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Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces

Journal of Inequalities and Applications20152015:104

https://doi.org/10.1186/s13660-015-0627-2

Received: 18 November 2014

Accepted: 10 March 2015

Published: 19 March 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.

Keywords

multilinear fractional operatorsrough kernelsMorrey-type spaces

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 [27] 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.

Declarations

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).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
School of Computer Science and Technique, Henan Polytechnic University, Jiaozuo, P.R. China
(2)
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, P.R. China

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© Wang and Si; licensee Springer. 2015