Open Access

The \(L_{p_{1} r_{1}}\times L_{p_{2} r_{2}}\times\dots\times L_{p_{k}r_{k}}\) boundedness of rough multilinear fractional integral operators in the Lorentz spaces

Journal of Inequalities and Applications20152015:71

https://doi.org/10.1186/s13660-015-0584-9

Received: 5 November 2014

Accepted: 30 January 2015

Published: 26 February 2015

Abstract

In this paper, we prove the O’Neil inequality for the k-linear convolution operator in the Lorentz spaces. As an application, we obtain the necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator \(M_{\Omega,\alpha}(\mathbf{f})\) and the k-linear fractional integral operator \(I_{\Omega,\alpha}(\mathbf{f})\) with rough kernels from the spaces \(L_{p_{1} r_{1}}\times L_{p_{2} r_{2}}\times\cdots\times L_{p_{k} r_{k}}\) to \(L_{q s}\), where \(n/(n+\alpha) \le p< q<\infty\), \(0 < r \le s < \infty\), p is the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\) and r is the harmonic mean of \(r_{1},r_{2},\ldots,r_{k}>0\).

Keywords

O’Neil inequality k-linear convolution rearrangement estimate k-sublinear fractional maximal function k-linear fractional integral harmonic mean Lorentz space

MSC

42B20 42B25 42B35 47G10

1 Introduction

Fractional maximal and fractional integral operators are two important operators in harmonic analysis and partial differential equations. Multilinear maximal operator and multilinear fractional integral operator and related topics have been areas of research of many mathematicians such as Coifman and Grafakos [1], Grafakos [2, 3], Grafakos and Kalton [4], Kenig and Stein [5], Ding and Lu [6], Guliyev and Nazirova [7, 8], Ragusa [9] and others.

Let \(k\geq2\) be an integer and \(\theta_{j}\) (\(j=1,2,\ldots,k\)) be fixed, distinct and nonzero real numbers, and let \(\mathbf{f}=(f_{1},\ldots,f_{k})\). The k-linear convolution operator \(\mathbf{f} \otimes g\) is defined by
$$(\mathbf{f} \otimes g ) (x)=\int_{\mathbb {R}^{n}} f_{1} ( x- \theta_{1}y ) \cdots f_{k} ( x-\theta_{k}y ) g(y)\,dy. $$
Let \(\Omega\in L_{s}({S}^{n-1})\), \(s \ge1\) and Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), and let \(0<\alpha<n\), where \({S}^{n-1}\) is the unit sphere in \(\mathbb {R}^{n}\). The k-sublinear fractional maximal function with rough kernel is defined by
$$M_{\Omega, \alpha} (\mathbf{f}) (x)=\sup_{r>0} \frac{1}{r^{n-\alpha}} \int_{|y|< r} \bigl|\Omega(y)\bigr| \bigl|f_{1} ( x-\theta_{1}y ) \ldots f_{k} ( x-\theta _{k}y )\bigr|\,dy, $$
and the k-linear fractional integral with rough kernel is defined by
$$ I_{\Omega, \alpha}(\mathbf{f}) (x)=\int_{\mathbb {R}^{n}} \frac{\Omega (y)}{|y|^{n-\alpha}} f_{1} ( x-\theta_{1}y ) \cdots f_{k} ( x-\theta_{k}y )\,dy. $$

This paper consists of four sections. In Section 2, some lemmas needed to facilitate the proofs of our theorems and the O’Neil inequality for rearrangements of the k-linear convolution operator \(\mathbf{f} \otimes g\) proved in [7] are given. In Section 3, we prove the O’Neil inequality for the k-linear convolution operator in the Lorentz spaces. Finally, in Section 4, we obtain rearrangement estimates for the multilinear fractional maximal function and multilinear fractional integral with rough kernels. We prove the boundedness of the multilinear fractional maximal operator \(M_{\Omega, \alpha}\) and the multilinear fractional integral operator \(I_{\Omega, \alpha}\) with rough kernels from the spaces \(L_{p_{1} r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k} r_{k}}\) to \(L_{q s}\), \(n/(n+\alpha) \le p< q<\infty\), \(0 < r \le s \le\infty\), where p and r are the harmonic means of \(p_{1},p_{2},\ldots,p_{k}>1\) and \(r_{1},r_{2},\ldots ,r_{k}>0\), respectively. We show that the conditions on the parameters ensuring the boundedness cannot be weakened.

2 Preliminaries

We need the following two generalized Hardy inequalities (see [10]) which are to be used in the proof of Theorem 3.1.

We denote by \(\mathfrak{M}(\mathbb {R}^{n})\) the set of all extended real-valued measurable functions on \(\mathbb {R}^{n}\). When v is a non-negative measurable function on \((0,\infty)\), we say that v is a weight. We denote \(W(t)=\int_{0}^{t} w(\tau)\,d\tau\), \(V(t)=\int_{0}^{t} v(\tau)\,d\tau\) and \(U(r,t)=\int_{t}^{r} u(\tau)\,d\tau\). For simplicity we suppose that \(0< V(t)<\infty\), \(0< W(t)<\infty\) for all \(t>0\) and \(V(\infty)=\infty\), \(W(\infty)=\infty\).

Lemma 2.1

[11]

Let \(0 < r \le s < \infty\) and let v, w be weights. Then the inequality
$$ \biggl(\int_{0}^{\infty} \bigl(g(t) \bigr)^{s} w(t)\,dt \biggr)^{1/s} \le C \biggl(\int _{0}^{\infty} \bigl(g(t) \bigr)^{r} v(t)\,dt \biggr)^{1/r} $$
(2.1)
holds for all non-negative and non-increasing g on \((0,\infty)\) if and only if
$$ A_{1} \equiv\sup_{t>0} W^{1/s}(t) V^{-1/r}(t)< \infty, $$
and the best constant C in (2.1) equals \(A_{1}\).

Lemma 2.2

[11, 12]

Let \(r, s \in(0,\infty)\) and let v, w be weights.

(i) Let \(1 < r \le s < \infty\). Then the inequality
$$ \biggl(\int_{0}^{\infty} \biggl( \frac{1}{t}\int_{0}^{t} g(\tau )\,d\tau \biggr)^{s} w(t)\,dt \biggr)^{1/s} \le C \biggl(\int _{0}^{\infty} \bigl(g(t) \bigr)^{r} v(t)\,dt \biggr)^{1/r} $$
(2.2)
holds for all non-negative and non-increasing g on \((0,\infty)\) if and only if \(A_{1}<\infty\),
$$ A_{2} \equiv\sup_{t>0} \biggl(\int _{t}^{\infty} \frac{w(\tau )}{\tau^{s}}\,d\tau \biggr)^{1/s} \biggl(\int_{0}^{t} \frac{v(\tau) \tau^{r'}}{V^{r'}(\tau)}\,d\tau \biggr)^{1/r'} < \infty, $$
and the best constant C in (2.2) satisfies \(C \thickapprox A_{1} + A_{2}\).
(ii) Let \(0 < r \le1\), \(r \le s\). Then (2.2) holds if and only if \(A_{1}<\infty\),
$$ A_{3} \equiv\sup_{t>0} t \biggl(\int _{t}^{\infty} \frac{w(\tau )}{\tau^{s}}\,d\tau \biggr)^{1/s} V^{-1/r}(t) < \infty, $$
and the best constant C in (2.2) satisfies \(C \thickapprox A_{1} + A_{3}\).

Lemma 2.3

[13]

Let \(r, s \in(0,\infty)\) and let u, v, w be weight functions.

(i) Let \(1 < r \le s < \infty\). Then the inequality
$$ \biggl(\int_{0}^{\infty} \biggl(\int _{t}^{\infty} g(\tau) u(\tau)\,d\tau \biggr)^{s} w(t)\,dt \biggr)^{1/s} \le C \biggl(\int _{0}^{\infty} \bigl(g(t) \bigr)^{r} v(t)\,dt \biggr)^{1/r} $$
(2.3)
holds for all non-negative and non-increasing g on \((0,\infty)\) if and only if
$$ A_{4} \equiv\sup_{t>0} \biggl(\int _{0}^{t} U^{s}(t,\tau) w(\tau)\,d\tau \biggr)^{1/s} V^{-1/r}(t) < \infty, $$
also
$$ A_{5} \equiv\sup_{t>0} W^{1/s}(t) \biggl(\int_{t}^{\infty} U^{r'}(\tau,t) V^{-r'} (\tau) v(\tau)\,d\tau \biggr)^{1/r'} < \infty, $$
and the best constant C in (2.3) satisfies \(C \thickapprox A_{4} + A_{5}\).

(ii) Let \(0 < r \le1\), \(r \le s\). Then (2.3) holds if and only if \(A_{4}<\infty\) and the best constant C in (2.3) equals \(A_{4}\).

Lemma 2.4

[13]

Let \(r \in(0,\infty)\) and let u, v, w be weight functions.

(i) Let \(1 < r < \infty\). Then the inequality
$$ \sup_{t>0} \biggl(\int_{t}^{\infty} g(\tau) u(\tau)\,d\tau \biggr) w(t) \le C \biggl(\int_{0}^{\infty} \bigl(g(t) \bigr)^{r} v(t)\,dt \biggr)^{1/r} $$
(2.4)
holds for all non-negative and non-increasing g on \((0,\infty)\) if and only if
$$ A_{6} \equiv\sup_{t>0} w(t) \biggl( \int_{t}^{\infty} U^{r'}(\tau,t) V^{-r'}(\tau) v(\tau)\,d\tau \biggr)^{1/r'} < \infty, $$
and the best constant C in (2.4) equals \(A_{6}\).
(ii) Let \(0 < r \le1\) and \(r \le s\). Then (2.4) holds if and only if
$$A_{7} \equiv\sup_{t>0} \sup _{0< \tau<t} U(\tau,t) w(\tau) V^{-1/r}(t) <\infty, $$
and the best constant C in (2.4) equals \(A_{7}\).

Lemma 2.5

[13]

Let \(r \in(0,\infty)\) and let u, v, w be weight functions.

(i) Let \(1 < r < \infty\). Then the inequality
$$ \sup_{t>0} \biggl(\int_{0}^{t} k(t,\tau) g(\tau) u(\tau)\,d\tau \biggr) w(t) \le C \biggl(\int _{0}^{\infty} \bigl(g(t) \bigr)^{r} v(t)\,dt \biggr)^{1/r} $$
(2.5)
holds for all non-negative and non-increasing g on \((0,\infty)\) if and only if
$$ A_{8} \equiv\sup_{t>0} w(t) \biggl( \int_{0}^{t} \biggl(\int_{s}^{t} k(t,\tau) V^{-1} (\tau)\,d\tau \biggr)^{r'} v(s)\,ds \biggr)^{1/r'} < \infty, $$
and the best constant C in (2.5) equals \(A_{8}\).
(ii) Let \(0 < r \le1\), \(r \le s\). Then (2.5) holds if and only if
$$ A_{9} \equiv\sup_{t>0} \sup _{\tau>0} K \bigl(t, \min(\tau ,t) \bigr) w(\tau) V^{-1/r}(t) < \infty, $$
and the best constant C in (2.5) equals \(A_{9}\).
Let g be a measurable function on \(\mathbb {R}^{n}\). The distribution function of g is defined by the equality
$$\lambda_{g}(t)= \bigl\vert \bigl\{ x\in \mathbb {R}^{n}: \bigl|g(x) \bigr|>t \bigr\} \bigr\vert ,\quad t\ge0. $$
We shall denote by \(L_{0}(\mathbb {R}^{n})\) the class of all measurable functions g on \(\mathbb {R}^{n}\), which are finite almost everywhere and such that \(\lambda _{g}(t)<\infty\) for all \(t>0\) (see [14]). If a function g belongs to \(L_{0}(\mathbb {R}^{n})\), then its non-increasing rearrangement is defined to be the function \(g^{\ast}\) which is non-increasing on \((0, \infty)\) equi-measurable with \(|g(x)|\):
$$\bigl\vert \bigl\{ t>0 : g^{\ast}(t)>\tau \bigr\} \bigr\vert = \lambda _{g}(\tau) $$
for all \(\tau\ge0\). Moreover, by the Hardy-Littlewood theorem (see [15], p.44) and for every \(f_{1}, f_{2} \in L_{0}(\mathbb {R}^{n})\),
$$ \int_{\mathbb {R}^{n}} \bigl|f_{1}(x)f_{2}(x) \bigr|\,dx \le\int_{0}^{\infty}f_{1}^{\ast }(t)f_{2}^{\ast}(t) \,dt. $$
Equi-measurable rearrangements of functions play an important role in various fields of mathematics. We give some of the main important properties (see, for example, [15]):
  1. (1)
    if \(0 < t < t+\tau\), then
    $$ ( g+h )^{\ast}(t+\tau)\leq g^{\ast}(t)+h^{\ast}(\tau), $$
     
  2. (2)
    if \(0< p<\infty\), then
    $$ \int_{\mathbb {R}^{n}}\bigl|g(x)\bigr|^{p}\,dx = \int _{0}^{\infty} \bigl( g^{\ast }(t) \bigr)^{p}\,dt, $$
     
  3. (3)
    for any \(t>0\) and for any set E,
    $$ \sup_{|E|=t}\int_{E}\bigl|g(x)\bigr|\,dx = \int _{0}^{t} g^{\ast}(\tau)\,d\tau. $$
     
We denote by \(WL_{p}(\mathbb {R}^{n})\) the weak \(L_{p}\) space of all measurable functions g with finite norm
$$\|f\|_{WL_{p}}=\sup_{t> 0} t^{1/p} f^{*}(t)< \infty,\quad 1\le p<\infty. $$
The function \(g^{**}: (0,\infty)\rightarrow[0,\infty]\) is defined as \(g^{\ast\ast}(t) = \frac{1}{t} \int_{0}^{t} f^{\ast}(s)\,ds\).

Definition 2.6

If \(0 < p,q < \infty\), then the Lorentz space \(L_{pq}(\mathbb {R}^{n})\) is the set of all classes of measurable functions f with the finite quasi-norm
$$\Vert f\Vert _{pq} \equiv \Vert f\Vert _{L_{pq}} = \biggl( \int_{0}^{\infty} \bigl( t^{1/p} f^{\ast}(t) \bigr)^{q} \frac{dt}{t} \biggr)^{1/q}. $$
If \(0< p\leq\infty\), \(q=\infty\), then \(L_{p\infty}(\mathbb {R}^{n})=WL_{p}(\mathbb {R}^{n})\).

If \(1 \le q \le p \) or \(p=q=\infty\), then the functional \(\Vert f\Vert _{pq}\) is a norm (see [16]). If \(p=q=\infty\), then the space \(L_{\infty\infty}(\mathbb {R}^{n})\) is denoted by \(L_{\infty}(\mathbb {R}^{n})\).

In the case \(1 < p, q < \infty\) we define
$$\Vert f\Vert _{(pq)} = \biggl( \int_{0}^{\infty} \bigl( t^{1/p} f^{\ast\ast}(t) \bigr)^{q} \frac{dt}{t} \biggr)^{1/q} $$
(with the usual modification if \(0< p\leq\infty\), \(q = \infty\)) which is a norm on \(L_{pq}(\mathbb {R}^{n})\) for \(1< p<\infty\), \(1\leq q\leq\infty\) or \(p=q=\infty\). If \(1< p\leq\infty\) and \(1\leq q\leq\infty\), then
$$\Vert f\Vert _{pq} \le \Vert f\Vert _{(pq)} \le p' \Vert f\Vert _{pq} $$
that is, the quasi-norms \(\Vert f\Vert _{pq}\) and \(\Vert f\Vert _{(pq)}\) are equivalent.

Lemma 2.7

[7]

Let \(f_{1}, f_{2}, \ldots, f_{k} \in L_{0}(\mathbb {R}^{n})\), \(k\geq2\). Then, for all \(x \in \mathbb {R}^{n}\) and nonzero real numbers \(\theta_{1}, \ldots, \theta_{k}\),
$$ \int_{\mathbb {R}^{n}}\bigl|f_{1}(x- \theta_{1} y)f_{2}(x-\theta_{2} y)\cdots f_{k}(x-\theta_{k} y)\bigr|\,dy \le C_{\theta} \int _{0}^{\infty}f_{1}^{\ast}(t)f_{2}^{\ast}(t) \cdots f_{k}^{\ast}(t)\,dt, $$
(2.6)
where \(C_{\theta} = \vert \theta_{1} \ldots\theta_{k} \vert ^{-n}\).
Let \(\mathbf{f}= ( f_{1},f_{2},\ldots,f_{k} ) \) and define
$$\mathbf{f}^{\ast}(t) = f_{1}^{\ast}(t) \cdots f_{k}^{\ast}(t),\qquad \mathbf{f}^{\ast\ast}(t) = \frac{1}{t} \int_{0}^{t} f_{1}^{\ast }(\tau) \cdots f_{k}^{\ast}( \tau)\,d\tau,\quad t>0. $$

In the following, we give the O’Neil inequality for rearrangements of the multilinear convolution operator \(\mathbf{f} \otimes g\) proved in [7].

Lemma 2.8

[7]

Let \(f_{1}, f_{2}, \ldots, f_{k}, g \in L_{0}(\mathbb{R}^{n})\). Then, for all \(0< t<\infty\), the following inequality holds:
$$ ( \mathbf{f} \otimes g )^{\ast\ast}(t)\leq C_{\theta} \biggl( t \mathbf{f}^{\ast\ast}(t) g^{\ast\ast}(t) + \int _{t}^{\infty} \mathbf{f}^{\ast}(s) g^{\ast}(s)\,ds \biggr). $$
(2.7)

Corollary 2.9

[7]

Let \(f_{1}, f_{2}, \ldots, f_{k} \in L_{0}(\mathbb {R}^{n})\) and \(g\in WL_{m}(\mathbb {R}^{n})\), \(1< m < \infty\). Then
$$\begin{aligned} (\mathbf{f} \otimes g )^{*}(t) &\le (\mathbf{f} \otimes g )^{**}(t) \\ &\le C_{\theta} \|g \|_{WL_{m}} \biggl( m' t^{-{1}/{m}} \int_{0}^{t} \mathbf{f}^{\ast}(\tau)\,d\tau+\int_{t}^{\infty} \tau^{-{1}/{m}} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr). \end{aligned}$$
(2.8)

Lemma 2.10

[7]

Let \(f_{1}, f_{2}, \ldots, f_{k}, g \in L_{0}(\mathbb {R}^{n})\). Then for any \(t>0\)
$$ ( \mathbf{f} \otimes g )^{\ast\ast}(t)\le C_{\theta} \int_{t}^{\infty} \mathbf{f}^{\ast\ast}(t) g^{\ast\ast}(t)\,dt. $$
(2.9)

Corollary 2.11

Let \(f_{1}, f_{2}, \ldots, f_{k} \in L_{0}(\mathbb {R}^{n})\) and \(g\in WL_{m}(\mathbb {R}^{n})\), \(1 < m < \infty\). Then
$$ (\mathbf{f} \otimes g )^{*}(t) \le (\mathbf{f} \otimes g )^{**}(t) \le m' C_{\theta} \|g \|_{WL_{m}} \int_{t}^{\infty} \tau^{-{1}/{m}} \mathbf{f}^{**}(\tau)\,d\tau. $$
(2.10)

3 O’Neil inequality for the multilinear convolutions in the Lorentz spaces

In this section, we prove the O’Neil inequality for the multilinear convolutions in the Lorentz spaces. It is said that p is the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\) if \({1}/{p}={1}/{p_{1}}+{1}/{p_{2}}+\cdots+{1}/{p_{k}}\). If \(f_{j}\in L_{p_{j}r_{j}}(\mathbb {R}^{n})\), \(j=1,2,\ldots,k\), then we say that \(\mathbf{f}\in L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots \times L_{p_{k}r_{k}}(\mathbb {R}^{n})\).

Theorem 3.1

(O’Neil inequality for k-linear convolution in the Lorentz spaces)

Suppose that \(1< m<\infty\), \(g\in WL_{m}(\mathbb {R}^{n})\), p and r are the harmonic means of \(p_{1},p_{2},\ldots,p_{k}>1\) and \(r_{1},r_{2},\ldots,r_{k}>0\), respectively. If \(1 < p < m'\), \(1< r \le s < \infty\) or \(m'/(1+m') \le p \le1\), \(0< r \le1\), \(r \le s < \infty\) or \(p = m'\), \(1< r < \infty\), \(s = \infty\) or \(p = m'\), \(0< r \le1\), \(s = \infty\) \(\mathbf{f}\in L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) and \(1/p-1/q=1/m'\), then \(\mathbf{f} \otimes g \in L_{qs}(\mathbb {R}^{n})\) and
$$\begin{aligned} \|\mathbf{f} \otimes g\|_{qs} \lesssim C_{\theta} K(p,q,r,s,m) \prod_{j=1}^{k} \Vert f_{j} \Vert _{p_{j}r_{j}} \|g\|_{WL_{m}}, \end{aligned}$$
where \(K(p,q,r,s,m)=\kappa\) and
$$\kappa\thickapprox \left \{ \begin{array}{@{}l@{\quad}l} m'\mathcal{A}_{1}+m'\mathcal{A}_{2}+\mathcal{A}_{4}+\mathcal{A}_{5}, &\textit{if } 1 < p < m', 1< r \le s < \infty,\\ m'\mathcal{A}_{1} + m'\mathcal{A}_{3}+\mathcal{A}_{4}, &\textit{if } \frac{m'}{1+m'} \le p \le1, 0<r\le1, r\le s<\infty\\ m'\mathcal{A}_{6} + m'\mathcal{A}_{8}, &\textit{if } p=m', 1< r < \infty, s=\infty,\\ m'\mathcal{A}_{7} + m'\mathcal{A}_{9}, &\textit{if } p=m', 0< r \le1, s=\infty \end{array} \right . $$
and
$$\begin{aligned} &\mathcal{A}_{1} = \biggl(\frac{m'q}{s(m'+q)} \biggr)^{1/s} \biggl(\frac {r}{p} \biggr)^{1/r}, \qquad \mathcal{A}_{2} = \frac{r}{p} \biggl(\frac{mq}{s(q-m)} \biggr)^{1/s} \biggl( \frac{p'}{r'} \biggr)^{1/r'}, \\ &\mathcal{A}_{3} = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac {mq}{s(q-m)} \biggr)^{1/s},\qquad \mathcal{A}_{4} = \bigl(m' \bigr)^{1+1/s} \biggl( \frac{r}{p} \biggr)^{1/r} \bigl(B \bigl(s+1,{sm'}/{q} \bigr) \bigr)^{1/s}, \\ &\mathcal{A}_{5} = \bigl(m' \bigr)^{1+1/r'} \frac{r}{p} \biggl(\frac{q}{s} \biggr)^{1/s} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'},\qquad \mathcal{A}_{6} = m' \biggl(\frac{r}{p} \biggr)^{1/r}, \\ &\mathcal{A}_{7} = \bigl(m' \bigr)^{1+1/r'} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'}, \\ &\mathcal{A}_{8} = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac {p}{p-r} \biggr)^{1+1/r'} \biggl(B \biggl(r'+1, \frac{r}{p-r} \biggr) \biggr)^{1/r'},\qquad \mathcal{A}_{9} = \biggl(\frac{r}{p} \biggr)^{1/r}. \end{aligned}$$
Here \(B(s,r)={ \int_{0}^{1} (1-\tau)^{s-1} \tau^{r-1}\,d\tau}\) is the beta function.

Proof

Let \(1< m<\infty\), \(m'/(1+m')\le p< m'\), \({1}/{p}-{1}/{q}={1}/{m'}\), p be the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\), r be the harmonic mean of \(r_{1},r_{2},\ldots,r_{k}>0\), \(0< r \le s \le\infty\) and \(\mathbf{f} \in L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots \times L_{p_{k}r_{k}}(\mathbb {R}^{n})\). By using inequality (2.8), we have
$$\begin{aligned} \Vert \mathbf{f} \otimes g\Vert _{qs}={}& \bigl\Vert ( \mathbf{f} \otimes g )^{\ast}(t) t^{{1}/{q}-{1}/{s}} \bigr\Vert _{L_{s}(0,\infty)} \\ \leq{}& C_{\theta} \biggl( \int_{0}^{\infty} \biggl(m' t^{-{1}/{m}} \int_{0}^{t} \mathbf{f}^{\ast}(\tau)\,d\tau+\int_{t}^{\infty} \tau^{-{1}/{m}} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr)^{s} t^{{s}/{q}-1}\,dt \biggr)^{1/s} \\ \leq{}& C_{\theta} m' \biggl( \int_{0}^{\infty} \biggl( \int_{0}^{t} \mathbf{f}^{\ast}( \tau)\,d\tau \biggr)^{s} t^{-{s}/{m}+ {s}/{q}-1}\,dt \biggr)^{1/s} \\ &{} +C_{\theta} \biggl( \int_{0}^{\infty} \biggl( \int_{t}^{\infty} \tau^{-{1}/{m}} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr)^{s} t^{{s}/{q}-1}\,dt \biggr)^{1/s}. \end{aligned}$$
Case I. Suppose that \(1< p<m'\) (equivalently \(m < q < \infty\)), \(1< r \le s < \infty\). From Lemma 2.2, for the validity of the inequality for \(1< r \le s < \infty\)
$$ \biggl( \int_{0}^{\infty} \biggl( \frac{1}{t}\int_{0}^{t} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr)^{s} t^{s-{s}/{m}+ {s}/{q}-1}\,dt \biggr)^{1/s}\leq C_{1} \biggl( \int_{0}^{\infty} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr)^{1/r} , $$
(3.1)
the necessary and sufficient condition is
$$\begin{aligned} \mathcal{A}_{1}&=\sup_{t>0} W^{1/s}(t) V^{-1/r}(t) = \biggl(\frac{m'q}{s(m'+q)} \biggr)^{1/s} \biggl( \frac{r}{p} \biggr)^{1/r} \sup_{t>0} t^{{1}/{m'}+{1}/{q}-{1}/{p}}< \infty \\ &\Leftrightarrow\quad{1}/{p}-{1}/{q}={1}/{m'} \mbox{ and } \mathcal{A}_{1} = \biggl(\frac{m'q}{s(m'+q)} \biggr)^{1/s} \biggl(\frac{r}{p} \biggr)^{1/r} \end{aligned}$$
and
$$\begin{aligned} \mathcal{A}_{2} & = \sup_{t>0} \biggl(\int _{t}^{\infty} \frac{w(\tau )}{\tau^{s}}\,d\tau \biggr)^{1/s} \biggl(\int_{0}^{t} \frac{v(\tau) \tau^{r'}}{V^{p'}(\tau)}\,d\tau \biggr)^{1/r'} \\ & = \frac{r}{p} \sup_{t>0} \biggl( \int _{t}^{\infty}\tau ^{-{s}/{m}+{s}/{q}-1}\,d\tau \biggr)^{1/s} \biggl( \int_{0}^{t} \tau^{r/p-1+r'-rr'/p}\,d\tau \biggr)^{1/r'} \\ & = \frac{r}{p} \biggl(\frac{mq}{s(q-m)} \biggr)^{1/s} \biggl( \frac {p'}{r'} \biggr)^{1/r'} \sup_{t>0} t^{-1/m+1/q-1/p'} < \infty \\ &\Leftrightarrow\quad{1}/{p}-{1}/{q}={1}/{m'} \mbox{ and } \mathcal{A}_{2} = \frac{r}{p} \biggl(\frac{mq}{s(q-m)} \biggr)^{1/s} \biggl(\frac {p'}{r'} \biggr)^{1/r'}. \end{aligned}$$
Note that the best constant \(C_{1}\) in (3.1) satisfies \(C_{1} \thickapprox\mathcal{A}_{1} + \mathcal{A}_{2}\). Furthermore, from Lemma 2.3 for the validity of the inequality for \(1< r \le s < \infty\)
$$ \biggl( \int_{0}^{\infty} \biggl( \int_{t}^{\infty }\tau^{{-{1}/{m}}} \mathbf{f}^{\ast} (\tau)\,d\tau \biggr)^{s} t^{{s}/{q}-1} \,dt \biggr) ^{1/s} \leq C_{2} \biggl( \int _{0}^{\infty } \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac {dt}{t} \biggr) ^{1/r} , $$
(3.2)
the necessary and sufficient condition is
$$\begin{aligned} \mathcal{A}_{4} & = m' \sup_{t>0} \biggl( \int_{0}^{t} \bigl(t^{1/m'}- \tau^{1/m'} \bigr)^{s} \tau^{s/q-1}\,d\tau \biggr)^{1/s} \biggl( \int_{0}^{t} \tau^{r/p-1}\,d\tau \biggr) ^{-1/r} \\ & = m' \biggl(\frac{r}{p} \biggr)^{1/r} \sup _{t>0} \biggl( \int_{0}^{t} \bigl(t^{1/m'}- \tau^{1/m'} \bigr)^{s} \tau^{s/q-1}\,d\tau \biggr)^{1/s} t^{-1/p} \\ & = \bigl(m' \bigr)^{1+1/s} \biggl(\frac{r}{p} \biggr)^{1/r} \bigl(B \bigl(s+1,{sm'}/{q} \bigr) \bigr)^{1/s} \sup_{t>0} t^{-1/m'+1/q-1/p}< \infty \\ &\Leftrightarrow\quad{1}/{p}-{1}/{q}={1}/{m'} \mbox{ and } \mathcal{A}_{4} = \bigl(m' \bigr)^{1+1/s} \biggl( \frac{r}{p} \biggr)^{1/r} \bigl(B \bigl(s+1,{sm'}/{q} \bigr) \bigr)^{1/s} \end{aligned}$$
and
$$\begin{aligned} \mathcal{A}_{5} & = \sup_{t>0} W^{1/s}(t) \biggl(\int_{t}^{\infty} U^{r'}(\tau,t) V^{-r'} (\tau) v(\tau)\,d\tau \biggr)^{1/r'} \\ & = \frac{m'r}{p} \biggl(\frac{q}{s} \biggr)^{1/s} \sup _{t>0} t^{1/q} \biggl( \int_{t}^{\infty} \bigl(\tau^{1/m'}-t^{1/m'} \bigr)^{r'} \tau^{-rr'/p+r/p-1}\,d\tau \biggr)^{1/r'} \\ & = \frac{m'r}{p} \biggl(\frac{q}{s} \biggr)^{1/s} \biggl( \int_{1}^{\infty} \bigl(\lambda^{1/m'}-1 \bigr)^{r'} \lambda^{-r'/p-1}\,d\lambda \biggr)^{1/r'} \sup _{t>0} t^{1/q+1/m'-1/p} \\ & = \bigl(m' \bigr)^{1+1/r'} \frac{r}{p} \biggl( \frac{q}{s} \biggr)^{1/s} \biggl( \int_{0}^{1} \bigl(1-\lambda^{1/m'} \bigr)^{r'} \lambda^{-r'/m'+r'/p-1}\,d \lambda \biggr)^{1/r'} \sup_{t>0} t^{1/q+1/m'-1/p} \\ & = \bigl(m' \bigr)^{1+1/r'} \frac{r}{p} \biggl( \frac{q}{s} \biggr)^{1/s} \biggl( \int_{0}^{1} (1-\tau )^{r'} \tau^{-r'+r'm'/p-1}\,d\tau \biggr)^{1/r'} \sup _{t>0} t^{1/q+1/m'-1/p} \\ & = \bigl(m' \bigr)^{1+1/r'} \frac{r}{p} \biggl( \frac{q}{s} \biggr)^{1/s} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'} \sup_{t>0} t^{1/q+1/m'-1/p}< \infty \\ &\Leftrightarrow\quad{1}/{p}-{1}/{q}={1}/{m'} \mbox{ and } \mathcal{A}_{5} = \bigl(m' \bigr)^{1+1/r'} \frac{r}{p} \biggl(\frac{q}{s} \biggr)^{1/s} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'}. \end{aligned}$$
Note that the best constant \(C_{2}\) in (3.2) satisfies \(C_{2} \thickapprox\mathcal{A}_{4} + \mathcal{A}_{5}\).
Case II. Let \(m'/(1+m')\le p \le1\), \(0 < r \le1\) and \(r \le s < \infty\). From Lemma 2.3, for the validity of inequality (3.1), the necessary and sufficient condition is \(\mathcal{A}_{1}<\infty\) and
$$\begin{aligned} \mathcal{A}_{3} & = \sup_{t>0} t \biggl(\int _{t}^{\infty} \frac{w(\tau )}{\tau^{s}}\,d\tau \biggr)^{1/s} V^{-1/r}(t) \\ & = \biggl(\frac{r}{p} \biggr)^{1/r} \sup_{t>0} t \biggl( \int_{t}^{\infty} \tau^{-{s}/{m}+{s}/{q}-1}\,d\tau \biggr)^{1/s} t^{-1/p} \\ & = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac{mq}{s(q-m)} \biggr)^{1/s} \sup_{t>0} t^{1-1/m+1/q-1/p} \\ &\Leftrightarrow\quad{1}/{p}-{1}/{q}={1}/{m'} \mbox{ and } \mathcal{A}_{3} = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac{mq}{s(q-m)} \biggr)^{1/s}. \end{aligned}$$
Note that the best constant \(C_{1}\) in (3.1) satisfies \(C_{1} \thickapprox\mathcal{A}_{1} + \mathcal{A}_{3}\). From Lemma 2.3, for the validity of inequality (3.2), the necessary and sufficient condition is \(\mathcal {A}_{4}<\infty\). Consequently, using inequalities (3.1), (3.2) and applying the Hölder inequality, we obtain
$$\begin{aligned} \Vert \mathbf{f} \otimes g \Vert _{qs} & \leq C_{\theta} \bigl(m'C_{1}+C_{2} \bigr) \biggl( { \int _{0}^{\infty}} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r}\frac{dt}{t} \biggr)^{1/r} \|g\|_{WL_{m}} \\ & = C_{\theta} K(p,q,r,s,m) \Biggl( { \int_{0}^{\infty}} \prod_{j=1}^{k} \bigl(f_{j}^{\ast}(t) t^{1/p_{j}} \bigr)^{r}\frac{dt}{t} \Biggr)^{1/r}\|g \|_{WL_{m}} \\ & \le C_{\theta} K(p,q,r,s,m) \prod_{j=1}^{k} \biggl( { \int_{0}^{\infty}} \bigl(f_{j}^{\ast}(t) t^{1/p_{j}} \bigr)^{r_{j}}\frac{dt}{t} \biggr)^{1/r_{j}}\|g \|_{WL_{m}} \\ & = C_{\theta} K(p,q,r,s,m) \prod_{j=1}^{k} \Vert f_{j}\Vert _{p_{j}r_{j}}\|g\|_{WL_{m}}. \end{aligned}$$
Case III. Let \(p=m'\), \(q=s=\infty\), \(1< r<\infty\) or \(p=m'\), \(q=s=\infty\), \(0< r\le1\) and \(\mathbf{f}\in L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\). By using inequality (2.8), we have
$$\begin{aligned} \|\mathbf{f} \otimes g \|_{\infty} &=\sup_{t>0} ( \mathbf{f} \otimes g )^{*}(t) \\ & \le C_{\theta} \sup_{t>0} \biggl( m' t^{-{1}/{m}} { \int_{0}^{t} \mathbf{f}^{*}(\tau)\,d\tau+\int_{t}^{\infty}\tau^{-{1}/{m}}} \mathbf{f}^{*}(\tau)\,d\tau \biggr) \|g \|_{WL_{m}} \\ & \le C_{\theta} m' \sup_{t>0} \biggl( t^{-{1}/{m}} { \int_{0}^{t}} \mathbf{f}^{*}(\tau)\,d\tau \biggr) + \sup_{t>0} \biggl({ \int_{t}^{\infty}} \tau ^{-{1}/{m}} \mathbf{f}^{*}(\tau)\,d\tau \biggr) \|g \|_{WL_{m}} \\ & \le C_{\theta} m' \biggl( { \int_{0}^{\infty }} \bigl(t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr) \|g \|_{WL_{m}}. \end{aligned}$$
From Lemma 2.5, for the validity of the inequality for \(1< r<\infty\)
$$\sup_{t>0} \biggl( t^{-{1}/{m}} { \int _{0}^{t}}\mathbf{f}^{*}(\tau)\,d\tau \biggr) \leq C_{3} \biggl( { \int_{0}^{\infty}} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr)^{1/r} , $$
(3.3)
the necessary and sufficient condition is
$$\begin{aligned} \mathcal{A}_{8} & = \biggl(\frac{r}{p} \biggr)^{1/r} \sup_{t>0} t^{-1/m} \biggl(\int_{0}^{t} \biggl(\int_{s}^{t} \tau^{-r/p}\,d\tau \biggr)^{r'} s^{r/p-1}\,ds \biggr)^{1/r'} \\ & = \biggl(\frac{r}{p} \biggr)^{1/r} \frac{r}{p-r}\sup _{t>0} t^{-1/m} \biggl( \int_{0}^{t} \bigl(t^{1-r/p}- \tau^{1-r/p} \bigr)^{r'} \tau^{r/p-1}\,d\tau \biggr)^{1/r'} \\ & = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac{p}{p-r} \biggr)^{1+1/r'} \biggl( \int_{0}^{1} \bigl(1-\tau^{1-r/p} \bigr)^{s} \tau^{r/p-1}\,d\tau \biggr)^{1/r'} \sup_{t>0} t^{-1/m-1/p+1} \\ & = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac{p}{p-r} \biggr)^{1+1/r'} \biggl(B \biggl(r'+1,\frac{r}{p-r} \biggr) \biggr)^{1/r'} \sup_{t>0} t^{-1/m-1/p+1}< \infty \\ &\Leftrightarrow\quad p={m'} \mbox{ and } \mathcal{A}_{8} = \biggl(\frac{r}{p} \biggr)^{1/r} \biggl(\frac{p}{p-r} \biggr)^{1+1/r'} \biggl(B \biggl(r'+1,\frac {r}{p-r} \biggr) \biggr)^{1/r'}. \end{aligned}$$
From Lemma 2.5, for the validity of the inequality for \(0< r\le1\)
$$ \sup_{t>0} \biggl( t^{-{1}/{m}} \int _{0}^{t} \mathbf{f}^{*}(\tau )\,d\tau \biggr) \leq C_{3} \biggl( \int_{0}^{\infty} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr)^{1/r} , $$
(3.4)
the necessary and sufficient condition is
$$\begin{aligned} \mathcal{A}_{9} & = \sup_{t>0} \sup _{\tau>0} K \bigl(t, \min(\tau,t) \bigr) w(\tau) V^{-1/r}(t) = \biggl(\frac{r}{p} \biggr)^{1/r} \sup _{t>0} t^{1/m'-1/p} < \infty \\ &\Leftrightarrow\quad p={m'} \mbox{ and } \mathcal{A}_{9} = \biggl(\frac{r}{p} \biggr)^{1/r}. \end{aligned}$$
From Lemma 2.4, for the validity of the inequality for \(1< r<\infty\)
$$\sup_{t>0} \biggl( \int_{t}^{\infty} \tau^{-1/m} \mathbf {f}^{*}(\tau)\,d\tau \biggr) \leq C_{3} \biggl( \int_{0}^{\infty} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr)^{1/r} , $$
(3.5)
the necessary and sufficient condition is \(\mathcal{A}_{6}\)
$$\begin{aligned} \mathcal{A}_{6} & = \sup_{t>0} \biggl(\int _{t}^{\infty} U^{r'}(\tau,t) V^{-r'}( \tau) v(\tau)\,d\tau \biggr)^{1/r'} \\ & = \biggl( \int_{t}^{\infty} \bigl( \tau^{1/m'}-t^{1/m'} \bigr)^{r'} \tau^{-rr'/p+r/p-1}\,d \tau \biggr)^{1/r'} \\ & = \biggl( \int_{1}^{\infty} \bigl( \lambda^{1/m'}-1 \bigr)^{r'} \lambda ^{-r'/p-1}\,d\lambda \biggr)^{1/r'} \sup_{t>0} t^{1/m'-1/p} \\ & = \frac{m'r}{p} \biggl( \int_{0}^{1} \bigl(1-\lambda^{1/m'} \bigr)^{r'} \lambda^{-r'/m'+r'/p-1}\,d \lambda \biggr)^{1/r'} \sup_{t>0} t^{1/m'-1/p} \\ & = \frac{m'r}{p} \biggl( \int_{0}^{1} (1- \tau )^{r'} \tau^{-r'+r'm'/p-1}\,d\tau \biggr)^{1/r'} \sup _{t>0} t^{1/m'-1/p} \\ & = \frac{m'r}{p} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'} \sup_{t>0} t^{1/m'-1/p}< \infty \\ &\Leftrightarrow\quad p=m' \mbox{ and } \mathcal{A}_{6} = \frac {m'r}{p} \bigl(B \bigl(r'+1,{r'm'}/{p}-r' \bigr) \bigr)^{1/r'}. \end{aligned}$$
Furthermore, from Lemma 2.4, for the validity of the inequality for \(0< r\le1\)
$$ \sup_{t>0} \biggl( \int_{t}^{\infty} \tau^{-1/m} \mathbf {f}^{*}(\tau)\,d\tau \biggr) \leq C_{3} \biggl( \int_{0}^{\infty} \bigl( t^{1/p} \mathbf{f}^{\ast}(t) \bigr)^{r} \frac{dt}{t} \biggr)^{1/r} , $$
(3.6)
the necessary and sufficient condition is
$$\begin{aligned} \mathcal{A}_{7} & = \sup_{t>0} \sup _{0< \tau<t} U(\tau ,t) w(\tau) V^{-1/r}(t) \\ & = m' \sup_{t>0} \sup_{0<\tau<t} \bigl(t^{1/m'}- \tau^{1/m'} \bigr) \biggl( \int _{0}^{t} \tau^{r/p-1}\,d\tau \biggr) ^{-1/r} \\ & = m' \biggl(\frac{r}{p} \biggr)^{1/r} \sup _{t>0} \sup_{0<\tau<t} \bigl(t^{1/m'}- \tau^{1/m'} \bigr) t^{-1/p} \\ & = m' \biggl(\frac{r}{p} \biggr)^{1/r} \sup _{t>0} t^{1/m'-1/p} < \infty \\ &\Leftrightarrow\quad p={m'} \mbox{ and } \mathcal{A}_{7} = m' \biggl(\frac{r}{p} \biggr)^{1/r}. \end{aligned}$$
Thus the proof of Theorem 3.1 is completed. □

Corollary 3.2

[8]

Suppose that \(1< m<\infty\), \(g\in WL_{m}(\mathbb {R}^{n})\) and p is the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\). If \(m'/(1+m') \le p < m'\), \(\mathbf{f}\in L_{p_{1}}\times L_{p_{2}}\times\cdots\times L_{p_{k}}(\mathbb {R}^{n})\) and q satisfy \(1/p-1/q =1/m'\), then \(\mathbf{f} \otimes g \in L_{q}(\mathbb {R}^{n})\) and
$$\begin{aligned} \|\mathbf{f} \otimes g\|_{q}\le C_{\theta} K(p,q,m) \prod_{j=1}^{k} \Vert f_{j}\Vert _{p_{j}} \|g\|_{WL_{m}}, \end{aligned}$$
where in the case \(1 < p=r < m'\), \(q=s\)
$$\begin{aligned} K(p,q,m) ={}& m' \biggl(\frac{m'}{m'+q} \biggr)^{1/q} + m' \biggl(\frac {m}{q-m} \biggr)^{1/q}\\ &{} + \bigl(m' \bigr)^{1+1/q} \bigl(B \bigl(q+1,m' \bigr) \bigr)^{1/q} + \bigl(m' \bigr)^{1+1/p'} \bigl(B \bigl(p'+1,{p'm'}/{p}-p' \bigr) \bigr)^{1/p'}, \end{aligned}$$
and in the case \(m'/(1+m') \le p=r \le1\), \(m< q=s\)
$$K(p,q,m)=m' \biggl(\frac{m'}{m'+q} \biggr)^{1/q} + \bigl(m'+1 \bigr) \biggl(\frac{m}{q-m} \biggr)^{1/q} + \bigl(m' \bigr)^{1+1/q} \bigl(B \bigl(q+1,m' \bigr) \bigr)^{1/q}. $$

4 The \(L_{p_{1} r_{1}}\times L_{p_{2} r_{2}}\times\cdots\times L_{p_{k} r_{k}}\) boundedness of rough multilinear fractional integral operators

In this section, we prove the Sobolev type theorem for the rough multilinear fractional integral \(I_{\Omega,\alpha}\mathbf{f}\).

Lemma 4.1

Let \(0<\alpha<n\), Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\) and
$$g(x)=\frac{\Omega(x)}{|x|^{n-\alpha}}. $$
Then \(g\in WL_{n/(n-\alpha)}(\mathbb {R}^{n})\) and
$$ \|g\|_{WL_{n/(n-\alpha)}}=n^{\alpha/n-1} \| \Omega \|_{L_{n/(n-\alpha)}}, $$
(4.1)
where
$$\| \Omega\|_{L_{n/(n-\alpha)}}= \biggl(\int_{{S}^{n-1}} \bigl|\Omega \bigl(x' \bigr)\bigr|^{n/(n-\alpha)}\,d\sigma \bigl(x' \bigr) \biggr)^{(n-\alpha)/n}. $$

Proof

Note that
$$g^{*}(t)=(nt)^{\alpha/n-1} \| \Omega\|_{L_{n/(n-\alpha)}},\qquad g^{**}(t)=\frac{n}{\alpha} g^{*}(t), $$
therefore \(g\in WL_{n/(n-\alpha)}(\mathbb {R}^{n})\) and equality (4.1) is valid. □

Lemma 4.2

Suppose that \(0<\alpha<n\), \(\Omega\in L_{s}({S}^{n-1})\) and \(s\ge1\). Then
$$ M_{\Omega, \alpha} \mathbf{f}(x) \le I_{|\Omega|, \alpha} \bigl(| \mathbf{f}|\bigr) (x), $$
(4.2)
where \(|\mathbf{f}|=(|f_{1}|, \ldots, |f_{k}|)\).

Proof

Indeed, for all \(r>0\), we have
$$\begin{aligned} I_{|\Omega|, \alpha} \bigl(|f|\bigr) (x) &\ge\int_{E(0,r)} \frac{|\Omega(y)|}{|y|^{n-\alpha}} \bigl|f_{1} ( x-\theta _{1}y ) \ldots f_{k} ( x-\theta_{k}y )\bigr|\,dy \\ & \ge\frac{1}{r^{n-\alpha}} \int_{E(0,r)} \bigl|\Omega(y)\bigr| \bigl|f_{1} ( x-\theta_{1}y ) \ldots f_{k} ( x- \theta_{k}y )\bigr|\,dy, \end{aligned}$$
where \(E(0,r)\) is the open ball centered at the origin of radius r. Taking supremum over all \(r>0\), we get (4.2). □

By Lemmas 2.8 and 4.2, we obtain a pointwise rearrangement estimate of the rough k-sublinear fractional maximal integral \(M_{\Omega, \alpha} \mathbf{f}\) and k-linear fractional integral \(I_{\Omega, \alpha} \mathbf{f}\).

Lemma 4.3

[7]

Suppose that Ω is homogeneous of degree zero on \(\mathbb {R}^{n}\) and \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), \(0<\alpha<n\). Then the following inequalities hold:
$$\begin{aligned}& \begin{aligned}[b] (I_{\Omega, \alpha} \mathbf{f} )^{\ast}(t)&\le (I_{\Omega, \alpha} \mathbf{f} )^{\ast\ast}(t) \\ & \le C_{\theta} n^{{\alpha}/{n}-1} \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \biggl(\frac{n}{\alpha} t^{{\alpha}/{n}-1} \int_{0}^{t} \mathbf{f}^{\ast}(\tau)\,d\tau+\int_{t}^{\infty} \tau^{{\alpha}/{n}-1} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr), \end{aligned}\\& \begin{aligned}[b] (M_{\Omega, \alpha} \mathbf{f} )^{\ast}(t) &\le (M_{\Omega, \alpha} \mathbf{f} )^{\ast\ast}(t) \\ & \le C_{\theta} n^{{\alpha}/{n}-1} \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \biggl(\frac{n}{\alpha} t^{{\alpha}/{n}-1} \int_{0}^{t} \mathbf{f}^{\ast}(\tau)\,d\tau+\int_{t}^{\infty} \tau^{{\alpha}/{n}-1} \mathbf{f}^{\ast}(\tau)\,d\tau \biggr). \end{aligned} \end{aligned}$$

From Theorem 3.1 and Lemma 4.3, we get the following.

Theorem 4.4

Let Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), \(0<\alpha<n\), p and r be the harmonic means of \(p_{1},p_{2},\ldots,p_{k}>1\) and \(r_{1},r_{2},\ldots,r_{k}>0\), respectively, and \(0< r \le s \le\infty\), q satisfy \({1}/{q}={1}/{p}-{\alpha}/{n}\). If \(1 < p < n/\alpha\), \(1< r \le s < \infty\) or \(n/(n+\alpha) \le p \le1\), \(0< r \le s < \infty\) or \(p=n/\alpha\), \(r=1\), then \(I_{\Omega, \alpha}\) is a bounded operator from \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\) and
$$\begin{aligned} \|I_{\Omega, \alpha} \mathbf{f}\|_{qs}\le C_{\theta} n^{{\alpha}/{n}-1} K\bigl(p,q,r,s,n/(n-\alpha)\bigr) \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \prod_{j=1}^{k}\|f_{j}\|_{p_{j}r_{j}}. \end{aligned}$$

Corollary 4.5

[8]

Let Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), \(0<\alpha<n\), p be the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\), and q satisfy \({1}/{q}={1}/{p}-{\alpha}/{n}\). Then \(I_{\Omega, \alpha}\) is a bounded operator from \(L_{p_{1}}\times L_{p_{2}}\times\cdots\times L_{p_{k}}(\mathbb {R}^{n})\) to \(L_{q}(\mathbb {R}^{n})\) for \(n/(n+\alpha)\le p < n/\alpha\) (equivalently \(1\le q < \infty\)) and
$$\begin{aligned} \|I_{\Omega, \alpha} \mathbf{f}\|_{q}\le C_{\theta} n^{{\alpha}/{n}-1} K\bigl(p,q,n/(n-\alpha)\bigr) \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \prod_{j=1}^{k}\|f_{j}\|_{p_{j}}. \end{aligned}$$

Corollary 4.6

[8]

Let Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), \(0<\alpha<n\), p be the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\), and q satisfy \({1}/{q}={1}/{p}-{\alpha}/{n}\). Then \(M_{\Omega, \alpha}\) is a bounded operator from \(L_{p_{1}}\times L_{p_{2}}\times\cdots\times L_{p_{k}}(\mathbb {R}^{n})\) to \(L_{q}(\mathbb {R}^{n})\) for \(n/(n+\alpha)\le p \le n/\alpha \) (equivalently \(1 \le q \le\infty\)) and
$$\begin{aligned} \|M_{\Omega, \alpha} \mathbf{f}\|_{q}\le C_{\theta} n^{{\alpha}/{n}-1} K\bigl(p,q,n/(n-\alpha)\bigr) \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \prod_{j=1}^{k}\|f_{j}\|_{p_{j}}, \end{aligned}$$
when \(n/(n+\alpha)\le p < n/\alpha\), and
$$\begin{aligned} \|M_{\Omega, \alpha} \mathbf{f}\|_{\infty}\le C_{\theta} \Vert \Omega \Vert _{L_{n/(n-\alpha)} } \prod_{j=1}^{k}\|f_{j}\|_{p_{j}}, \quad p=n/\alpha. \end{aligned}$$

Finally, in the following theorem we obtain the necessary and sufficient conditions for the rough k-linear fractional integral operator \(I_{\Omega,\alpha} \) to be bounded from the Lorentz spaces \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\), \(n/(n+\alpha) \le p< q<\infty\), \(0 < r \le s < \infty\).

Theorem 4.7

Let \(0<\alpha<n\), Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), p and r be the harmonic means of \(p_{1},p_{2},\ldots,p_{k}>1\) and \(r_{1},r_{2},\ldots,r_{k}>0\), respectively. If \(1 < p < n/\alpha\), \(1< r \le s < \infty\) or \(n/(n+\alpha) \le p \le1\), \(0< r \le s < \infty\), then the condition \(1/p-1/q=\alpha/n\) is necessary and sufficient for the boundedness of \(I_{\Omega,\alpha} \) from \(L_{p_{1}r_{1}} \times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\).

Proof

Sufficiency of the theorem follows from Theorem 4.4.

Necessity. Suppose that the operator \(I_{\Omega,\alpha} \) is bounded from \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots \times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\), and \(n/(n+\alpha)\leq p< n/\alpha \) (equivalently \(1\leq q<\infty\)). Define \(\mathbf{f}_{t} (x)=:\mathbf{f}(tx)\) for \(t>0\) and \(\|\mathbf {f}\|_{pr} =\prod_{j=1}^{k}\|f_{j}\|_{p_{j}r_{j}}\). Then it can be easily shown that
$$\begin{aligned} \|\mathbf{f}_{t}\|_{pr}= \prod_{j=1}^{k}\bigl\| (f_{j})_{t}\bigr\| _{p_{j}r_{j}} = \prod_{j=1}^{k} t^{-{n}/{p_{j}}} \|f_{j}\|_{p_{j}r_{j}} =t^{-n/p} \| \mathbf{f}\|_{pr} \end{aligned}$$
and
$$I_{\Omega,\alpha} \mathbf{f}_{t}(x)=t^{-\alpha}I_{\Omega,\alpha} \mathbf{f}(tx),\qquad \Vert I_{\Omega,\alpha} \mathbf{f}_{t}\Vert _{qs} =t^{-\alpha-n/q} \Vert I_{\Omega,\alpha } \mathbf{f} \Vert _{qs} . $$
Since the operator \(I_{\Omega,\alpha} \) is bounded from \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\), we have
$$\Vert I_{\Omega,\alpha} \mathbf{f} \Vert _{qs}\leq C \|\mathbf {f}\|_{pr}, $$
where C is independent of f. Then we get
$$\begin{aligned}& \Vert I_{\Omega,\alpha} \mathbf{f} \Vert _{qs}= t^{\alpha +n/q} \Vert I_{\Omega,\alpha} \mathbf{f}_{t} \Vert _{qs} \le C t^{\alpha+{n}/{q}}\|\mathbf{f}_{t}\|_{pr} = C t^{\alpha +{n}/{q}-{n}/{p}} \|f\|_{pr}. \end{aligned}$$
If \(1/p <1/q+ {\alpha}/n\), then for all \(\mathbf{f} \in L_{p_{1}r_{1}} \times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) we have \(\Vert I_{\Omega,\alpha} \mathbf{f} \Vert _{L_{q,s} }=0\) as \(t\rightarrow0\). If \(1/p>1/q+{\alpha}/n\), then for all \(\mathbf{f} \in L_{p_{1}r_{1}} \times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) we have \(\Vert I_{\Omega,\alpha} \mathbf{f} \Vert _{qs}=0\) as \(t\rightarrow\infty\). Therefore we get \(1/p=1/q+{\alpha}/n\). □

Corollary 4.8

[8]

Let \(0<\alpha<n\), p be the harmonic mean of \(p_{1},p_{2},\ldots,p_{k}>1\), Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\) and \(\Omega\in L_{{n/(n-\alpha)}}(S^{n-1})\). If \(n/(n+\alpha) \le p < n/\alpha\), then the condition \(1/p-1/q=\alpha/n\) is necessary and sufficient for the boundedness of \(I_{\Omega,\alpha} \) from \(L_{p_{1}} \times L_{p_{2}}\times\cdots\times L_{p_{k}}(\mathbb {R}^{n})\) to \(L_{q}(\mathbb {R}^{n})\).

Remark 4.9

Note that the sufficiency part of Corollary 4.8 was proved in [7] and in the case \(\Omega\equiv1\) in [2], and in the case \(\Omega\in L_{s}({S}^{n-1})\), \(s>n/(n-\alpha)\) in [6].

Theorem 4.10

Let \(0<\alpha<n\), Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\), \(\Omega\in L_{n/(n-\alpha)}({S}^{n-1})\), p and r be the harmonic means of \(p_{1},p_{2},\ldots,p_{k}>1\) and \(r_{1},r_{2},\ldots,r_{k}>0\), respectively. If \(1 < p < n/\alpha\), \(1< r \le s < \infty\) or \(n/(n+\alpha) \le p \le1\), \(0< r \le s < \infty\), then the condition \(1/p-1/q= \alpha/n\) is necessary and sufficient for the boundedness of \(M_{\Omega,\alpha} \) from \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times \cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\).

Proof

Sufficiency part of the theorem follows from Theorem 4.7 and Lemma 4.2.

Necessity. Suppose that the operator \(M_{\Omega,\alpha} \) is bounded from \(L_{p_{1}r_{1}}\times L_{p_{2}r_{2}}\times\cdots\times L_{p_{k}r_{k}}(\mathbb {R}^{n})\) to \(L_{qs}(\mathbb {R}^{n})\), and \(n/(n+\alpha)\leq p< n/\alpha\), \(0< r \le s < \infty\). Then we have
$$M_{\Omega,\alpha} f_{t}(x)=t^{-\alpha} M_{\Omega,\alpha} f(tx) $$
and
$$\begin{aligned}& \Vert M_{\Omega,\alpha} f_{t}\Vert _{qs} = t^{-\alpha-\frac{n}{q}} \Vert M_{\Omega,\alpha} f\Vert _{qs}. \end{aligned}$$
By the same argument in Theorem 4.7, we obtain \(\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}\). □

Corollary 4.11

[8]

Let \(0<\alpha<n\), p be the harmonic mean of \(p_{1},p_{2},\ldots ,p_{k}>1\), Ω be homogeneous of degree zero on \(\mathbb {R}^{n}\) and \(\Omega\in L_{{n/(n-\alpha)}}(S^{n-1})\). If \(n/(n+\alpha) \le p \le n/\alpha\), then the condition \(1/p-1/q= \alpha/n\) is necessary and sufficient for the boundedness of \(M_{\Omega,\alpha} \) from \(L_{p_{1}}\times L_{p_{2}}\times\cdots \times L_{p_{k}}(\mathbb {R}^{n})\) to \(L_{q}(\mathbb {R}^{n})\).

Declarations

Acknowledgements

The research of V Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2014-9(15)-46/10/1 and by the grant of Presidium of Azerbaijan National Academy of Science 2015.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Institute of Mathematics and Mechanics
(2)
Department of Mathematics, Ahi Evran University
(3)
Department of Mathematics, Dumlupınar University
(4)
Khazar University

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