Let $(\mathbb{R}^{n})^{m}=\mathbb{R}^{n}\times\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}$ be the m-fold product space of $\mathbb{R}^{n}$, the multilinear fractional integrals on $\mathbb{R}^{n}$ are defined by

$ I_{\beta, m}(f_{1}, \cdots, f_{m})(x)=\int_{(\mathbb{R}^{n})^{m}}\frac{f_{1}(y_{1})\cdots f_{m}(y_{m})} {|(x-y_{1}, \cdots, x-y_{m})|^{mn-\beta}}dy_{1}\cdots dy_{m}, $

where $0 < \beta < mn, |(x-y_{1}, \cdots, x-y_{m})|=\sqrt{|x-y_{1}|^{2}+\cdots+|x-y_{m}|^{2}}.$

When $m=1$, $I_{\beta, m} = I_{\beta}$, where $I_{\beta}f(x)=\int_{\mathbb{R}^{n}}\frac{f(y)}{|x-y|^{n-\beta}}dy$. The famous Hardy-Littlewood-Sobolev theorem tells us that the fractional integral operator $I_{\beta}$ is a bounded operator from the usual Lebesgue spaces $L^{p_{1}}(\mathbb{R}^{n})$ to $L^{p_{2}}(\mathbb{R}^{n})$ when $1 < p_{1} < p_{2} < \infty$ and $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\beta}{n}$. Kenig and Stein [1] as well as Grafakos and Kalton [2] considered the boundedness of a family of related multilinear fractional integrals. Lan [3] presented the boundedness of multilinear fractional integral operators on weak type Hardy spaces. Recently, the paper [4] by Yasuo considered the boundedness of multilinear fractional integral operators on Herz spaces.

It is well known that function spaces with variable exponents were intensively studied during the past 20 years, due to their applications to PDE with non-standard growth conditions and so on, we mention e.g. ([5, 6]). A great deal of work has been done to extend the theory of fractional integral on the classical Lebesgue spaces to the variable exponent case, see ([7-9]). However, these articles do not consider the behavior of $I_{\beta}$ when $p^{+}>\frac{\beta}{n}$. Recently, Ramseyer, Salinas and Viviani [10] studied that Lipschitz type smoothness of fractional integral on variable exponent spaces, when $p^{+}>\frac{\beta}{n}$. Hence, when $p^{+}>\frac{\beta}{n}$, it will be an interesting problem whether we can establish the boundedness of multilinear fractional integral from Lebesgue spaces $L^{p(\cdot)}$ into Lipschitz-type spaces with variable exponents. The main purpose of this paper is to answer the above problem.

To meet the requirements in the next sections, here, the basic elements of the theory of the Lebsegue spaces with variable exponent are briefly presented.

Let $p(\cdot):\Omega\rightarrow[1, \infty)$ be a measurable function. The variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ is defined by

$ L^{p(\cdot)}(\Omega):=\left\{f \text{is measurable}: \int_{\Omega}\left|\frac{f(x)}{\lambda}\right|^{p(x)}dx < \infty \text{for some constant} \lambda>0\right\}. $

$L^{p(\cdot)}(\Omega)$ is a Banach space with the norm defined by

$ \|f\|_{L^{p(\cdot)}(\Omega)}:=\inf\{\lambda>0:\int_{\Omega}\left|\frac{f(x)}{\lambda}\right|^{p(x)}dx\leq1\}. $

We denote ${p^ - }: = {\text{ess}}\mathop {\inf }\limits_{x \in \Omega } p(x),{p^ + }: = {\text{ess}}\mathop {\sup }\limits_{x \in \Omega } p(x).$

Let $\mathcal{P}(\mathbb{R}^{n})$ be the set of measurable function $p(\cdot)$ on $\mathbb{R}^{n}$ with value in $[1, \infty)$ such that $1 < p_{-}(\mathbb{R}^{n})\leq p(\cdot)\leq p_{+}(\mathbb{R}^{n}) < \infty.$

We say a function $p(\cdot) : \mathbb{R}^{n} \longrightarrow \mathbb{R}$ is locally log-Hölder continuous, if there exists a constant $C$ such that

$ |p(x)-p(y)|\leq\frac{C}{\text{log}(e+1/|x-y|)} $

for all $x, y\in\mathbb{R}^{n}$. If, for some $p(\infty)\in\mathbb{R}$ and $C>0$, there holds $|p(x)-p(\infty)|\leq\frac{C}{\text{log}(e+|x|)}, $ for all $x\in\mathbb{R}^{n}$, then we say $p(\cdot)$ is log-Hölder continuous at infinity.

The notation $\mathcal{P}^{\log}(\mathbb{R}^{n})$ is used for all those exponents $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$ which are locally log-Hölder continuous and log-Hölder continuous at infinity with $p(\infty) :=\lim\limits_{|x|\rightarrow\infty}p(x)$. Moreover, we can easily show that $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$ implies $p^{\prime}(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n})$, where $\frac{1}{p}+\frac{1}{p'}=1$.

For brevity, $C$ always means a positive constant independent of the main parameters and may change from one occurrence to another.$B(x, r)=\{y\in\mathbb{R}^{n} : |x-y| < r\}$, $B_{k}=B(x_{0}, 2^{k}R)$, $A_{k}=B_{k}\setminus B_{k-1}$ and $\chi_{A_{k}}=\chi_{k}$ be the characteristic function of the set $A_{k}$ for $k\in \mathbb{Z}$.$|S|$ denotes the Lebesgue measure of $S$. $f\sim g$ means $C^{-1}g\leq f\leq Cg$.

Definition 1.1 [10] Given an exponent function $p(\cdot)$ we say that a measurable function $f$ belongs $L^{p(\cdot), \infty}$ if there exists a constant $C$ such that $\int_{\mathbb{R}^{n}}t^{p(x)}\chi_{\{|f|>t\}}(x)dx\leq C, $ for every $t>0$.

It is not difficult to see that

$ [f]_{p(\cdot), \infty}=\text{inf}\left\{\lambda>0 :\sup\limits_{t>0}\int_{\mathbb{R}^{n}}\left(\frac{t}{\lambda}\right)^{p(x)} \chi_{\{|f|>t\}}(x)dx\leq 1\right\} $

is a quasi-norm in $L^{p(\cdot), \infty}$.

Definition 1.2 [10] Given $0 < \beta < n$ and an exponent function $p(\cdot)$ with $1 < p^{-}\leq p^{+} < \infty$ we say that a locally integrable function $f$ belongs to $Lip_{\beta, p(\cdot)}$ if there exists a constant $C$ such that

$ \frac{1}{|B|^{\frac{\beta}{n}}\|\chi_{B}\|_{p^{\prime}(\cdot)}}\int_{B}|f-m_{B}f|dx\leq C $

(1.1)

for every ball $B\subset\mathbb{R}^{n}$, with $m_{B}f=\frac{1}{|B|}\int_{B}f$. The least constant $C$ in (1.1) will be denoted by $\|f\|_{Lip_{\beta, p(\cdot)}}$.

Remark It is easy to see that in definition the average can be replaced by a constant in the following sense

$ \frac{1}{2}\|f\|_{Lip_{\beta, p(\cdot)}}\leq \sup\limits_{B\in\mathbb{R}^{n}}\inf\limits_{c\in\mathbb{R}} \frac{1}{|B|^{\frac{\beta}{n}}\|\chi_{B}\|_{p^{\prime}(\cdot)}}\int_{B}|f-c|dx \leq\|f\|_{Lip_{\beta, p(\cdot)}}. $

In this paper, we consider the case of bilinear fractional integral.

Definition 1.3 [4]

$ \tilde{I}_{\beta}(f_{1}, f_{2})(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\left(\frac{1}{|(x-y_{1}, x-y_{1})|^{2n-\beta}} -\frac{\chi_{\{|(y_{1}, y_{2})|>1\}}}{|(y_{1}, y_{2})|^{2n-\beta}}\right)f_{1}(y_{1})f_{2}(y_{2})dy_{1}dy_{2}, $

where $0 < \beta < 2n$.

Now it is in this position to state our results.

Theorem 1.1 Let $0 < \beta < 2n$, $1 < p_{i}^{-}\leq p_{i}^{+} < \infty$ and $\frac{n}{p_{1}^{+}}+\frac{n}{p_{2}^{+}} < \beta < \frac{n}{p_{1}^{+}}+\frac{n}{p_{2}^{+}}+1$. Suppose that $p_{i}(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$ for $i=1, 2$ and $\frac{1}{p(x)}=\frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$, then $\tilde{I}_{\beta}(f_{1}, f_{2})$ is bounded from $L^{p_{1}(\cdot)}(\mathbb{R}^{n})\times L^{p_{2}(\cdot)}(\mathbb{R}^{n})$ into $Lip_{\beta, p(\cdot)}$.

Theorem 1.2 Let $0 < \beta < 2n$, $1 < p_{i}^{-}\leq p_{i}^{+} < \infty$ and $\frac{n}{p_{1}^{+}}+\frac{n}{p_{2}^{+}} < \beta < \frac{n}{p_{1}^{+}}+\frac{n}{p_{2}^{+}}+1$. Suppose that $p_{i}(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$ for $i=1, 2$, there exists a positive $r_{0}>1$ such that $p_{i}(x)\leq p_{i}(\infty)$ for $|x|>r_{0}$ and $\frac{1}{p(x)}=\frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$. Then $\tilde{I}_{\beta}(f_{1}, f_{2})$ is bounded from $L^{p_{1}(\cdot), \infty}(\mathbb{R}^{n})\times L^{p_{2}(\cdot), \infty}(\mathbb{R}^{n})$ into $Lip_{\beta, p(\cdot)}$.

Lemma 2.1 [11] If $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}), $ then for all $f\in L^{p(\cdot)}(\mathbb{R}^{n})$ and all $g\in L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})$ we have

$ \int_{\mathbb{R}^{n}}|f(x)g(x)|dx\leq r_{p}\|f\|_{p(\cdot)}\|g\|_{p^{\prime}(\cdot)}, $

where $r_{p}:=1+1/{p^{-}}-1/{p^{+}}.$

Lemma 2.2 [10] Let $p(\cdot)$ be an exponent function in $\mathcal{P}^{log}(\mathbb{R}^{n})$ such that $1 < p^{-}\leq p^{+} < \infty$ and $p(x)\leq p(\infty)$ for $|x|> r_{0}$ with $r_{0}>1$. Then there exists a positive constant $C$ depending on $r_{0}$ and the constants associated $\mathcal{P}^{log}(\mathbb{R}^{n})$, such that $\int_{B}|f(x)|dx\leq C[f]_{p(\cdot), \infty}\|\chi_{B}\|_{p^{\prime}(\cdot)}, $ for every ball $B$ and $f\in L^{p(\cdot), \infty}$.

The following lemma see Corollary 4.5.9 in [12].

Lemma 2.3 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$, then for every ball $B\subset\mathbb{R}^{n}$, we have

$ \|\chi_{B}\|_{p(\cdot)}\sim |B|^{\frac{1}{p(x)}}, \text{if} |B|\leq 2^{n}, x\in B $

and

$ \|\chi_{B}\|_{p(\cdot)}\sim |B|^{\frac{1}{p(\infty)}}, \text{if} |B|\geq 1. $

We remark that the Lemma 2.4 were showed in [13] and we will give the proof of it.

Lemma 2.4 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ and $x_{2}\in 2B(x_{1}, r)$, then we have

$ \|\chi_{B(x_{1}, r)}\|_{p(\cdot)}\sim\|\chi_{B(x_{2}, r)}\|_{p(\cdot)}. $

Proof We consider two cases, by Lemma 2.3.

Case 1 $|B|\geq 1$.

$ \|\chi_{B(x_{1}, r)}\|_{p(\cdot)}\sim|B(x_{1}, r)|^{\frac{1}{p(\infty)}} \sim|B(x_{2}, r)|^{\frac{1}{p(\infty)}}\sim\|\chi_{B(x_{2}, r)}\|_{p(\cdot)}. $

Case 2 $|B|\leq 1$.

$ \|\chi_{B(x_{1}, r)}\|_{p(\cdot)}\sim|B(x_{1}, r)|^{\frac{1}{p(x^{\prime})}} =|B(x_{1}, r)|^{\frac{1}{p(x^{\prime})}}|B(x_{2}, r)|^{-\frac{1}{p(x^{\prime\prime})}} \|\chi_{B(x_{2}, r)}\|_{p(\cdot)}\\ \sim r^{\frac{1}{p(x^{\prime})}-\frac{1}{p(x^{\prime\prime})}} \|\chi_{B(x_{2}, r)}\|_{p(\cdot)}\leq C\|\chi_{B(x_{2}, r)}\|_{p(\cdot)}, $

where we denote that $x^{\prime}\in B(x_{1}, r)$ and $x^{\prime\prime}\in B(x_{2}, r)$.

Indeed, since $x_{2}\in 2B(x_{1}, r), x^{\prime}\in B(x_{1}, r)$ and $x^{\prime\prime}\in B(x_{2}, r)$ we note that $|x^{\prime}-x^{\prime\prime}|\leq 4r$, we make use of local-Hölder continuity of $p(x)$ and get,

$ \left|\frac{1}{p^{\prime}(x^{\prime})}-\frac{1}{p^{\prime}(x^{\prime\prime})}\right|\log\frac{1}{r} \leq \frac{\log\frac{1}{r}}{\log(e+\frac{1}{|x^{\prime}-x^{\prime\prime}|})}\\ \leq \frac{\log\frac{1}{r}}{\log(e+\frac{1}{4r})} \leq C. $

Lemma 2.5 Let $p_{i}(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ for $i=1, 2$ and $\frac{1}{p(x)}=\frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$, then for every ball $B=B(x, r)\subset\mathbb{R}^{n}$, we have

$ \|\chi_{B}\|_{p(\cdot)}\sim \|\chi_{B}\|_{p_{1}(\cdot)}\|\chi_{B}\|_{p_{2}(\cdot)}, $

(2.1)

$ \|\chi_{B}\|_{p^{\prime}(\cdot)}\sim r^{-n}\|\chi_{B}\|_{p^{\prime}_{1}(\cdot)}\|\chi_{B}\|_{p^{\prime}_{2}(\cdot)}. $

(2.2)

Proof We will give the proof of the inequality (2.2), the argument for the inequality (2.1) is similar, we omit the details here. We consider two cases, by Lemma 2.3

Case 1 $|B|\leq 1$.

$ \|\chi_{B}\|_{p^{\prime}_{1}(\cdot)}\|\chi_{B}\|_{p^{\prime}_{2}(\cdot)}\sim|B|^{\frac{1}{p^{\prime}_{1}(x)}} |B|^{\frac{1}{p^{\prime}_{2}(x)}}\sim r^{n}|B|^{\frac{1}{p^{\prime}(x)}}\sim r^{n}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Case 2 $|B|\geq 1$.

$ \|\chi_{B}\|_{p^{\prime}_{1}(\cdot)}\|\chi_{B}\|_{p^{\prime}_{2}(\cdot)}\sim|B|^{\frac{1}{p^{\prime}_{1}(\infty)}} |B|^{\frac{1}{p^{\prime}_{2}(\infty)}}\sim r^{n}|B|^{\frac{1}{p^{\prime}(\infty)}}\sim r^{n}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Lemma 2.6 Let $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^{n})$, then there exists a constant $C>0$ such that for all balls $B$ and all measurable subsets $S=B(x_{0}, r_{0})\subset B=B(x_{1}, r_{1})$,

$ \frac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\leq C\left(\frac{|S|}{|B|}\right)^{1-\frac{1}{p^{-}}}, $

(2.3)

$ \frac{\|\chi_{B}\|_{p^{\prime}(\cdot)}}{\|\chi_{S}\|_{p^{\prime}(\cdot)}}\leq C\left(\frac{|B|}{|S|}\right)^{1-\frac{1}{p^{+}}}. $

(2.4)

Remark We can easily show that the inequality (2.4) implies $\|\chi_{2B}\|_{p^{\prime}(\cdot)} \leq C\|\chi_{B}\|_{p^{\prime}(\cdot)}$.

Proof We will prove the inequality (2.3), the argument for the inequality (2.4) is similar, we omit the details here. We consider three cases, by Lemma 2.3

(1)$|S| < |B| < 1$, $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(x_{S})}}} {|B|^{\frac{1}{p^{\prime}(x_{S})}}}|B|^{\frac{1}{p^{\prime}(x_{S})}-\frac{1}{p^{\prime}(x_{B})}}\leq C\left(\dfrac{|S|}{|B|}\right)^{\frac{1}{(p^{\prime})^{+}}}=C\left(\dfrac{|S|}{|B|}\right)^{1-\frac{1}{p^{-}}}$, where we denote that $x_{S}\in S$ and $x_{B}\in B$.

Indeed, since $|x_{B}-x_{S}|\leq 2r_{1}$, we make use of local-Hölder continuity of $p^{\prime}(x)$ and get

$ \left|\frac{1}{p^{\prime}(x_{S})}-\frac{1}{p^{\prime}(x_{B})}\right|\log\frac{1}{r_{1}} \leq \frac{\log\frac{1}{r_{1}}}{\log(e+\frac{1}{|x_{S}-x_{B}|})}\\ \leq \frac{\log\frac{1}{r_{1}}}{\log(e+\frac{1}{2r_{1}})} \leq C. $

(2)$|S| < 1 < |B|$, $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(x_{S})}}} {|B|^{\frac{1}{p^{\prime}(\infty)}}}\leq\left(\dfrac{|S|}{|B|}\right)^{\frac{1}{(p^{\prime})^{+}}} =\left(\dfrac{|S|}{|B|}\right)^{1-\frac{1}{p^{-}}}.$

(3)$1\leq |S| < |B|$, $\dfrac{\|\chi_{S}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\sim\dfrac{|S|^{\frac{1}{p^{\prime}(\infty)}}} {|B|^{\frac{1}{p^{\prime}(\infty)}}}\leq\left(\dfrac{|S|}{|B|}\right)^{\frac{1}{(p^{\prime})^{+}}} =\left(\dfrac{|S|}{|B|}\right)^{1-\frac{1}{p^{-}}}.$

Lemma 2.7 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ such that $1 < p^{-}\leq p^{+} < \infty$, $B=B(x_{0}, R)$ and $k < n-n/p_{-}$, then there exists a constant $C>0$ such that

$ \int_{|x_{0}-y|\leq R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\leq CR^{-k}\|f\|_{p(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Proof Using Lemma 2.1, we obtain

$ \int_{|x_{0}-y|\leq R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy=\sum\limits_{i=-\infty}^{0}\int_{A_{i}}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\\ \leq C\sum\limits_{i=-\infty}^{0}(2^{i}R)^{-k}\|f\|_{p(\cdot)}\|\chi_{B_{i}}\|_{p^{\prime}(\cdot)}. $

Lemma 2.6 gives

$ \int_{|x_{0}-y|\leq R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy \leq C\sum\limits_{i=-\infty}^{0}(2^{i}R)^{-k}\|f\|_{p(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)} \frac{\|\chi_{B_{i}}\|_{p^{\prime}(\cdot)}}{\|\chi_{B}\|_{p^{\prime}(\cdot)}}\\ \leq C\sum\limits_{i=-\infty}^{0}R^{-k}(2^{i})^{n-\frac{n}{p_{-}}-k}\|f\|_{p(\cdot)} \|\chi_{B}\|_{p^{\prime}(\cdot)}\\ \leq CR^{-k}\|f\|_{p(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Lemma 2.8 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ such that $1 < p^{-}\leq p^{+} < \infty$, $B=B(x_{0}, R)$ and $k>n-n/p_{+}$, then there exists a constant $C>0$ such that

$ \int_{|x_{0}-y|> R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\leq CR^{-k}\|f\|_{p(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Proof Applying Lemma 2.1, we derive the estimate

$ \int_{|x_{0}-y|> R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy=\sum\limits_{i=1}^{\infty}\int_{A_{i}}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\\ \leq C\sum\limits_{i=1}^{\infty}(2^{i}R)^{-k}\|f\|_{p(\cdot)}\|A_{i}\|_{p^{\prime}(\cdot)}. $

Lemma 2.6 implies that

$ \int_{|x_{0}-y|> R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy \leq C\sum\limits_{i=1}^{\infty}R^{-k}(2^{i})^{n-\frac{n}{p_{+}}-k}\|f\|_{p(\cdot)} \|\chi_{B}\|_{p^{\prime}(\cdot)}\\ \leq CR^{-k}\|f\|_{p(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Lemma 2.9 Suppose $p_{i}(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$, for $i=1, 2$, $B=B(x_{0}, R)$ and $\frac{1}{p(x)}=\frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$, then

$ M:=\int_{B}\int_{2B}\int_{2B}\frac{|f_{1}(y_{1})||f_{2}(y_{2})|}{|(x-y_{1}, x-y_{2})|^{2n-\beta}}dxdy_{1}dy_{2} \leq C|B|^{\frac{\beta}{n}}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p'(\cdot)}. $

Proof For $y_{1}, y_{2}\in 2B$, one can obtain the following inequality in [4].

$ \int_B {\frac{1}{{|(x - {y_1},x - {y_2}){|^{2n - \beta }}}}} dx = \left\{ \begin{gathered} C{R^{\beta - n}},{\text{ if }}n < \beta < 2n, \hfill \\ C\log \frac{{8R}}{{|{y_1} - {y_2}|}},{\text{ if }}n = \beta ,{\text{ }} \hfill \\ \frac{C}{{|{y_1} - {y_2}{|^{n - \beta }}}},{\text{ if }}0 < \beta < n. \hfill \\ \end{gathered} \right. $

When $n < \beta < 2n$, using Lemma 2.1 and 2.5, we obtain

$ M\leq CR^{\beta-n}\int_{2B}|f_{1}(y_{1})|dy_{1}\int_{2B}|f_{2}(y_{2})|dy_{2}\\ \leq CR^{\beta-n}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{2B}\| _{p^{\prime}_{1}(\cdot)}\|\chi_{2B}\|_{p^{\prime}_{2}(\cdot)}\\ \leq CR^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

When $0 < \beta < n$, we write

$ M\leq C\int_{2B}\int_{2B}\frac{|f_{1}(y_{1})||f_{2}(y_{2})|}{|y_{1}-y_{2}|^{n-\beta}}dy_{1}dy_{2}\\ =C\sum\limits_{i=-\infty}^{1}\sum\limits_{j=-\infty}^{1}\int_{A_{i}}\int_{A_{j}}\frac{|f_{1}(y_{1})| |f_{2}(y_{2})|}{|y_{1}-y_{2}|^{n-\beta}}dy_{1}dy_{2}\\ =C\sum\limits_{i=-\infty}^{1}\left(\sum\limits_{j=-\infty}^{i-2}+\sum\limits_{j=i-1}^{i+1}+\sum\limits_{j=i+2}^{1}\right) \int_{2B}\int_{2B}\frac{|f_{1}\chi_{i}(y_{1})||f_{2}\chi_{j}(y_{2})|}{|y_{1}-y_{2}|^{n-\beta}}dy_{1}dy_{2}\\ := D_{1}+D_{2}+D_{3}, $

when $j>-1$ we define $D_{3}=0$.

First we estimate $D_{1}$.

For $y_{1}\in A_{i}, y_{2}\in A_{j}$, we have $|y_{1}-y_{2}|\geq |y_{1}|-|y_{2}|>2^{i-2}R$. Then

$ D_{1}\leq C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta-n}\left(\sum\limits_{j=-\infty}^{i-2} \int|f_{1}\chi_{i}(y_{1})|dy_{1}\int|f_{2}\chi_{j}(y_{2})|dy_{2}\right)\\ = C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta-n} \int|f_{1}\chi_{i}(y_{1})|dy_{1}\int_{B_{i-2}}|f_{2}(y_{2})|dy_{2}. $

Now Lemma 2.1 yields

$ D_{1}\leq C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta-n}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B_{i}}\| _{p^{\prime}_{1}(\cdot)}\|\chi_{B_{i}}\|_{p^{\prime}_{2}(\cdot)}. $

By Lemma 2.5, we get

$ D_{1}\leq C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)} \|\chi_{B_{i}}\|_{p^{\prime}(\cdot)}\\ \leq CR^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Next we estimate $D_{2}$.

Noting that $|y_{1}-y_{2}|\leq|y_{1}|+|y_{2}| < C2^{i}R$ for $y_{1}\in A_{i}, y_{2}\in A_{j}$, using Lemma 2.1 and 2.7, we have

$ D_{2}\leq C\sum\limits_{i=-\infty}^{1}\sum\limits_{j=i-1}^{i+1}\int|f_{1}\chi_{i}(y_{1})|dy_{1} \int\frac{|f_{2}\chi_{j}(y_{2})|}{|y_{1}-y_{2}|^{n-\beta}}dy_{2}\\ \leq C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta-n}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B_{i}}\| _{p^{\prime}_{1}(\cdot)}\|\chi_{B(y_{1}, 2^{i}R)}\|_{p^{\prime}_{2}(\cdot)}. $

By Lemma 2.4 and 2.5, we arrive at the inequality

$ D_{2}\leq C\sum\limits_{i=-\infty}^{1}(2^{i}R)^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)} \|\chi_{B_{i}}\|_{p^{\prime}(\cdot)}\\ \leq CR^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Finally we estimate $D_{3}$.

We note $y_{1}\in A_{i}, y_{2}\in A_{j}$, $|y_{1}-y_{2}|\geq |y_{1}|-|y_{2}|>2^{j-2}R$ and derive

$ D_{3}\leq C\sum\limits_{i=-\infty}^{1}\sum\limits_{j=i+2}^{1}(2^{j}R)^{\beta-n} \int|f_{1}\chi_{i}(y_{1})|dy_{1}\int|f_{2}\chi_{j}(y_{2})|dy_{2}\\ =C\sum\limits_{j=-\infty}^{1}\sum\limits_{i=-\infty}^{j-2}(2^{j}R)^{\beta-n} \int|f_{1}\chi_{i}(y_{1})|dy_{1}\int|f_{2}\chi_{j}(y_{2})|dy_{2}\\ =C\sum\limits_{j=-\infty}^{1}(2^{j}R)^{\beta-n} \int_{B_{j-2}}|f_{1}(y_{1})|dy_{1}\int|f_{2}\chi_{j}(y_{2})|dy_{2}. $

Hence, we apply Lemma 2.1 and 2.5 and obtain

$ D_{3}\leq C\sum\limits_{j=-\infty}^{1}(2^{j}R)^{\beta-n}\|f_{1}\|_{p_{1}(\cdot)}\|\chi_{B_{j}}\| _{p^{\prime}_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B_{j}}\|_{p^{\prime}_{2}(\cdot)}\\ \leq C\sum\limits_{j=-\infty}^{1}(2^{j}R)^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B_{j}}\|_{p^{\prime}(\cdot)}\\ \leq CR^{\beta}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

When $\beta=n$, the proof is similar. Therefore we omit the details. We use the following inequality

$ \sum\limits_{i=-\infty}^{1}\left(\log\frac{8}{2^{i-2}}\right)2^{in}=\log2\sum\limits_{i=-\infty}^{1}(5-i)2^{in}\leq C. $

As long as we change the conclusion of Lemma 2.1 into the conclusion of Lemma 2.2 in the proof of Lemma 2.7-2.9, we can obtain the corresponding conclusions in $L^{p(\cdot), \infty}$ space.

Corollary 2.1 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ such that $1 < p^{-}\leq p^{+} < \infty$, $B=B(x_{0}, R)$, $p(x)\leq p(\infty)$ for $|x|>r_{0}$ with $r_{0}>1$ and $k < n-n/p_{-}$, then there exists a constant $C>0$ such that

$ \int_{|x_{0}-y|\leq R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\leq CR^{-k}[f]_{p(\cdot), \infty}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Corollary 2.2 Let $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ such that $1 < p^{-}\leq p^{+} < \infty$, $B=B(x_{0}, R)$, $p(x)\leq p(\infty)$ for $|x|>r_{0}$ with $r_{0}>1$ and $k>n-n/p_{+}$, then there exists a constant $C>0$ such that

$ \int_{|x_{0}-y|> R}\frac{|f(y)|}{|x_{0}-y|^{k}}dy\leq CR^{-k}[f]_{p(\cdot), \infty}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Corollary 2.3 Let $p_{i}(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^{n})$ such that $1 < p_{i}^{-}\leq p_{i}^{+} < \infty$, $p_{i}(x)\leq p_{i}(\infty)$ for $i=1, 2$ and $|x|>r_{0}$ with $r_{0}>1$. Suppose $B=B(x_{0}, R)$ and $\frac{1}{p(x)}=\frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$, then

$ M:=\int_{B}\int_{2B}\int_{2B}\frac{|f_{1}(y_{1})||f_{2}(y_{2})|}{|(x-y_{1}, x-y_{2})|^{2n-\beta}}dxdy_{1}dy_{2} \leq C|B|^{\frac{\beta}{n}}[f_{1}]_{p_{1}(\cdot), \infty}[f_{2}]_{p_{2}(\cdot), \infty}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

We will give the proof of the Theorem 1.1 below. In Corollary 2.1-Corollary 2.3, we obtain the corresponding results in $L^{p(\cdot), \infty}$ space.The argument for Theorem 1.2 is similar, we omit the details here.

Proof of Theorem 1.1 We write

$ f_{i}(x)=f_{i}\chi_{2B}(x)+f_{i}\chi_{\mathbb{R}^{n}\setminus2B}(x) (i=1, 2). $

And we need to estimate four terms $\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{2B}), \tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B}), \tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})$ and $\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})$.

First we estimate $\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{2B})$.

Let $c=-\int\int_{|(y_{1}, y_{2})|\geq 1}\frac{f_{1}\chi_{2B}(y_{1})f_{2}\chi_{2B}(y_{2})}{|(y_{1}, y_{2})|^{2n-\beta}}dy_{1}dy_{2}$. By Lemma 2.9, we get

$ \int_{B}|\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{2B})(x)-c|dx=\int_{B}|I_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{2B})(x)|dx\\ \leq C|B|^{\frac{\beta}{n}}\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p'(\cdot)}. $

Hence, we arrive at the inequality

$ \frac{1}{|B|^{\frac{\beta}{n}}\|\chi_{B}\|_{p'(\cdot)}}\int_{B}|\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{2B})(x)-c|dx \leq C\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}. $

Next we estimate $\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B})$ and $\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})$.

We only estimate $\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B})$ and the estimate for $\tilde{I}_{\beta}(f_{1}\chi_{2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})$ is similar, we omit the details here.

Let $c=\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B})(x_{0})$, then for $x\in B$, we have

$ |\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B})(x)-c|\leq CR\int\int_{|x_{0}-y_{1}|>2R} \frac{|f_{1}\chi_{\mathbb{R}^{n}\setminus2B}(y_{1})f_{2}\chi_{2B}(y_{2})|}{|(x_{0}-y_{1}, x_{0}-y_{2})|^{2n-\beta+1}}dy_{1}dy_{2}\\ \leq CR\int_{|x_{0}-y_{1}|>2R}\frac{|f_{1}(y_{1})|}{|x_{0}-y_{1}|^{2n-\beta+1}}dy_{1} \int|f_{2}\chi_{2B}(y_{2})|dy_{2}. $

Applying Lemma 2.8, 2.1 and 2.5, we obtain

$ |\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{2B})(x)-c| \leq CR^{-2n+\beta}\|f\|_{p_{1}(\cdot)}\|\chi_{B}\|_{p_{1}^{\prime}(\cdot)}\|f\|_{p_{2}(\cdot)}\|\chi_{2B}\|_{p_{2}^{\prime}(\cdot)}\\ \leq CR^{-n+\beta}\|f\|_{p_{1}(\cdot)}\|f\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}. $

Thus we get that

$ \frac{1}{|B|^{\frac{\beta}{n}}\|\chi_{B}\|_{p'(\cdot)}}\int_{B}|\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B} , f_{2}\chi_{2B})(x)-c|dx\leq C\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}. $

Finally we estimate $\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})$.

Let $c=\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})(x_{0})$, then for $x\in B$,

$ |\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}\chi_{\mathbb{R}^{n}\setminus2B})(x)-c|\leq CR\int\int \frac{|f_{1}\chi_{\mathbb{R}^{n}\setminus2B}(y_{1})f_{2}\chi_{\mathbb{R}^{n}\setminus2B}(y_{2})|} {|(x_{0}-y_{1}, x_{0}-y_{2})|^{2n-\beta+1}}dy_{1}dy_{2}. $

By Lemma 2.8and 2.5, we have

$ |\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B}, f_{2}(x)-c| \leq CR\int_{|x_{0}-y_{1}|>2R}\frac{|f_{1}(y_{1})|}{|x_{0}-y_{1}|^{n-s_{1}}}dy_{1} \int_{|x_{0}-y_{2}|>2R}\frac{|f_{2}\chi_{2B}(y_{2})|}{|x_{0}-y_{2}|^{n-s_{2}}}dy_{2}\\ \leq CR\|f\|_{p_{1}(\cdot)}\|\chi_{2B}\|_{p_{1}^{\prime}(\cdot)}R^{s_{1}-n}\|f\|_{p_{2}(\cdot)} \|\chi_{2B}\|_{p_{2}^{\prime}(\cdot)}R^{s_{2}-n}\\ \leq CR^{-n+\beta}\|f\|_{p_{1}(\cdot)}\|f\|_{p_{2}(\cdot)}\|\chi_{B}\|_{p^{\prime}(\cdot)}, $

where we can take $s_{1}$ and $s_{2}$ such that $s_{1} < n/p_{1}^{+}, s_{2} < n/p_{2}^{+}$ and $s_{1}+s_{2}=\beta-1$.

Hence, we obtain

$ \frac{1}{|B|^{\frac{\beta}{n}}\|\chi_{B}\|_{p'(\cdot)}}\int_{B}|\tilde{I}_{\beta}(f_{1}\chi_{\mathbb{R}^{n}\setminus2B} , f_{2}\chi_{\mathbb{R}^{n}\setminus2B})(x)-c|dx\leq C\|f_{1}\|_{p_{1}(\cdot)}\|f_{2}\|_{p_{2}(\cdot)}. $

Consequently we have proved the Theorem 1.1.