1 Introduction

Kenig and Stein [1] studied the boundedness of multilinear fractional integral operator \(\mathcal{I}_{\alpha,m}\), \(0<\alpha< mn\), on Lebesgue spaces.

$$\mathcal{I}_{\alpha,m}\vec{f}(x)=\int_{(\mathbb{R}^{n})^{m}} \frac {f_{1}(y_{1})f_{2}(y_{2})\cdots f_{m}(y_{m})}{|(x-y_{1},x-y_{2},\ldots ,x-y_{m})|^{mn-\alpha}}\,dy_{1}\cdots \,dy_{m}, $$

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

$$\begin{aligned}& \mathcal{I}_{\Omega,\alpha}\vec{f}(x)=\int_{(\mathbb{R}^{n})^{m}} \frac {\Omega(\vec{y})}{|\vec{y}|^{mn-\alpha}}\prod_{i=1}^{m}f_{i}(x-y_{i}) \,d\vec{y}; \\& \mathcal{M}_{\Omega,\alpha}\vec{f}(x)=\sup_{r>0} \frac{1}{r^{mn-\alpha }}\int_{|\vec{y}|< r}\bigl|\Omega(\vec{y})\bigr|\prod _{i=1}^{m}\bigl|f_{i}(x-y_{i})\bigr| \,d\vec{y}, \end{aligned}$$

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

$$I_{\alpha}f(x) =\int_{\mathbb{R}^{n}} \frac{f(x-y)}{|y|^{n-\alpha}}\,dy,\qquad M_{\alpha}f(x) =\sup_{r>0}\frac {1}{r^{n-\alpha}}\int _{|y|\leq r} f(x-y)\,dy. $$

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

[12, 13]

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

$$\begin{aligned}& \Vert f\Vert _{L^{p,\lambda}(\mathbb{R}^{n})}=\sup_{x\in\mathbb {R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}}\int_{B(x,t)}\bigl|f(y)\bigr|^{p}\,dy \biggr)^{\frac{1}{p}}, \\& \Vert f\Vert _{WL^{p,\lambda}(\mathbb{R}^{n})}=\sup_{r>0}\ r\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{t^{\lambda}} \bigl|\bigl\{ y\in B(x,t):\bigl|f(y)\bigr|>r \bigr\} \bigr| \biggr)^{\frac{1}{p}}, \end{aligned}$$

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

$$\begin{aligned}& \Vert f\Vert _{\widetilde{L}^{p,\lambda}(\mathbb{R}^{n})} =\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac{1}{[t]_{1}^{\lambda}}\int_{B(x,t)}\bigl|f(y)\bigr|^{p}\,dy \biggr)^{\frac{1}{p}}, \\& \Vert f\Vert _{W\widetilde{L}^{p,\lambda}(\mathbb{R}^{n})}=\sup_{r>0} r\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[t]_{1}^{\lambda}} \bigl|\bigl\{ y\in B(x,t):\bigl|f(y)\bigr|>r \bigr\} \bigr| \biggr)^{\frac{1}{p}}, \end{aligned}$$

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

$$\widetilde{L}^{p,\lambda}\bigl(\mathbb{R}^{n}\bigr) \subset_{\succ}L^{p,\lambda }\bigl(\mathbb{R}^{n}\bigr)\cap L^{p}\bigl(\mathbb{R}^{n}\bigr), \qquad\max \bigl\{ \Vert f \Vert _{L^{p,\lambda}}, \Vert f\Vert _{L^{p}} \bigr\} \leq \Vert f \Vert _{\widetilde{L}^{p,\lambda}}. $$

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

[12, 14]

Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).

  1. (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 \(L^{q,\lambda}(\mathbb{R}^{n})\).

  2. (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 \(0\leq\lambda_{j}< n-\frac{\alpha p_{j}}{m}\).

  1. (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 \(L^{q,\lambda}(\mathbb{R}^{n})\).

  2. (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.

  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 \(\widetilde{L}^{q,\lambda}(\mathbb{R}^{n})\).

  2. (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

[12, 14]

Let \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).

  1. (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 \(L^{q,\lambda}(\mathbb{R}^{n})\).

  2. (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 \(x\in\mathbb{R}^{n}\),

$$\bigl\vert \mathcal{I}_{\Omega,\alpha}\vec{f}(x)\bigr\vert \leq C \bigl[ \mathcal {M}_{\Omega,\alpha+\epsilon}\vec{f}(x) \bigr]^{\frac{1}{2}} \bigl[ \mathcal{M}_{\Omega,\alpha-\epsilon}\vec{f}(x) \bigr]^{\frac{1}{2}}. $$

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 \(x\in\mathbb{ R}^{n}\),

$$\mathcal{M}_{\Omega,\alpha}\vec{f}(x)\leq C\prod_{i={1}}^{m} \bigl[M_{\frac{\alpha s'}{m} }f_{j}^{s'} \bigr]^{\frac{1}{s'}}(x). $$

Proof

Since \(\Omega\in L^{s}(\mathbb{S}^{mn-1})\), using the Hölder inequality, we obtain

$$\begin{aligned} &\frac{1}{r^{mn-\alpha}}\int_{|\vec{y}|< r}\bigl\vert \Omega(\vec{y})\bigr\vert \prod_{j={1}}^{m} \bigl\vert f_{j}(x-y_{j})\bigr\vert \,d\vec{y} \\ &\quad\leq \frac{1}{r^{mn-\alpha}} \biggl(\int_{|\vec{y}|<r}\bigl|\Omega(\vec {y})\bigr|^{s}\,d\vec{y} \biggr)^{\frac{1}{s}} \Biggl(\int _{|\vec{y}|<r}\prod_{j={1}}^{m}\bigl|f_{j}(x-y_{j})\bigr|^{s'} \,d\vec{y} \Biggr)^{\frac{1}{s'}} \\ &\quad\leq C\sup_{r>0}\frac{1}{r^{mn(1-\frac{1}{s})-\alpha}} \Biggl(\int _{|\vec {y}|<r}\prod_{j={1}}^{m}\bigl|f_{j}(x-y_{j})\bigr|^{s'} \,d\vec{y} \Biggr)^{\frac {1}{s'}} \\ &\quad\leq C\sup_{r>0} \Biggl(\frac{1}{r^{mn-\alpha s'}}\int _{|\vec {y}|<r}\prod_{j={1}}^{m}\bigl|f_{j}(x-y_{j})\bigr|^{s'} \,d\vec{y} \Biggr)^{\frac {1}{s'}} \\ &\quad\leq C \Biggl(\sup_{r>0}\frac{1}{r^{mn-\alpha s'}}\int _{|\vec{y}|<r}\prod_{j={1}}^{m}\bigl|f_{j}(x-y_{j})\bigr|^{s'} \,d\vec{y} \Biggr)^{\frac{1}{s'}} \\ &\quad\leq C \Biggl(\sup_{r>0}\frac{1}{r^{mn-\alpha s'}}\int _{|y_{1}|<r}\cdots\int_{|y_{m}|<r}\prod _{j={1}}^{m}\bigl|f_{j}(x-y_{j})\bigr|^{s'} \,d\vec{y} \Biggr)^{\frac{1}{s'}} \\ &\quad\leq C \prod_{j={1}}^{m} \biggl(\sup _{r>0}\frac{1}{r^{n-\alpha s'/m}}\int_{|y_{j}|<r}\bigl\vert f_{j}(x-y_{j})\bigr\vert ^{s'} \,dy_{j} \biggr)^{\frac{1}{s'}} \\ &\quad=C\prod_{j={1}}^{m} \bigl[M_{\frac{\alpha s'}{m}}f_{j}^{s'} \bigr]^{\frac {1}{s'}}(x). \end{aligned}$$

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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f} \|_{L^{q,\lambda}(\mathbb {R}^{n})}&=\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{t^{\lambda}} \int _{B(x,t)}\bigl\vert \mathcal{M}_{\Omega,\alpha}\vec{f}(y)\bigr\vert ^{q}\,dy \biggr)^{\frac{1}{q}} \\ &\leq C\sup_{x\in\mathbb{R}^{n},t>0} \Biggl(\frac{1}{t^{\lambda}} \int _{B(x,t)}\Biggl\vert \prod_{j={1}}^{m} \bigl[M_{ \frac{\alpha s'}{m} }f_{j}^{s'}(y)\bigr]^{\frac{1}{s'}} \Biggr\vert ^{q}\,dy \Biggr)^{\frac{1}{q}} \\ &\leq C \prod_{j=1}^{m}\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{t^{\lambda_{j}}} \int_{\mathbb{R}^{n}} \bigl\vert M_{ \frac{\alpha s'}{m}}f_{j}^{s'}(y)\bigr\vert ^{\frac{q_{j}}{s'}}\,dy \biggr)^{\frac{1}{q_{j}}} \\ &\leq C \prod_{j=1}^{m} \bigl\| f_{j}^{s'}\bigr\| _{L^{ p_{j}/s',\lambda_{j}}(\mathbb{R}^{n})}^{1/ s'} \\ &=C \prod_{j=1}^{m}\| f_{j} \| _{L^{ p_{j},\lambda_{j} }(\mathbb{R}^{n})}, \end{aligned}$$

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

$$ \mathcal{M}_{\Omega,\alpha}{\vec{f}_{\epsilon}}(y)= \epsilon^{-\alpha }\mathcal{M}_{\Omega,\alpha}{\vec{f}}(\epsilon y). $$
(2.1)

Thus

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}{\vec{f}_{\epsilon}}\|_{L^{q,\lambda}} &= \epsilon^{-\alpha}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{t^{\lambda}}\int _{B(x,t)}\bigl\vert \mathcal{M}_{\Omega,\alpha}{\vec {f}}( \epsilon y)\bigr\vert ^{q}\,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-n/q}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{t^{\lambda}} \int_{B(x,\epsilon t)}\bigl\vert \mathcal{M}_{\Omega,\alpha }{\vec{f}}(y) \bigr\vert ^{q}\,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-n/q+\lambda/q}\sup_{x\in\mathbb {R}^{n},t>0} \biggl(\frac{1}{(\epsilon t)^{\lambda}} \int_{B( x,\epsilon t)}\bigl\vert \mathcal{M}_{\Omega,\alpha}{\vec{f}}(y) \bigr\vert ^{q}\,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-(n-\lambda)/q}\|\mathcal{M}_{\Omega,\alpha}{\vec {f}} \|_{L^{q,\lambda}}. \end{aligned}$$

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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{L^{q,\lambda}}&=\epsilon ^{\alpha+(n-\lambda)/q}\|\mathcal{M}_{\Omega,\alpha}{\vec{f}_{\epsilon}} \|_{L^{q,\lambda}} \\ &\leq C\epsilon^{\alpha+(n-\lambda)/q}\prod_{j=1}^{m} \bigl\| f_{j}(\epsilon\cdot )\bigr\| _{L^{p_{j},\lambda_{j}}} \\ &=C\epsilon^{\alpha+(n-\lambda)/q} \prod_{j=1}^{m} \sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{t^{\lambda_{j}}}\int_{B(x,t)}\bigl|f_{j}( \epsilon y)\bigr|^{p_{j}}\,dy \biggr)^{1/{p_{j}}} \\ &= C\epsilon^{\alpha+(n-\lambda)/q} \prod_{j=1}^{m} \epsilon^{-n/{p_{j}}}\sup_{x\in\mathbb {R}^{n},t>0} \biggl(\frac{1}{t^{\lambda_{j}}}\int _{B( x,\epsilon t)}\bigl|f_{j}(y)\bigr|^{p_{j}}\,dy \biggr)^{1/{p_{j}}} \\ &= C\epsilon^{\alpha+(n-\lambda)/q} \prod_{j=1}^{m} \epsilon^{(\lambda_{j}-n)/{p_{j}}}\sup_{x\in\mathbb {R}^{n},t>0} \biggl(\frac{1}{(\epsilon t)^{\lambda_{j}}}\int _{B( x,\epsilon t)}\bigl|f_{j}(y)\bigr|^{p_{j}}\,dy \biggr)^{1/{p_{j}}} \\ &= C\epsilon^{\alpha+(n-\lambda)/q-(n-\lambda)/{p}} \prod_{j=1}^{m} \|f_{j}\|_{L^{p_{j},\lambda_{j}}}, \end{aligned}$$

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

$$\mathcal{M}_{\Omega,\alpha}\vec{f}(x)\leq C\prod_{j={1}}^{m} \bigl[M_{\frac{\alpha s'}{m} }f_{j}^{s'} \bigr]^{\frac{1}{s'}}(x) \leq C \prod_{j={1}}^{m} \bigl[M_{\frac{\alpha p_{j}s'}{mp} }f_{j}^{\frac{p_{j}s'}{p}} \bigr]^{\frac{p}{p_{j}s'}}(x)=C \prod_{j={1}}^{m} \bigl[M_{\frac{\alpha p_{j}}{m} }f_{j}^{p_{j}} \bigr]^{\frac{1}{p_{j}}}(x). $$

For any \(\beta>0\), let \(\varepsilon_{0}=\beta\), \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon _{m-1}>0\) and \(\varepsilon_{m}=1\) such that

$$ \biggl(\frac{\varepsilon_{j}}{\varepsilon_{j-1}} \biggr)^{p_{j}q_{j}}=\frac { [\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j},{\lambda_{j}}}} ]^{{q}}}{\beta ^{q}\|f_{j}\|^{p_{j}}_{L^{p_{j},{\lambda_{j}}}}},\quad j=1,2, \ldots,m, $$

where \(q_{j}\) is given by \(1-\frac{1}{q_{j}}=\frac{\alpha p_{j}}{m(n-\lambda_{j})}\). Hence, we have

$$\begin{aligned} \bigl\{ y\in B(x,t):\bigl\vert \mathcal{M}_{\Omega,\alpha}\vec{f}(y)\bigr\vert >C\beta \bigr\} \subset\bigcup_{j=1}^{m} \biggl\{ y\in B(x,t): \bigl[M_{\frac{\alpha p_{j}}{m} }f_{j}^{p_{j}} \bigr]^{\frac{1}{p_{j}}}(y)>\frac{\varepsilon _{j-1}}{\varepsilon_{j}t^{(\lambda-\lambda_{j})/p_{j}q_{j}}} \biggr\} . \end{aligned}$$

Then, by Lemma 2.1, we have

$$\begin{aligned} &\bigl\vert \bigl\{ y\in B(x,t):\bigl\vert \mathcal{M}_{\Omega,\alpha} \vec{f}(y)\bigr\vert >\beta\bigr\} \bigr\vert \\ &\quad\leq C\sum_{j=1}^{m}\biggl\vert \biggl\{ y\in B(x,t): \bigl[M_{\frac{\alpha p_{j}}{m} }f_{j}^{p_{j}} \bigr]^{\frac{1}{p_{j}}}(y)>\frac{\varepsilon _{j-1}}{\varepsilon_{j}t^{(\lambda-\lambda_{j})/p_{j}q_{j}}} \biggr\} \biggr\vert \\ &\quad\leq C\sum_{j=1}^{m}\biggl\vert \biggl\{ y\in B(x,t):M_{\frac{\alpha p_{j}}{m} }f_{j}^{p_{j}}(y)> \biggl( \frac{\varepsilon_{j-1}}{\varepsilon_{j} t^{(\lambda -\lambda_{j})/p_{j}q_{j}}} \biggr)^{{p_{j}}} \biggr\} \biggr\vert \\ &\quad\leq C\sum_{j=1}^{m}t^{\lambda_{j}} \biggl(\frac{\varepsilon_{j}t^{(\lambda -\lambda_{j})/p_{j}q_{j}}}{\varepsilon_{j-1}} \biggr)^{{p_{j}q_{j}}} \bigl\| {f_{j}}^{p_{j}} \bigr\| _{L^{1,{\lambda_{j}}}}^{q_{j}} \\ &\quad=C\sum_{j=1}^{m}t^{\lambda} \biggl( \frac{\varepsilon _{j}}{\varepsilon_{j-1}} \biggr)^{{p_{j}q_{j}}} \|f_{j}\|_{L^{p_{j},{\lambda_{j}}}}^{p_{j}} \\ &\quad=C t^{\lambda}\Biggl(\frac{1}{\beta}\prod _{j=1}^{m}\|f_{j}\|_{L^{p_{j},{\lambda _{j}}}} \Biggr)^{q}. \end{aligned}$$

Hence, we obtain the following inequality:

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{WL^{q,\lambda}}=\sup _{\beta>0}\ \beta\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}} \bigl|\bigl\{ y\in B(x,t):\bigl|\mathcal{M}_{\Omega,\alpha}\vec{f}(y)\bigr|> \beta\bigr\} \bigr| \biggr)^{\frac{1}{q}} \leq C\prod_{j=1}^{m}\|f_{j} \|_{L^{p_{j},\lambda_{j}}}. \end{aligned}$$

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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}{\vec{f}_{\epsilon}}\|_{WL^{q,\lambda}} &=\sup _{r>0} r\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{t^{\lambda}} \int_{\{y\in B(x,t):|\mathcal{M}_{\Omega,\alpha}{\vec {f}_{\epsilon}}(y)|>r\}}\,dy \biggr)^{\frac{1}{q}} \\ &=\sup_{r>0} r\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac {1}{t^{\lambda}}\int_{\{y\in B( x,t):|\mathcal{M}_{\Omega,\alpha}{\vec {f}}(\epsilon y)|>r\epsilon^{\alpha}\}}\,dy \biggr)^{\frac{1}{q}} \\ &=\epsilon^{-n/q}\sup_{r>0}\ r\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{t^{\lambda}}\int_{\{y\in B( x,\epsilon t):|\mathcal{M}_{\Omega,\alpha}{\vec{f}}(y)|>r\epsilon^{\alpha}\} } \,dy \biggr)^{\frac{1}{q}} \\ &=\epsilon^{-\alpha-n/q+\lambda/q}\sup_{r>0} r\epsilon^{\alpha}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{(\epsilon t)^{\lambda}} \int_{\{y\in B( x,\epsilon t):|\mathcal{M}_{\Omega,\alpha}{\vec{f}}(y)|>r\epsilon^{\alpha}\} } \,dy \biggr)^{\frac{1}{q}} \\ &=\epsilon^{-\alpha-(n-\lambda)/q}\|\mathcal{M}_{\Omega,\alpha}{\vec {f}} \|_{WL^{q,\lambda}}. \end{aligned}$$

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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}{\vec{f}}\|_{WL^{q,\lambda}}&=\epsilon ^{\alpha+(n-\lambda)/q}\|\mathcal{M}_{\Omega,\alpha}{\vec{f}_{\epsilon}} \|_{WL^{q,\lambda}} \\ &\leq C\epsilon^{\alpha+(n-\lambda)/q}\prod_{j=1}^{m} \bigl\| f_{j}(\epsilon\cdot )\bigr\| _{L^{p_{j},\lambda_{j}}} \\ &\leq C\epsilon^{\alpha+(n-\lambda)/q-(n-\lambda)/p} \prod_{j=1}^{m} \|f_{j}\|_{L^{p_{j},\lambda_{j}}}, \end{aligned}$$

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

$$\frac{1}{\tilde{q}_{1}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots+ \frac {1}{p_{m}}-\frac{\alpha+\epsilon}{n-\lambda}=\frac{1}{p}-\frac{\alpha +\epsilon}{n-\lambda}, $$

and

$$\frac{1}{\tilde{q}_{2}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots+ \frac {1}{p_{m}}-\frac{\alpha-\epsilon}{n-\lambda}=\frac{1}{p}-\frac{\alpha -\epsilon}{n-\lambda}. $$

Now if each \(p_{j}>s'\), then Theorem 1.1(i) implies that

$$\begin{aligned}& \|\mathcal{M}_{\Omega,\alpha+\epsilon}\vec{f}\|_{L^{\tilde{q}_{1},\lambda }(\mathbb{R}^{n})}\leq \|f_{j} \|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}, \qquad \|\mathcal{M}_{\Omega,\alpha-\epsilon}\vec{f}\|_{L^{\tilde{q}_{2},\lambda }(\mathbb{R}^{n})}\leq \|f_{j} \|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}. \end{aligned}$$

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

$$\begin{aligned} &\|\mathcal{I}_{\Omega,\alpha}\vec{f}\|_{L^{q,\lambda}(\mathbb{R}^{n})}\\ &\quad=\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{t^{\lambda}} \int _{B(x,t)}\bigl\vert \mathcal{I}_{\Omega,\alpha}\vec{f}(y)\bigr\vert ^{q}\,dy \biggr)^{\frac{1}{q}} \\ &\quad\leq C\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}}\int _{\mathbb{R}^{n}} \bigl[\mathcal{M}_{\Omega,\alpha+\epsilon}\vec {f}(x) \bigr]^{\frac{q}{2}} \bigl[\mathcal {M}_{\Omega,\alpha-\epsilon}\vec{f}(x) \bigr]^{\frac{q}{2}}\,dx \biggr)^{\frac{1}{q}} \\ &\quad\leq C\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}}\int _{\mathbb{R}^{n}} \bigl[\mathcal{M}_{\Omega,\alpha+\epsilon}\vec {f}(x) \bigr]^{\tilde{q}_{1}}\,dx \biggr)^{\frac{1}{2\tilde{q}_{1}}}\sup_{x\in\mathbb {R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}} \int_{\mathbb{R}^{n}} \bigl[\mathcal {M}_{\Omega,\alpha-\epsilon}\vec{f}(x) \bigr]^{\tilde{q}_{2}}\,dx \biggr) ^{\frac{1}{2\tilde{q}_{2}}} \\ &\quad\leq C\|\mathcal{M}_{\Omega,\alpha+\epsilon}\vec{f}\|_{L^{\tilde {q}_{1},\lambda}(\mathbb{R}^{n})}^{1/2}\| \mathcal {M}_{\Omega,\alpha-\epsilon}\vec{f}\|_{L^{\tilde{q}_{2},\lambda}(\mathbb {R}^{n})}^{1/2} \\ &\quad\leq C\prod_{j=1}^{m}\| f_{j} \|_{L^{ p_{j},\lambda_{j} }(\mathbb{R}^{n})}. \end{aligned}$$

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

$$\begin{aligned} &\bigl\vert \bigl\{ y\in B(x,t):\bigl\vert \mathcal{I}_{\Omega,\alpha}\vec{f}(y)\bigr\vert >\beta\bigr\} \bigr\vert \\ &\quad\leq C\bigl\vert \bigl\{ y\in B(x,t):C \bigl[\mathcal{M}_{\Omega,\alpha+\epsilon } \vec{f}(x) \bigr]^{\frac{1}{2}} \bigl[\mathcal{M}_{\Omega,\alpha -\epsilon}\vec{f}(x) \bigr]^{\frac{1}{2}}>\beta\bigr\} \bigr\vert \\ &\quad\leq C\bigl\vert \bigl\{ y\in B(x,t):\sqrt{C} \bigl[\mathcal{M}_{\Omega,\alpha +\epsilon} \vec{f}(x) \bigr]^{\frac{1}{2}}>\mu\bigr\} \bigr\vert \\ &\qquad{}+\bigl\vert \bigl\{ y\in B(x,t):\sqrt{C} \bigl[\mathcal{M}_{\Omega,\alpha-\epsilon}\vec {f}(x) \bigr]^{\frac{1}{2}}>\beta/ \mu\bigr\} \bigr\vert \\ &\quad\leq C\bigl\vert \bigl\{ y\in B(x,t):\mathcal{M}_{\Omega,\alpha+\epsilon}\vec {f}(x)>C \mu^{2} \bigr\} \bigr\vert +\bigl\vert \bigl\{ y\in B(x,t): \mathcal{M}_{\Omega,\alpha -\epsilon}\vec{f}(x)>C\beta^{2}/ \mu^{2}\bigr\} \bigr\vert \\ &\quad\leq C t^{\lambda}\Biggl[ \Biggl( \frac{1}{\mu^{2}}\prod _{j=1}^{m}\| f_{j} \|_{L^{ p_{j},\lambda_{j} }} \Biggr)^{\tilde{q}_{1}}+C \Biggl( \frac{\mu^{2}}{\beta ^{2}}\prod _{j=1}^{m}\| f_{j} \|_{L^{ p_{j},\lambda_{j} }} \Biggr)^{\tilde {q}_{2}} \Biggr] \\ &\quad\leq C t^{\lambda}\Biggl(\frac{1}{\beta}\prod _{j=1}^{m}\|f_{j}\|_{L^{p_{j},{\lambda _{j}}}} \Biggr)^{q}. \end{aligned}$$

Hence, we obtain the following inequality:

$$\begin{aligned} \|\mathcal{I}_{\Omega,\alpha}\vec{f}\|_{WL^{q,\lambda}}&=\sup _{\beta>0}\ \beta\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac{1}{t^{\lambda}} \bigl|\bigl\{ y\in B(x,t):\bigl|\mathcal{I}_{\Omega,\alpha}\vec{f}(y)\bigr|> \beta\bigr\} \bigr| \biggr)^{\frac{1}{q}}\leq C\prod_{j=1}^{m}\|f_{j} \|_{L^{p_{j},\lambda_{j}}}. \end{aligned}$$

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 \(0<\alpha<n\), \(1\leq p<{n}/\alpha\), \(0\leq\lambda< n-\alpha p\).

  1. (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 \(\widetilde{L}^{q,\lambda}\).

  2. (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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f}_{\epsilon}\|_{\widetilde {L}^{q,\lambda}} &= \epsilon^{-\alpha}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{[t]_{1}^{\lambda}}\int _{B(x,t)}\bigl|\mathcal{M}_{\Omega,\alpha}\vec {f}(\epsilon y)\bigr|^{q}\,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-n/q}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{[t]_{1}^{\lambda}} \int_{B(\epsilon x,\epsilon t)}\bigl|\mathcal{M}_{\Omega ,\alpha}\vec{f}(y)\bigr|^{q} \,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-n/q}\sup_{t>0} \biggl(\frac{[\epsilon t]_{1}}{[t]_{1}} \biggr)^{\lambda/q}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[\epsilon t]_{1}^{\lambda}}\int _{B(\epsilon x,\epsilon t)}\bigl|\mathcal{M}_{\Omega,\alpha}\vec{f}(y)\bigr|^{q} \,dy \biggr)^{1/q} \\ &=\epsilon^{-\alpha-n/q}[\epsilon]_{1,+}^{\frac{\lambda}{q}}\|\mathcal {M}_{\Omega,\alpha}\vec{f}\|_{\widetilde{L}^{q,\lambda}} \end{aligned}$$

and

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f}_{\epsilon}\|_{W\widetilde {L}^{q,\lambda}} ={}&\sup _{r>0}\ r\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[t]_{1}^{\lambda}} \int_{\{y\in B(x,t):|\mathcal{M}_{\Omega,\alpha}\vec{f}_{\epsilon}(y)|>r\} }\,dy \biggr)^{\frac{1}{q}} \\ ={}&\sup_{r>0} r\sup_{x\in\mathbb{R}^{n},t>0} \biggl( \frac {1}{[t]_{1}^{\lambda}} \int_{\{y\in B( x,t):|\mathcal{M}_{\Omega,\alpha}\vec{f}(\epsilon y)|>r\epsilon^{\alpha}\}}\,dy \biggr)^{\frac{1}{q}} \\ ={}&\epsilon^{-n/q}\sup_{r>0}\ r\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[t]_{1}^{\lambda}}\int_{\{ y\in B(\epsilon x,\epsilon t):|\mathcal{M}_{\Omega,\alpha}\vec{f}(y)|>r\epsilon^{\alpha}\}} \,dy \biggr)^{\frac{1}{q}} \\ ={}&\epsilon^{-\alpha-n/q}\sup_{t>0} \biggl(\frac{[\epsilon t]_{1}}{[t]_{1}} \biggr)^{\lambda/q} \\ &{} \times\sup_{r>0}\ r\epsilon^{\alpha}\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[\epsilon t]_{1}^{\lambda}}\bigl\vert \bigl\{ y\in B( \epsilon x,\epsilon t):\bigl|\mathcal{M}_{\Omega,\alpha}\vec{f}(y)\bigr|>r \epsilon^{\alpha}\bigr\} \bigr\vert \biggr)^{\frac{1}{q}} \\ &=\epsilon^{-\alpha-n/q}[\epsilon]_{1,+}^{\frac{\lambda}{q}}\|\mathcal {M}_{\Omega,\alpha}\vec{f}\|_{W\widetilde{L}^{q,\lambda}}. \end{aligned}$$

(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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{\widetilde{L}^{q,\lambda}} ={}&\epsilon^{\alpha+n/q}[ \epsilon]_{1,+}^{-\frac{\lambda}{q}}\|\mathcal {M}_{\Omega,\alpha} \vec{f}_{\epsilon}\|_{\widetilde{L}^{q,\lambda}} \\ \leq{}& C\epsilon^{\alpha+n/q}[\epsilon]_{1,+}^{-\frac{\lambda}{q}}\prod _{j=1}^{m}\bigl\| f_{j}(\epsilon\cdot) \bigr\| _{\widetilde{L}^{p_{j},\lambda_{j}}} \\ ={}&C\epsilon^{\alpha+n/q}[\epsilon]_{1,+}^{-\frac{\lambda}{q}} \prod _{j=1}^{m}\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{[t]_{1}^{\lambda_{j}}}\int_{B(x,t)}\bigl|f_{j}(\epsilon y)\bigr|^{p_{j}}\,dy \biggr)^{1/{p_{j}}} \\ ={}& C\epsilon^{\alpha+n/q}[\epsilon]_{1,+}^{-\frac{\lambda}{q}} \prod _{j=1}^{m}\epsilon^{-n/{p_{j}}}\sup _{x\in\mathbb{R}^{n},t>0} \biggl(\frac{1}{[t]_{1}^{\lambda_{j}}}\int_{B(\epsilon x,\epsilon t)}\bigl|f_{j}(y)\bigr|^{p_{j}} \,dy \biggr)^{1/{p_{j}}} \\ \leq{}& C\epsilon^{\alpha+n/q}[\epsilon]_{1,+}^{-\frac{\lambda}{q}} \prod _{j=1}^{m}\epsilon^{-n/{p_{j}}}\sup _{t>0} \biggl(\frac{[\epsilon t]_{1}}{[t]_{1}} \biggr)^{\lambda_{j}/p_{j}} \\ &{} \times\sup_{x\in\mathbb{R}^{n},t>0} \biggl(\frac {1}{[\epsilon t]^{\lambda_{j}}}\int _{B(\epsilon x,\epsilon t)}\bigl|f_{j}(y)\bigr|^{p_{j}}\,dy \biggr)^{1/{p_{j}}} \\ \leq{}& C\epsilon^{\alpha+n/q-n/p}[\epsilon]_{1,+}^{-\frac{\lambda}{q}} [ \epsilon]_{1,+}^{\frac{\lambda}{p}} \prod_{j=1}^{m} \|f_{j}\|_{\widetilde{L}^{p_{j},\lambda_{j}}} \\ \leq{}& C\epsilon^{\alpha+n/q-n/p}[\epsilon]_{1,+}^{\frac{\lambda}{p}-\frac {\lambda}{q}} \prod _{j=1}^{m}\|f_{j} \|_{\widetilde{L}^{p_{j},\lambda_{j}}}, \end{aligned}$$

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

$$\begin{aligned} \|\mathcal{M}_{\Omega,\alpha}\vec{f}\|_{W\widetilde{L}^{q,\lambda}} &=\epsilon^{\alpha+n/q}[ \epsilon]_{1,+}^{-\frac{\lambda}{q}}\|\mathcal {M}_{\Omega,\alpha} \vec{f}_{\epsilon}\|_{W\widetilde{L}^{q,\lambda}} \\ &\leq C\epsilon^{\alpha+n/q}[\epsilon]_{1,+}^{-\frac{\lambda}{q}}\prod _{j=1}^{m}\bigl\| f_{j}(\epsilon\cdot) \bigr\| _{\widetilde{L}^{p_{j},\lambda_{j}}} \\ &\leq C\epsilon^{\alpha+n/q-n/p}[\epsilon]_{1,+}^{\frac{\lambda}{p}-\frac {\lambda}{q}} \prod _{j=1}^{m}\|f_{j} \|_{\widetilde{L}^{p_{j},\lambda_{j}}}, \end{aligned}$$

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, \(1/q=1/p-\alpha/{n}\), \(\mu/q=\lambda/p\).

  1. (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

    $$\begin{aligned} \|M_{\Omega,\alpha}\vec{f}\|_{L^{q,\mu}(\mathbb{R}^{n})}\leq C\prod _{j=1}^{m}\|f_{j}\|_{L^{p_{j},\lambda_{j}}(\mathbb{R}^{n})}. \end{aligned}$$
  2. (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.

  1. (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

    $$\|W\cdot M_{\Omega,\alpha}\vec{f}\|_{L^{p,\lambda}(\mathbb{R}^{n})} \leq C \|W\|_{L^{(n-\lambda)/\alpha,\lambda}(\mathbb {R}^{n})} \|f_{1}\|_{L^{p_{1},\lambda_{1}}(\mathbb{R}^{n})}\times\cdots\times \|f_{m} \|_{L^{p_{m},\lambda_{m}}(\mathbb{R}^{n})}. $$
  2. (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.