morrey spaces.
Lemma 3.3
Let\(0<\alpha<n\), \(\Omega\in L^{s}(\mathbb{S}^{n-1})\)with\(1< s\leq\infty\), pbe the harmonic mean of\(p_{1},\ldots,p_{m}>1\), \(0\leq\lambda< n-\alpha p\), \(s'\leq p<{n}/\alpha\)and satisfy (1.1), then there exists a positive constantCsuch that
For any \(\delta>0\), we do the same decomoposition of \(T_{\Omega,\alpha}\) as in the proof of Lemma 3.1, then we only need to estimate \(F_{\sigma}(x,\delta)\). We also choose the same σ during the proof of Lemma 3.1, then we get
Similarly to the proof of sufficiency in Theorem 1.1, by the boundedness of \(M_{\Omega}\) in Theorem 2.4, we will get the sufficiency. Now, we give only the proof of necessity.
Let \([\epsilon]_{1,+}=\max\{1,\epsilon\}\), by (3.3), for \({\mathbf {f}}_{\epsilon}(x)\) with \(\epsilon>0\), we get
(i) Assume that \(T_{\Omega,\alpha}\) is bounded from \(\widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\) to \(\widetilde{L}^{q,\lambda}\), we get
When \(1/p<1/q+\alpha/n\), then for all \({\mathbf {f}} \in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
When \(1/p>1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Therefore we get \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\).
(ii) Assume that \(T_{\Omega,\alpha}\) is bounded from \(\widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\) to \(W\widetilde{L}^{q,\lambda}\), we have
When \(1/p<1/q+\alpha/n\), then for all \({\mathbf {f}} \in \widetilde{L}^{p_{1},\lambda}\times\cdots\times \widetilde{L}^{p_{m},\lambda}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{W\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
When \(1/p>1/q+\alpha/(n-\lambda)\), then for all \({\mathbf {f}} \in \widetilde{L}^{p_{1},\lambda}\times\cdots\times \widetilde{L}^{p_{m},\lambda}\), we have \(\|T_{\Omega,\alpha}{\mathbf{f}}\|_{W\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Consequently, we get \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\).
Next, we prove conclusions (i) and (ii) hold for \(M_{\Omega,\alpha}\). By the same arguments as above we get the necessity part and the sufficiency part follows from the conclusion of \(T_{\Omega,\alpha}\) and Lemma [9].
Therefore, we complete the proof of Theorem 1.3. □
Finally, we would like to remark that our theorems generalize the relevant results in [13‐15].
Acknowledgements
The research of the first author was supported by the Soft Science Foundation of Ningbo City (No. 2015A10001). The second author was supported by the National Natural Science Foundation of China (No. 11401175 and No. 11501169). The third author was supported by the National Natural Science Foundation of China (No. 11171306). Thanks are also given to the anonymous referees for careful reading and suggestions.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.