Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint cases

Abstract

In this paper, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that $p_{i}\ne 1$ for all $i=1,\ldots ,m$ and extend some known results.

1 Introduction

Let ${\mathbb {R}}^n$ be the n-dimensional Euclidean space and $({\mathbb {R}}^n)^m={\mathbb {R}}^n\times \ldots {\mathbb {R}}^n.$ The multilinear fractional operator is defined by

$$\begin{aligned} I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)=\int _{({\mathbb {R}}^n)^{m}} \frac{f_{1}(y_{1})\ldots f_{m}(y_{m})}{|(x-y_{1}), \ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}, \end{aligned}$$

(1.1)

where $0<\alpha <mn,m\in {\mathbb {N}}.$ 

The multilinear fractional integral $I_{\alpha ,m}$ is a natural generalization of the classical fractional integral. Kenig and Stein [9] as well as Grafakos and Kalton [4] considered the boundedness of a family of related fractional integrals. In 2012, Iida, Sato, Sawano and Tanaka in [6] obtained the boundedness property of the Adams type for multilinear fractional integral operators. Let $0<\alpha <mn$, $1<p_{1},p_{2},\ldots ,p_{m}<\infty , 0<p\le p_{0}<\frac{n}{\alpha }, 0<q\le q_{0}<\infty .$ Suppose that

$$\begin{aligned} \frac{1}{p}=\frac{1}{p_{1}}+\ldots +\frac{1}{p_{m}}, \frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha }{n} ~\text {and}~\frac{q}{q_{0}}=\frac{p}{p_{0}}. \end{aligned}$$

Then there exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} \Vert I_{\alpha ,m}(\vec {f})\Vert _{{\mathcal {M}}^{q_{0}}_{q}}&\le C\sup _{Q\in {\mathcal {Q}}}|Q|^{1/p_{0}}\prod _{i=1}^{m} \bigg (\frac{1}{|Q|}\int _{Q}|f_{i}(y_{i})|^{p_{i}}dy_{i} \bigg )^{1/p_{i}}\\&=:\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}_{\vec {P}}^{p_{0}}}, \end{aligned} \end{aligned}$$

(1.2)

where ${\mathcal {M}}^{q_{0}}_{q}$ stands for the Morrey spaces. The right-hand side of (1.2) is named by “the multi-Morrey norm”, which is strictly smaller than m-fold product of the Morrey norms.

An interesting question arises. Can we obtain the similar results for the endpoint cases, that is $p_{0}=\frac{n}{\alpha }$, $p_{0}>\frac{n}{\alpha }$ and $p_{i} = 1$ for some $i\in \{1,\ldots , m\}$. In this paper, we prove that the questions above have a affirmative solution.

To state the main results of this paper, we need first to recall some necessary notations and notion.

Definition 1

For $0< q\le p <\infty$, we say that a function f belongs to Morrey space ${\mathcal {M}}^{p}_{q}$ if

$$\begin{aligned} \Vert f\Vert _{{\mathcal {M}}^{p}_{q}}:=\sup _{x\in {\mathbb {R}}^{n},r>0}\frac{1}{|B(x,r)|^{1/q-1/p}} \bigg (\int _{B(x,r)}|f(y)|^{q}dy\bigg )^{1/q}<\infty ; \end{aligned}$$

a function f belongs to weak Morrey space $WM_{q}^{p}$ if

$$\begin{aligned} \Vert f\Vert _{W{\mathcal {M}}_{q}^{p}}:=\sup _{x\in {\mathbb {R}}^{n},r>0}\frac{1}{|B(x,r)|^{1/q-1/p}} \sup _{\lambda>0}\bigg (\lambda ^{q}|\{y\in B(x,r):|f(y)|>\lambda \}|\bigg )^{1/q}<\infty . \end{aligned}$$

Morrey spaces seem to describe precisely the boundedness property of fractional integral operators. Morrey spaces describe local regularity more precisely than $L^{p}$ spaces and can be seen as a complement of $L^{p}$. In fact, $L^p=M^{p}_{p}\subset M^{p}_{q}$ and $WM^{p}_{p}=L^{p,\infty }$ for $0<q \le p<\infty$.

Recall that the fractional integral operator (or the Riesz potential) $I_{\alpha }$, $0<\alpha <n$, is given by

$$\begin{aligned} I_{\alpha }(f)(x)=\int _{{\mathbb {R}}^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy. \end{aligned}$$

The inequality

$$\begin{aligned} \Vert I_{\alpha }(f)\Vert _{{\mathcal {M}}^{q_{0}}_{q}}\le C\Vert f\Vert _{{\mathcal {M}}^{p_{0}}_{p}} \end{aligned}$$

(1.3)

with $1/q_{0}=1/p_{0}-\alpha /n$ and $1/q=1/p-\alpha /n$ was obtained by Spanne. The result was improved by Adams [1](see also [2]), the inequality (1.3) holds if $1/q_{0}=1/p_{0}-\alpha /n$ and $q/q_{0}=p/p_{0}$. And, Olsen [10] showed by an example that the result of Adams is optimal. Tang [13] extended the result to the multilinear fractional integral operator ${\mathcal {I}}_\alpha$. They also established some endpoint estimates for the multilinear fractional integral.

In the endpoint case $p_{0} = n/\alpha$, the exponential integrability of $I_{\alpha ,m}$ was proved by Strichartz [11] for $m = 1$, by Tang [13] for $m\ge 2$.

Theorem

(cf. [13]) Let $m\in {\mathbb {N}}$, $0<\alpha <mn, 1/p=1/p_{1}+\ldots +1/p_{m}=\alpha /n$ with $1<p_{i}<\infty$ for $i=1,\ldots , m$. Let B be a ball of radius R in ${\mathbb {R}}^n$ and let $f_{j}\in L^{p_{j}}(B)$ be supported in B. Then there exist constants $k_{1},k_{2}$ depending only on $n,m,\alpha ,p$ and the $p_{j}$ such that

$$\begin{aligned} \int _{B}\exp \bigg (k_{1}\Big (\frac{|I_{\alpha ,m}(f_{1}, \ldots ,f_{m})(x)|}{\prod _{j=1}^{k}\Vert f_{j}\Vert _{L^{p_{j}}(B)}} \Big )^{n/(mn-\alpha )}\bigg )dx\le k_{2}R^{n}. \end{aligned}$$

(1.4)

For the case $p_{0}\ge n/\alpha$, we also study the boundedness for multilinear fractional integrals on spaces as BMO space and Lipschitz spaces. Campanato spaces are a useful tool in the regularity theory of PDEs due to their better structures, which allows us to give an integral characterization of the spaces of Hölder continuous functions.

Definition 2

Let $0<p<\infty$ and $-n/p<\beta <n$. A locally integrable function f is said to belong to Campanato space ${\mathcal {C}}_{\beta ,q}$ if there exists a constant $C > 0$ such that for any ball $B\subset {\mathbb {R}}^n$,

$$\begin{aligned} \frac{1}{|B|^{\beta /n}}\bigg (\frac{1}{|B|}\int _{B}|f(x) -f_{B}|^{p}dx\bigg )^{1/p}\le C, \end{aligned}$$

where $f_{B}=\frac{1}{|B|}\int _{B}f(x)dx$ and the minimal constant C is defined by $\Vert f\Vert _{{\mathcal {C}}_{\beta ,p}}$.

The Lipschitz (Hölder) and Campanato spaces are related by the following equivalences:

$$\begin{aligned} \Vert f\Vert _{Lip_{\beta }}:=\sup _{x,h\in {\mathbb {R}}^n,h\ne 0}\frac{|f(x+h)-f(x)|}{|h|^{\beta }}\approx \Vert f\Vert _{{\mathcal {C}}_{\alpha ,q}},\quad 0<\alpha <1. \end{aligned}$$

The equivalence can be found in [3] for $q=1$, [7] for $1<q<\infty$ and [15] for $0<q<1$. Specially, ${\mathcal {C}}_{0,q}={{\textrm{BMO}}}$, the spaces of bounded mean oscillation. The crucial property of ${{\textrm{BMO}}}$ functions is the John-Nirenberg inequality [8],

$$\begin{aligned} |\{x\in Q: |f(x)-f_{Q}|>\lambda \}|\le c_{1}|Q|e^{-\frac{c_{2}\lambda }{\Vert f\Vert _{{\textrm{BMO}}}}}, \end{aligned}$$

where $c_{1}$ and $c_{2}$ depend only on the dimension. A well-known immediate corollary of the John-Nirenberg inequality as follows:

$$\begin{aligned} \Vert f\Vert _{{\textrm{BMO}}}\approx \sup _{Q}\frac{1}{|Q|}\Big (\int _{Q}|f(x)-f_{Q}|^{p}dx\Big )^{1/q}, \end{aligned}$$

for all $1<q<\infty$. In fact, the equivalence also holds for $0<q<1$. See, for example, the work of Strömberg [12](or [5] and [16] for the general case). In addition, we also proved in [14] that $f\in {{\textrm{BMO}}}$ if and only if for $0<q<\infty$,

$$\begin{aligned} \sup _{Q}\frac{\big \Vert (f-f_{Q})\chi _{Q} \big \Vert _{L^{q,\infty }}}{|Q|^{1/q}}<\infty . \end{aligned}$$

(1.5)

The main result of this paper are stated as follows.

Theorem 1.1

Let $m\in {\mathbb {N}}$, $0<\alpha <mn$ and $I_{\alpha ,m}$ be as in (1.1). Let $\vec {P}=(p_{1},\ldots ,p_{m}), 1\le p_{1},\ldots ,p_{m}<\infty$ with $1/p=1/p_{1}+\ldots +1/p_{m}$ and $0<p\le p_{0}<\infty$. If $p_{0}=n/\alpha$ and $f_{i}\in L^{p_{i}}$ with compact support, then for any ball $B=B(x_{0},R)$, there exist constants ${\tilde{C}}, k_{1},k_{2}$ such that $f_{1},\ldots , f_{m}$ is supported in $B(x_{0},{\tilde{C}}R)$ and

$$\begin{aligned} \frac{1}{|B|}\int _{B}\exp \Big (\frac{k_{1}|I_{\alpha ,m}(f_{1}, \ldots ,f_{m})(x)|}{\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}} _{\vec {P}}}}\Big )dx\le k_{2}({\tilde{C}}+1)^{\alpha p}. \end{aligned}$$

(1.6)

Theorem 1.2

Let $m\in {\mathbb {N}}$, $(m-1)n<\alpha <mn$ and $I_{\alpha ,m}$ be as in (1.1). Let $\vec {P}=(p_{1},\ldots ,p_{m}), 1\le p_{1},\ldots ,p_{m}<\infty , 1/p=1/p_{1}+\ldots +1/p_{m}$ and $0<p\le p_{0}<\infty$. If $p_{0}=n/\alpha ,$ then

$$\begin{aligned} \Vert I_{\alpha ,m}(f_{1},\ldots ,f_{m})\Vert _{{\textrm{BMO}}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

For the case $n/\alpha<p_{0}<\infty$, we obtain the result as follows. We remark that when $p_{0}=\infty$ in Theorem 1.3, the conclusion also holds. Indeed, the proof is similar to the case $p_{0} < \infty$, moreover the proof is simpler.

Theorem 1.3

Let $m\in {\mathbb {N}}$, $0<\alpha <mn$ and $I_{\alpha ,m}$ be as in (1.1). Let $\vec {P}=(p_{1},\ldots ,p_{m}), 1\le p_{1},\ldots ,p_{m}<\infty$ and $0<p\le p_{0}<\infty$. If $p_{0}>n/\alpha$ and $0<\alpha -n/p_{0}<1,$ then

$$\begin{aligned} \Vert I_{\alpha ,m}(f_{1},\ldots ,f_{m})\Vert _{Lip_{\alpha -n/p_{0}}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {p}}}. \end{aligned}$$

If there is at least $p_{i}$ is equiv to 1, one has the following weak type estimate.

Theorem 1.4

Let $m\in {\mathbb {N}}$, $0<\alpha <mn$ and $I_{\alpha ,m}$ be as in (1.1). Let $\vec {P}=(p_{1},\ldots ,p_{m})$, $1\le p_{j}<\infty$ for $j=1,\ldots , m$, $1/p=1/p_{1}+\ldots +1/p_{m}$, $1/q=1/q_{1}+\ldots +1/q_{m}$ with $1/q=1/p-\alpha /n$ and $1/q_{0}=1/p_{0}-\alpha /n$. If $p_{0}<n/\alpha$ and there is at least one $q_{i}$ which is equal to 1, we have

$$\begin{aligned} \Vert I_{\alpha ,m}(f_{1},\ldots ,f_{m})\Vert _{W{\mathcal {M}}^{q_{0}}_{q}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

Let |E| denote the Lebesgue measure of a measurable set $E\subset {\mathbb {R}}^n$. Throughout this paper, the letter C denotes constants which are independent of main variables and may change from one occurrence to another. B(x, r) denotes a ball centered at x, with side length r.

2 Proofs of Theorems 1.1–1.2

Proof of Theorem 1.1

Let us first assume $\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}=1.$ For some $\delta >0$, we have

$$\begin{aligned}&|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\\&\quad \le \int _{|(x-y_{1},\ldots , x-y_{m})|\le \delta } \frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(x-y_{1}, \ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}\\&\qquad +\int _{|(x-y_{1},\ldots , x-y_{m})|> \delta } \frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(x-y_{1}, \ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}\\&\quad =:F_{1}+F_{2}. \end{aligned}$$

For $F_{1}$, by a direct argument, we see that

$$\begin{aligned} F_{1}{} & {} =\sum _{j=1}^{m}\int _{2^{-j}\delta <|(x-y_{1}, \ldots , x-y_{m})|\le 2^{-j+1}\delta } \frac{|f(y_{1})|\ldots |f(y_{m})|}{|(x-y_{1}, \ldots , x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}\nonumber \\{} & {} \le \sum _{j=1}^{m}(2^{-j}\delta )^{\alpha -mn} \int _{|(x-y_{1},\ldots , x-y_{m})|\le 2^{-j+1} \delta }|f(y_{1})|\ldots |f(y_{m})|dy_{1}\ldots dy_{m}\nonumber \\{} & {} \le \sum _{j=1}^{m}(2^{-j}\delta )^{\alpha -mn} (2^{-j+1}\delta )^{mn}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x)\nonumber \\{} & {} \le \frac{2^{mn}\delta ^{\alpha }}{2^{\alpha } -1}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x),\nonumber \\ \end{aligned}$$

(2.1)

where ${\mathcal {M}}$ denotes the multilinear Hardy-Littlewood maximal function

$$\begin{aligned} {\mathcal {M}}(f_{1},\ldots ,f_{m})(x)=\sup _{B\ni x}\prod _{i=1}^{m}\frac{1}{|B|}\int _{B}|f_{i}(y_{i})|dy_{i}. \end{aligned}$$

For any $x\in B=B(x_{0},R)$, there exist a ball ${\tilde{B}}=B(x_{0}, {\tilde{C}}R)$ with ${\tilde{C}}>0$ such that the functions $f_{1}, \ldots , f_{m}$ are supported in ${\tilde{B}}$. We conclude that for any $y_{i}\in {\tilde{B}}$, $|x-y_{i}|\le |x-x_{0}|+|x_{0}-y_{i}|\le ({\tilde{C}}+1)R$ and

$$\begin{aligned} |(x-y_{1},\ldots ,x-y_{m})|\le ({\tilde{C}}+1)\sqrt{m}R, \end{aligned}$$

this shows that

$$\begin{aligned} F_{2}{} & {} \le \int _{\delta<|(x-y_{1},\ldots ,x-y_{m})|\le ({\tilde{C}}+1)\sqrt{m}R}\frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\cdots dy_{m}\nonumber \\{} & {} \le \sum _{j=1}^{J}\int _{2^{j}\delta <|(x-y_{1},\ldots ,x-y_{m})|\le 2^{j+1}\delta }\frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}\nonumber \\{} & {} \le \sum _{j=1}^{J}(2^{j}\delta )^{\alpha -mn}\int _{|(x-y_{1},\ldots ,x-y_{m})|\le 2^{j+1}\delta }|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|dy_{1}\ldots dy_{m}\nonumber \\{} & {} \le \sum _{j=1}^{J}(2^{j}\delta )^{\alpha -mn}\prod _{i=1}^{m}\int _{|x-y_{i}|\le 2^{j+1}\delta }|f_{i}(y_{i})|dy_{i}\nonumber \\{} & {} \le \sum _{j=1}^{J}(2^{j}\delta )^{\alpha -mn}(2^{j+1} \delta )^{\frac{n}{p'_{i}}}\prod _{i=1}^{m}\Big (\int _{|x-y_{i}|\le 2^{j+1}\delta }|f_{i}(y_{i})|^{p_{i}}dy_{i}\Big )^{1/p_{i}}\nonumber \\{} & {} \le 2^{mn-\alpha }J, \nonumber \\ \end{aligned}$$

(2.2)

where $J=\log \frac{({\tilde{C}}+1)\sqrt{m}R}{\delta }.$

Combining (2.1) and (2.2), we obtain for any $x\in B$ and $0<\delta \le ({\tilde{C}}+1)\sqrt{m}R$,

$$\begin{aligned} |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\le \frac{2^{mn}}{2^{\alpha }-1}\delta ^{\alpha }{\mathcal {M}}(f_{1},\ldots ,f_{m})(x)+ 2^{mn-\alpha }\log \frac{({\tilde{C}}+1)\sqrt{m}R}{\delta }. \end{aligned}$$

(2.3)

In particular, the choice of $\delta =({\tilde{C}}+1)\sqrt{m}R$ yields for all $x\in B$,

$$\begin{aligned} |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\le \frac{2^{mn}}{2^{\alpha }-1}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x). \end{aligned}$$

Therefore, the election of

$$\begin{aligned} \delta =\delta (x)=\epsilon \Big (\frac{|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|}{ \frac{2^{mn}}{2^{\alpha }-1}{\mathcal {M}}(f_{1}, \ldots ,f_{m})(x)}\Big )^{\frac{1}{\alpha }} \end{aligned}$$

for all $\epsilon <1$. Now, (2.3) implies that

$$\begin{aligned}{} & {} |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\nonumber \\{} & {} \quad \le \epsilon ^{\alpha }|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\nonumber \\{} & {} \qquad + \frac{2^{mn-\alpha }}{\alpha p}\log \frac{\big (\frac{2^{mn} \sqrt{m}}{\epsilon (2^{\alpha }-1)}\big )^{p}({\tilde{C}}+1)^{\alpha p} R^{\alpha p}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x)^{p}}{|I_{\alpha ,m} (f_{1},\ldots ,f_{m})(x)|^{p}} \nonumber \\ \end{aligned}$$

(2.4)

If we use the notation $k_{1}=\frac{(1-\epsilon )^{\alpha }\alpha p}{2^{mn-\alpha }}$ and $C_{1}=\big (\frac{2^{mn}\sqrt{m}}{\epsilon (2^{\alpha }-1)}\big )^{p}$, (2.4) is equivalent to

$$\begin{aligned} k_{1}|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\le \log \frac{C_{1} ({\tilde{C}}+1)^{\alpha p}R^{\alpha p}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x)^{p}}{|I_{\alpha ,m} (f_{1},\ldots ,f_{m})(x)|^{p}}. \end{aligned}$$

(2.5)

By exponentiating (2.5), we get

$$\begin{aligned} \exp \{k_{1}|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\}\le \frac{C_{1}({\tilde{C}}+1)^{\alpha p}R^{\alpha p}{\mathcal {M}}(f_{1},\ldots ,f_{m})(x)^{p}}{|I_{\alpha ,m} (f_{1},\ldots ,f_{m})(x)|^{p}}. \end{aligned}$$

(2.6)

Let $B_{1}=\{x\in B: |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\ge 1\}$ and $B_{2}=B\backslash B_{1}$. By the inequality (2.6),

$$\begin{aligned} \int _{B_{1}}\exp \{k_{1}|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)|\}dx&\le C_{1}({\tilde{C}}+1)^{\alpha p}R^{\alpha p}\int _{B_{1}} {\mathcal {M}}(f_{1},\ldots ,f_{m})(x)^{p}dx\\&\le C_{1}({\tilde{C}}+1)^{\alpha p} R^{n}\Vert {\mathcal {M}}(f_{1}, \ldots ,f_{m})\Vert ^{p}_{{\mathcal {M}}^{p_{0}}_{p}}\\&\le C_{2}({\tilde{C}}+1)^{\alpha p}R^n, \end{aligned}$$

where $C_{2}=C_{1}\Vert {\mathcal {M}}\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}\rightarrow {\mathcal {M}}^{p_{0}}_{p}}$. On the other hand,

$$\begin{aligned} \int _{B_{2}}\exp \{k_{1}|I_{\alpha ,m}(f_{1},\ldots ,f_{m}) (x)|\}dx\le \exp \{k_{1}\}|B_{2}|. \end{aligned}$$

(2.7)

Thus, adding the integrals above over $B_1$ and $B_2$,

$$\begin{aligned} \frac{1}{|B|}\int _{B}\exp \{k_{1}|I_{\alpha ,m}(f_{1}, \ldots ,f_{m})(x)|\}dx\le k_{2}({\tilde{C}}+1)^{\alpha p}, \end{aligned}$$

where $k_{2}=(\exp \{k_{1}\}+C_{2})\omega _{n}$ and $\omega _{n}=|B(0,1)|$.

For the general $\vec {f}=(f_{1},\ldots ,f_{m})$, we can use the natation ${\tilde{f}}_{1}=f_{1}/\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}} _{\vec {P}}}$. Obviously,

$$\begin{aligned} I_{\alpha ,m}(\vec {f}_{1},f_{2},\ldots ,f_{m})=I_{\alpha ,m} (f_{1},f_{2},\ldots ,f_{m})/\Vert (f_{1},\ldots ,f_{m}) \Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

Therefore, we obtain the desired inequality (1.6). $\square$

Proof of Theorem 1.2

The inequality (1.5) implies that for $0<q<\infty ,$ $f\in {{\textrm{BMO}}}$ if and only if there exists a constant $c_{B}$ related to the ball B,  such that

$$\begin{aligned} \sup _{B}\frac{\big \Vert |f(x)-c_{B}|\chi _{B} \big \Vert _{L^{q,\infty }}}{|B|^{1/q}}<\infty . \end{aligned}$$

Then, we need only to prove that for any ball $B=B(x_{0},r)$,

$$\begin{aligned} \frac{\big \Vert |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)-c_{B}| \chi _{B}\big \Vert _{L^{q,\infty }}}{|B|^{1/q}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

(2.8)

Set $\Omega (x_{0},r):=\{(y_{1},\ldots ,y_{m}): |x_{0}-y_{1}|+\ldots +|x_{0}-y_{m}|\le r\},$ 

$$\begin{aligned} {{\textrm{I}}}_{1}^{B}(x)&:=\int _{\Omega (x_{0},2^mr)}\frac{f_{1}(y_{1}) \ldots f_{m}(y_{m})}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m},\\ {{\textrm{I}}}_{2}^{B}(x)&:=\int _{\Omega ^{c}(x_{0},2^mr)}\frac{f_{1}(y_{1}) \ldots f_{m}(y_{m})}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m} \end{aligned}$$

and

$$\begin{aligned} c_{B}:=\frac{1}{|B|}\int _{B}I_{2}^{B}(z)dz. \end{aligned}$$

Then $I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)=I_{1}^{B}(x)+I_{2}^{B}(x).$

Since $\Omega (x_{0},2r)\subset B(x_{0},2r)\times \ldots \times B(x_{0},2r)$ and the boundedness of $I_{\alpha ,m}$ from $L^{p_{1}}\times \ldots \times L^{p_{m}}$ to $L^{q,\infty }$ with $1/p_{1}+\ldots +1/p_{m}-1/q=\alpha /n$, we arrive at

$$\begin{aligned} \Vert {{\textrm{I}}}_{1}^{B}\Vert _{L^{q,\infty }}{} & {} \le \big \Vert |I_{\alpha ,m} (|f_{1}|\chi _{4B},\ldots ,|f_{m}|\chi _{4B})|\big \Vert _{L^{q,\infty }}\nonumber \\{} & {} \le C\prod _{i=1}^{m}\Vert f_{1}\chi _{4B}\Vert _{L^{p_{i}}}\nonumber \\{} & {} \le C|B|^{-1/q}\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}.\nonumber \\ \end{aligned}$$

(2.9)

On the other hand, for $x,z\in B, (y_{1},\ldots ,y_{m})\in \Omega ^{c}(x_{0},2^mr)$, by the direct calculation, we get

$$\begin{aligned} \begin{aligned}&\bigg |\frac{1}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }} -\frac{1}{|(z-y_{1},\ldots ,z-y_{m})|^{mn-\alpha }}\bigg |\\&\le \frac{C|x-z|}{|(z-y_{1},\ldots ,z-y_{m})|^{mn-\alpha +1}} \le \frac{Cr}{|(z-y_{1},\ldots ,z-y_{m})|^{mn-\alpha +1}}. \end{aligned} \end{aligned}$$

Using this observation, we see that

$$\begin{aligned} \begin{aligned}&|{{\textrm{I}}}_{2}^{B}(x)-c_{B}|\\&\quad \le \frac{Cr}{|B|}\int _{B}\int _{\Omega ^{c}(x_{0},2^mr)} \frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(z-y_{1}, \ldots ,z-y_{m})|^{mn-\alpha +1}}dy_{1}\ldots dy_{m}dz\\&\quad \le \frac{Cr}{|B|}\int _{B}\sum _{k=1}^{\infty }\int _{\Omega (x_{0}, 2^{(k+1)m}r)\backslash \Omega (x_{0},2^{km}r)} \frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(z-y_{1}, \ldots ,z-y_{m})|^{mn-\alpha +1}}dy_{1}\ldots dy_{m}dz\\&\quad \le C\sum _{k=1}^{\infty }\frac{r}{(2^{km}r)^{mn-\alpha +1}} \prod _{i=1}^{m}\int _{B(x_{0},2^{(k+1)m}r)}|f_{i}(y_{i})|dy_{i}\\&\quad \le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}\nonumber \\ \end{aligned}$$

(2.10)

The inequality (2.10) shows that

$$\begin{aligned} \big \Vert |{{\textrm{I}}}_{2}^{B}(x)-c_{B}|\chi _{B}\big \Vert _{L^{q,\infty }}\le C|B|^{1/q}\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

(2.11)

Combining with the estimates (2.9) and (2.11), we obtain the inequality  (2.8). $\square$

3 Proofs of Theorems 1.3–1.4

Proof of Theorem 1.3

For any $x_{1},x_{2}\in {\mathbb {R}}^n,$ and $r=|x_{1}-x_{2}|$, we write

$$\begin{aligned} \Omega (x_{1},r):=\{(y_{1},\ldots ,y_{m}): |x_{1}-y_{1}|+\ldots +|x_{1}-y_{m}|\le r\}, \end{aligned}$$

it follows that

$$\begin{aligned} \begin{aligned}&|I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x_{1})-I_{\alpha ,m}(f_{1}, \ldots ,f_{m})(x_{2})|\\&\quad \le \Big |\int _{\Omega (x_{1},2^{m}r)}\frac{f_{1}(y_{1}) \ldots f_{m}(y_{m})dy_{1}\ldots dy_{m}}{|(x_{1}-y_{1}, \ldots ,x_{1}-y_{m})|^{mn-\alpha }}\Big |\\&\qquad +\Big |\int _{\Omega (x_{1},2^{m}r)}\frac{f_{1}(y_{1}) \ldots f_{m}(y_{m})dy_{1}\ldots dy_{m}}{|(x_{2}-y_{1},\ldots ,x_{2}-y_{m})|^{mn-\alpha }}\Big |\\&\qquad +\Big |\int _{\Omega ^{c}(x_{1},2^{m}r)}\frac{f_{1}(y_{1}) \ldots f_{m}(y_{m})dy_{1}\ldots dy_{m}}{|(x_{1}-y_{1}, \ldots ,x_{1}-y_{m})|^{mn-\alpha }}\\&\qquad -\int _{\Omega ^{c}(x_{1},2^{m}r)} \frac{f_{1}(y_{1})\ldots f_{m}(y_{m})dy_{1}\ldots dy_{m}}{|(x_{2}-y_{1}, \ldots ,x_{2}-y_{m})|^{mn-\alpha }}\Big |\\&\quad =:{{\textrm{II}}}_{1}+{{\textrm{II}}}_{2}+{{\textrm{II}}}_{3}. \end{aligned} \end{aligned}$$

(3.1)

The same argument as (2.9), we have

$$\begin{aligned} {{\textrm{II}}}_{1}, {{\textrm{II}}}_{2}\le Cr^{\alpha -n/p_{0}}\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

(3.2)

For the last term ${{\textrm{II}}}_{3},$ we also obtain

$$\begin{aligned} \begin{aligned} {{\textrm{II}}}_{3}&\le Cr\int _{\Omega ^{c}(x_{1},2^mr)} \frac{|f_{1}(y_{1})|\ldots |f_{m}(y_{m})|}{|(x_{1}-y_{1}, \ldots ,x_{1}-y_{m})|^{mn-\alpha +1}}dy_{1}\ldots dy_{m}\\&\le C\sum _{k=1}^{\infty }\frac{r}{(2^{km}r)^{mn-\alpha +1}} \prod _{i=1}^{m}\int _{B(x_{0},2^{(k+1)m}r)}|f_{i}(y_{i})|dy_{i}\\&\le Cr^{\alpha -n/p_{0}}\Vert (f_{1},\ldots ,f_{m}) \Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned} \end{aligned}$$

(3.3)

Combining with the estimates (3.1),(3.2) and (3.3), we arrive at

$$\begin{aligned}{} & {} |I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x_{1})-I_{\alpha ,m} (f_{1},\ldots ,f_{m})(x_{2})|\\{} & {} \quad \le C|x_{1}-x_{2}|^{\alpha -n/p}\Vert (f_{1},\ldots ,f_{m}) \Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

Thus, Theorem 1.3 is proved. $\square$

Proof of Theorem 1.4

For any $x_{0}\in {\mathbb {R}}^n$ and $B=B(x_{0},r)$ centered at $x_{0}$ with radius r, set $\Omega (x_{0},r):=\{(y_{1},\ldots ,y_{m}): |x_{0}-y_{1}|+\cdots +|x_{0}-y_{m}|\le r\}.$ For any $x\in B$, we split $I_{\alpha ,m}(f_{1},\ldots ,f_{m})(x)$ with

$$\begin{aligned} {{\textrm{III}}}_{1}^{B}(x):=\int _{\Omega (x_{0},4r)}\frac{f_{1}(y_{1})\ldots f_{m}(y_{m})}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}, \end{aligned}$$

and

$$\begin{aligned} {{\textrm{III}}}_{2}^{B}(x):=\int _{\Omega ^{c}(x_{0},4r)} \frac{f_{1}(y_{1})\ldots f_{m}(y_{m})}{|(x-y_{1},\ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}. \end{aligned}$$

By the boundedness of $I_{\alpha ,m}$ from $L^{p_{1}}\times \cdots \times \cdots L^{p_{m}}$ to $L^{q,\infty }$ and the fact that

$$\begin{aligned} {{\textrm{III}}}_{1}^{B}(x)\le I_{\alpha }(|f^{0}_{1}|,\ldots ,|f^{0}_{m}|)(x) \end{aligned}$$

with $f^{0}_{i}=f_{i}\chi _{4B}$, we can easily get

$$\begin{aligned} \Vert {{\textrm{III}}}_{1}^{B}\Vert _{L^{q,\infty }(B(x_{0},r))}&\le \Vert I_{\alpha }(f^{0}_{1},\ldots ,f^{0}_{m})\Vert _{L^{q,\infty }(B(x_{0},r))}\\&\le \Vert I_{\alpha ,m}(|f^{0}_{1}|,\ldots ,|f^{0}_{m}|)\Vert _{L^{q,\infty }} \le C\prod _{i=1}^{m}\Vert f_{i}\Vert _{L^{p_{i}}(B(x_{0},4r))}. \end{aligned}$$

By $\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha }{n}$ and $\frac{1}{q}=\frac{1}{p_{1}}+\ldots +\frac{1}{p_{m}}-\frac{\alpha }{n}$, one has

$$\begin{aligned} |B|^{\frac{1}{q_{0}}-\frac{1}{q}}\Vert {{\textrm{III}}}_{1}^{B}\Vert _{L^{q,\infty }(B(x_{0},r))}\le C|4B|^{\frac{1}{p_{0}}-\frac{1}{p}}\prod _{i=1}^{m} \Vert f_{i}\Vert _{L^{p_{i}}(B(x_{0},4r))} \end{aligned}$$

and $\Vert {{\textrm{III}}}_{1}^{B}\Vert _{W{\mathcal {M}}^{q_{0}}_{q}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}$.

For the term ${{\textrm{III}}}^{B}_{2}$, Hölder inequality shows that

$$\begin{aligned} |{{\textrm{III}}}_{2}^{B}(x)|&\le \sum _{j=1}^{\infty } \int _{\Omega (x,2^{j+1}r)\backslash \Omega (x,2^{j}r)} \frac{|f_{1}(y_{1})\ldots f_{m}(y_{m})|}{|(x-y_{1}, \ldots ,x-y_{m})|^{mn-\alpha }}dy_{1}\ldots dy_{m}\\&\le \sum _{j=1}^{\infty }(2^{j}r)^{-mn+\alpha } \int _{\Omega (x,2^{j+1}r)\backslash \Omega (x,2^{j}r)}|f_{1}(y_{1})| \ldots |f_{m}(y_{m})|dy_{1}\ldots dy_{m}\\&\le C \sum _{j=1}^{\infty }(2^{j}r)^{-\frac{n}{p}+\alpha } \Vert f_{j}\Vert _{L^{p_{i}}(B(x,2^{j+1}r)}\\&\le C\sum _{j=1}^{\infty }(2^{j}r)^{\alpha -\frac{n}{p_{0}}} \Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}\\&\le C r^{\alpha -\frac{n}{p_{0}}}\Vert (f_{1},\ldots ,f_{m}) \Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}. \end{aligned}$$

Thus, $\Vert {{\textrm{III}}}_{2}^{B}\Vert _{W{\mathcal {M}}^{q_{0}}_{q}}\le C\Vert (f_{1},\ldots ,f_{m})\Vert _{{\mathcal {M}}^{p_{0}}_{\vec {P}}}$ and the theorem has been proved. $\square$