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Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces
Journal of Inequalities and Applications volume 2015, Article number: 104 (2015)
Abstract
In this paper, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator \(\mathcal{M}_{\Omega,\alpha}\) and the multilinear fractional integral operator \(\mathcal{I}_{\Omega,\alpha}\) with rough kernels on Morrey spaces and modified Morrey spaces, respectively. This extends some recent results of Guliyev, Hasnov and Zeren; the necessary and sufficient conditions for the boundedness of \(M_{\alpha}\) and \(I_{\alpha}\) on modified spaces are considered.
1 Introduction
Kenig and Stein [1] studied the boundedness of multilinear fractional integral operator \(\mathcal{I}_{\alpha,m}\), \(0<\alpha< mn\), on Lebesgue spaces.
we denote by \(\vec{f}\) the m-tuple \((f_{1}, f_{2}, \ldots,f_{m})\) and by m, n nonnegative integers with \(m\geq1\), \(n\geq2\). As one of the most important multilinear operators, the multilinear fractional integral operator has been widely studied; we refer the reader to [2–7] for an overview. In this paper, we study the necessary and sufficient conditions on the parameters for boundedness of the multilinear fractional maximal operator \(\mathcal{M}_{\Omega,\alpha}\) and the multilinear fractional integrals \(\mathcal{I}_{\Omega,\alpha}\) with rough kernels on Morrey spaces and modified Morrey spaces, respectively, whose definitions are given below.
Let \(0<\alpha<mn\), \(s>1\), \(\Omega\in L^{s}(\mathbb{S}^{mn-1})\) be a homogeneous function of degree zero on \(\mathbb{R}^{mn}\). The multilinear fractional integral operator and its corresponding maximal operator are, respectively, defined by
where \(d\vec{y}=dy_{1}\cdots dy_{m}\). If \(m=1\), \(\mathcal{I}_{\Omega,\alpha}\) is the homogeneous fractional integral operators (see [8]). If \(m=1\) and \(\Omega\equiv1\), \(\mathcal{I}_{\Omega,\alpha}\) and \(\mathcal {M}_{\Omega,\alpha}\) are the Riesz potential \(I_{\alpha}\) and the fractional maximal operator \(M_{\alpha}\) [9, 10] given by
In the theory of partial differential equations, Morrey spaces play an important role. Morrey spaces were introduced by Morrey [11] in 1938 in connection with certain problems in elliptic partial differential equations and the calculus of variation.
Definition 1.1
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, the sets of locally integrable functions \(f(x)\), \(x\in\mathbb{R}^{n}\), with the finite norms
respectively.
Definition 1.2
[14]
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, the sets of locally integrable functions \(f(x)\), \(x\in\mathbb{R}^{n}\), with the finite norms
respectively.
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 [14], we know
We list two remarkable results on Morrey spaces for \(I_{\alpha}\).
Theorem A
[13]
Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\), \(1/q=1/p-\alpha/n\), and \(\mu/q=\lambda/p\). Then for \(p>1\), the operator \(I_{\alpha}\) is bounded from \(L^{p,\lambda}(\mathbb{R}^{n})\) to \(L^{q,\mu}(\mathbb{R}^{n})\) and for \(p=1\), \(I_{\alpha}\) is bounded from \(L^{1,\lambda}(\mathbb{R}^{n})\) to \(WL^{q,\mu}(\mathbb{R}^{n})\).
Theorem B
Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).
-
(i)
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 \(WL^{q,\lambda }(\mathbb{R}^{n})\).
Motivated by these two results above, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator \(\mathcal{M}_{\Omega,\alpha}\) and the multilinear fractional integral operator \(\mathcal{I}_{\Omega,\alpha}\) with rough kernels on Morrey spaces and modified Morrey spaces, respectively. This extends a recent result of [14]; the necessary and sufficient conditions for the boundedness of \(M_{\alpha}\) and \(I_{\alpha}\) on modified spaces are considered. If we denote by p, q the harmonic mean of \(p_{1},\ldots,p_{m}>1\) and \(q_{1},\ldots,q_{m}>1\), then our results can be stated as follows.
Theorem 1.1
Let \(0<\alpha<mn\), \(1< s<\infty\) and \(\Omega\in L^{s}(\mathbb {S}^{mn-1})\). Suppose \(\frac{\lambda}{p}=\sum_{j=1}^{m} \frac{\lambda _{j}}{p_{j}}\), \(\frac{1}{q_{j}}=\frac{1}{p_{j}}-\frac{\alpha}{m(n-\lambda_{j})}\) and \(0\leq\lambda_{j}< n-\frac{\alpha p_{j}}{m}\).
-
(i)
If \(p>s'\) and \(\frac{\lambda}{q}=\sum_{j=1}^{m} \frac {\lambda_{j}}{q_{j}}\), then the condition \(1/p-1/q=\alpha/(n-\lambda)\) is necessary and sufficient for the boundedness of the operator \(\mathcal{M}_{\Omega,\alpha}\) 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 \(p=s'\) and \(\lambda\sum_{j=1}^{m}\frac{1}{ p_{j}q_{j}}=\sum_{j=1}^{m} \frac{\lambda_{j}}{p_{j}q_{j}}\), then the condition \(1/p-1/q=\alpha/(n-\lambda)\) is necessary and sufficient for the boundedness of the operator \(\mathcal{M}_{\Omega,\alpha}\) 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, the corresponding estimates for \(\mathcal{I}_{\Omega ,\alpha}\) hold.
Theorem 1.2
Let α, Ω, s, \(p_{j}\), \(\lambda_{j}\), p and λ be as in Theorem 1.1.
-
(i)
If \(p>s'\) and \(\frac{\lambda}{q}=\sum_{j=1}^{m} \frac {\lambda_{j}}{q_{j}}\), then the condition \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\) is necessary and sufficient for the boundedness of the operator \(\mathcal{M}_{\Omega ,\alpha}\) 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 \(p=s'\) and \(\lambda\sum_{j=1}^{m}\frac{1}{ p_{j}q_{j}}=\sum_{j=1}^{m} \frac{\lambda_{j}}{p_{j}q_{j}}\), then the condition \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\) is necessary and sufficient for the boundedness of the operator \(\mathcal{M}_{\Omega ,\alpha}\) 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, the corresponding estimates for \(\mathcal{I}_{\Omega ,\alpha}\) hold.
The organization of this paper is as follows: We will give the boundedness of \(\mathcal{M}_{\Omega,\alpha}\) and \(\mathcal{I}_{\Omega ,\alpha}\) on Morrey spaces and on modified Morrey spaces in Section 2 and Section 3, respectively. In Section 4, some applications are given.
2 Boundedness on Morrey spaces
In this section we study the boundedness of \(\mathcal{M}_{\Omega,\alpha }\) and \(\mathcal{I}_{\Omega,\alpha}\) on Morrey spaces. The following lemmas play an important role in the proof of Theorem 1.1.
Lemma 2.1
Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).
-
(i)
If \(p>1\), then the condition \(1/p-1/q=\alpha/(n-\lambda)\) is necessary and sufficient for the boundedness of the operator \(M_{\alpha}\) 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 \(M_{\alpha}\) from \(L^{1,\lambda}(\mathbb{R}^{n})\) to \(WL^{q,\lambda }(\mathbb{R}^{n})\).
Lemma 2.2
[15]
Let \(0<\alpha<mn\), and let \(f_{j}\in L^{p_{j}}(\mathbb{R}^{n})\) with \(1< p_{j}<\infty\) for \(j=1,2,\ldots,m\). For any \(0<\epsilon<\min\{\alpha,mn-\alpha\}\), there exists a constant \(C<\infty\) such that for any \(x\in\mathbb{R}^{n}\),
Lemma 2.3
Let \(0<\alpha<mn\), \(1\leq s'<\frac{mn}{\alpha}\), and let \(f_{j}\in L^{p_{j}}(\mathbb{R}^{n})\) with \(1< p_{j}< \infty\) for \(j=1,2,\ldots,m\). Then there exists a constant \(C<\infty\) such that for any \(x\in\mathbb{ R}^{n}\),
Proof
Since \(\Omega\in L^{s}(\mathbb{S}^{mn-1})\), using the Hölder inequality, we obtain
This completes the proof of the lemma. □
Proof of Theorem 1.1
We first prove Theorem 1.1 is true for \(\mathcal{M}_{\Omega,\alpha}\); then the proof for \(\mathcal {I}_{\Omega,\alpha}\) follows.
(i) Sufficiency. The case \(p>s'\). Since each \(p_{j}> s'\), by the Hölder inequality and Lemma 2.1 and Lemma 2.3, we have
where \(\frac{1}{q_{j}}=\frac{1}{p_{j}}-\frac{\alpha}{m(n-\lambda_{j})}\).
Necessity. Suppose that \(\mathcal{M}_{\Omega,\alpha}\) is bounded from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(L^{q,\lambda}\). Let \(\vec{f}_{\epsilon}(x)=(f_{1}(\epsilon x),\ldots, f_{m}(\epsilon x))\) for all \(\epsilon>0\). Then by changing of the variables, we see that
Thus
Since \(\mathcal{M}_{\Omega,\alpha}\) is bounded from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(L^{q,\lambda}\), we have
where C is independent of ϵ.
If \(1/p<1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{L^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
Also, if \(1/p>1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{L^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Therefore we get \(1/p=1/q+\alpha/(n-\lambda)\).
(ii) Sufficiency. The case \(p=s'\). We apply the Hölder inequality to Lemma 2.3 to obtain the fact
For any \(\beta>0\), let \(\varepsilon_{0}=\beta\), \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon _{m-1}>0\) and \(\varepsilon_{m}=1\) such that
where \(q_{j}\) is given by \(1-\frac{1}{q_{j}}=\frac{\alpha p_{j}}{m(n-\lambda_{j})}\). Hence, we have
Then, by Lemma 2.1, we have
Hence, we obtain the following inequality:
Necessity. Let \(\mathcal{M}_{\Omega,\alpha}\) be bounded from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(WL^{q,\lambda}\). By using (2.1), we obtain
By the boundedness of \(\mathcal{M}_{\Omega,\alpha}\) from \(L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\) to \(WL^{q,\lambda}\), we have
where C is independent of ϵ.
If \(1/p<1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{WL^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
Also, if \(1/p>1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in L^{p_{1},\lambda_{1}}\times\cdots\times L^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{WL^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Consequently, we get \(1/p=1/q+\alpha/(n-\lambda)\).
Now we prove the corresponding estimates for \(\mathcal{I}_{\Omega,\alpha }\) hold. By the same arguments as above we can get the necessity parts of Theorem 1.1(i) and (ii) for \(\mathcal {I}_{\Omega,\alpha}\). So we just give the sufficiency parts, respectively.
First we study the sufficiency of the condition in Theorem 1.1(i) for \(\mathcal{I}_{\Omega,\alpha}\).
Following the method used in [16], we choose a small positive number ϵ with \(0<\epsilon<\min\{ \alpha,\frac{m(n-\lambda_{j})}{p_{j}}-\alpha,\frac{n-\lambda}{p}-\alpha\}\). One can then see from the condition of Theorem 1.1 that \(1\leq s'< p_{j}<\frac{m(n-\lambda_{j})}{\alpha+\epsilon}\) and \(1\leq s'< p_{j}<\frac{m(n-\lambda_{j})}{\alpha-\epsilon}\), and we let
and
Now if each \(p_{j}>s'\), then Theorem 1.1(i) implies that
A simple calculation yields \(\frac{q}{2\tilde{q}_{1}}+\frac{q}{2\tilde {q}_{2}}=1\). Hence, using Lemma 2.2, the Hölder inequality and the above inequalities, we have
Now we study the sufficiency of the condition in Theorem 1.1(ii) for \(\mathcal{I}_{\Omega,\alpha}\).
For any \(\beta>0\), we denote \(\mu^{2}=\beta^{2-\frac{q}{\tilde{q}_{2}}} (\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j},{\lambda_{j}}}} )^{\frac{q}{\tilde {q}_{2}}-1}\). Then by Lemma 2.2, we have
Hence, we obtain the following inequality:
Thus we complete the proof of Theorem 1.1. □
3 Boundedness on modified Morrey spaces
In this section we study the boundedness of \(\mathcal{M}_{\Omega,\alpha }\) and \(\mathcal{I}_{\Omega,\alpha}\) on modified Morrey spaces. The following inequality for \(M_{\alpha}\) in Modified Morrey spaces is valid.
Lemma 3.1
[14] Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).
-
(i)
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 \(M_{\alpha}\) 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 \(M_{\alpha}\) from \(\widetilde{L}^{1,\lambda}\) to \(W\widetilde{L}^{q,\lambda}\).
We are ready to prove Theorem 1.2.
Proof
Similar to the proofs of sufficiency in Theorem 1.1, we will get the sufficiency parts for \(\mathcal{M}_{\Omega,\alpha}\) and \(\mathcal{I}_{\Omega,\alpha}\), respectively. Now, we give only the proof of necessity for \(\mathcal{M}_{\Omega,\alpha }\), since the main steps and the ideas are almost the same as \(\mathcal{I}_{\Omega,\alpha}\).
Let \([\epsilon]_{1,+}=\max\{1,\epsilon\}\). Then by (2.1), we obtain
and
(i) Let \(\mathcal{M}_{\Omega,\alpha}\) be bounded from \(\widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\) to \(\widetilde{L}^{q,\lambda}\). Then we have
where C is independent of ϵ.
If \(1/p<1/q+\alpha/n\), then for all \(\vec{f}\in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
Also, if \(1/p>1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Therefore we get \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\).
(ii) Let \(\mathcal{M}_{\Omega,\alpha}\) be bounded from \(\widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\) to \(W\widetilde{L}^{q,\lambda}\). Then we have
where C is independent of ϵ.
If \(1/p<1/q+\alpha/n\), then for all \(\vec{f} \in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{W\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow0\).
Also, if \(1/p>1/q+\alpha/(n-\lambda)\), then for all \(\vec{f} \in \widetilde{L}^{p_{1},\lambda_{1}}\times\cdots\times \widetilde{L}^{p_{m},\lambda_{m}}\), we have \(\|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{W\widetilde{L}^{q,\lambda}}=0\) as \(\epsilon\rightarrow\infty\).
Consequently, we get \(\alpha/n\leq1/p-1/q\leq\alpha/(n-\lambda)\).
This completes the proof of Theorem 1.2. □
4 Some applications
As an application, we first obtain a result parallel to Theorem A for the operator \(\mathcal{M}_{\Omega,\alpha}\) and \(\mathcal{I}_{\Omega,\alpha}\).
Corollary 4.1
Let α, Ω, s, \(p_{j}\), \(\lambda_{j}\), p, and λ be as in Theorem 1.1, \(1/q=1/p-\alpha/{n}\), \(\mu/q=\lambda/p\).
-
(i)
If \(p>s'\) and \(\frac{\lambda}{q}=\sum_{j=1}^{m} \frac {\lambda_{j}}{q_{j}}\), then there exists a constant \(C<\infty\) such that
$$\begin{aligned} \|M_{\Omega,\alpha}\vec{f}\|_{L^{q,\mu}(\mathbb{R}^{n})}\leq C\prod _{j=1}^{m}\|f_{j}\|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}. \end{aligned}$$ -
(ii)
If \(p=s'\) and \(\lambda\sum_{j=1}^{m}\frac{1}{ p_{j}q_{j}}=\sum_{j=1}^{m} \frac{\lambda_{j}}{p_{j}q_{j}}\), then there exists a constant \(C<\infty\) such that
$$\begin{aligned} \|M_{\Omega,\alpha}\vec{f}\|_{WL^{q,\mu}(\mathbb{R}^{n})}\leq C\prod _{j=1}^{m}\|f_{j}\|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}. \end{aligned}$$
Moreover, similar estimates hold for \(\mathcal{I}_{\Omega,\alpha}\).
Proof
The proof follows from similar steps in Corollary 3.1, [17], here we omit the proof. □
As another application, we obtain the Olsen inequality which is a multi-version of the results considered by Olsen in [18] in the study of the Schrödinger equation with perturbed potentials W on \(\mathbb{R}^{n}\). As a consequence of Theorem 1.1 and the Hölder inequality, we have the following.
Corollary 4.2
Let α, Ω, s, \(p_{j}\), \(\lambda_{j}\), p, and λ be as in Theorem 1.1, \(1/p-1/q=\alpha/(n-\lambda)\) and let \(W\in L^{(n-\lambda)/\alpha,\lambda}\). We get the following.
-
(i)
If \(p>s'\) and \(\frac{\lambda}{q}=\sum_{j=1}^{m} \frac {\lambda_{j}}{q_{j}}\), then there exists a constant \(C<\infty\) such that
$$\|W\cdot M_{\Omega,\alpha}\vec{f}\|_{L^{p,\lambda}(\mathbb{R}^{n})} \leq C \|W\|_{L^{(n-\lambda)/\alpha,\lambda}(\mathbb {R}^{n})} \|f_{1}\|_{L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})}\times\cdots\times \|f_{m} \|_{L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})}. $$ -
(ii)
If \(p=s'\) and \(\lambda\sum_{j=1}^{m}\frac{1}{ p_{j}q_{j}}=\sum_{j=1}^{m} \frac{\lambda_{j}}{p_{j}q_{j}}\), then there exists a constant \(C<\infty\) such that
$$\|W\cdot M_{\Omega,\alpha}\vec{f}\|_{WL^{p,\lambda}(\mathbb{R}^{n})} \leq C \|W\|_{WL^{(n-\lambda)/\alpha,\lambda}(\mathbb {R}^{n})} \|f_{1}\|_{L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})}\times\cdots\times \|f_{m} \|_{L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})}. $$
Moreover, similar estimates hold for \(\mathcal{I}_{\Omega,\alpha}\).
Remark 4.1
We point out that similar results in Corollary in 4.1 and 4.2 hold on modified Morrey spaces; we do not list them here.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11401175) and Doctor Foundation of Henan Polytechnic University (No. B2012-055).
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Wang, Z., Si, Z. Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces. J Inequal Appl 2015, 104 (2015). https://doi.org/10.1186/s13660-015-0627-2
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DOI: https://doi.org/10.1186/s13660-015-0627-2