Let $\psi$ be a function on ${\mathbb {R}}^n$ such that there exist positive constants $C$ and $\gamma$ satisfying

(a) $\psi\in L^1({\mathbb {R}}^n)\ {\rm and} \ \displaystyle\int_{{\mathbb {R}}^n}\psi(x)dx=0;$

(b) $|\psi(x)|\leq C(1+|x|)^{-n-1};$

(c) $|\psi(x+y)-\psi(x)|\leq C|y|^\gamma/(1+|x|)^{n+\gamma+1}\ {\rm for} \ 2|y|\leq |x|.$

For this $\psi$ and $\mu > 1$, the Littlewood-Paley's $g^\ast_\mu$ function is defined by

$ \begin{equation}\begin{split} g^\ast_\mu(f)(x)=\bigg(\displaystyle\int\displaystyle\int_{{\mathbb {R}}^{n+1}_+}\bigg(\frac{t}{t+|x-y|}\bigg)^{n\mu} |(\psi_t\ast f)(y)|^2\frac{dydt}{t^{n+1}}\bigg)^{1/2}, \end{split}\end{equation} $

(1.1)

where ${\mathbb {R}}^{n+1}_+=\{(y, t): y\in {\mathbb {R}}^n, t > 0\}$ and $\psi_t(x)=t^{-n}\psi(x/t).$

Given a positive integer $m$ and a vector $\vec{b}= (b_1, b_2, \cdots, b_m)$ of locally integrable functions, motivated by the work of Pérez and Trujillo-González [1] on multilinear operators, we define multilinear commutators of the Littlewood-Paley's $g^\ast_\mu$ function as follows:

$ \begin{equation}\begin{split} g^\ast_{\mu, \vec{b}}(f)(x)=\bigg(\displaystyle\int\displaystyle\int_{{\mathbb {R}}^{n+1}_+}\bigg(\frac{t}{t+|x-y|}\bigg)^{n\mu} \bigg|\displaystyle\int_{{\mathbb {R}}^n}\psi_t(y-z)\prod^m_{i=1}(b_i(x)-b_i(z))f(z)dz\bigg|^2\frac{dydt}{t^{n+1}}\bigg)^{1/2}. \end{split}\end{equation} $

(1.2)

In the case of $m=1$, we usually denote $g^\ast_{\mu, \vec{b}}$ by $[b, g^\ast_\mu]$.

A locally integrable function $b$ is said to be a ${\rm BMO}$ function, if it satisfies

$ \begin{equation*}\begin{split} \|b\|_{\ast}:=\sup\limits_{x\in\mathbb {R}^n, r>0}\frac{1}{|B|}\displaystyle\int_{B}|b(y)-b_{B}|\, dy<\infty, \end{split}\end{equation*} $

where and in the sequel $B$ is ball centered at $x$ and radius of $r$,

$ b_B=\frac{1}{|B|}\displaystyle\int_Bb(t)dt $

and $\|b\|_{\ast}$ is the norm in ${\rm BMO}(\mathbb R^n).$ For $b_i\in {\rm BMO}(\mathbb R^n)$, $i=1, 2, \cdots, m$, Xue and Ding [2] established the weighted $L^p$ and weighted weak $L({\rm log}L)$-type estimates for the multilinear commutators $g^\ast_{\mu, \vec{b}}$. Zhang et al. [3] obtained some boundedness results for $g^\ast_{\mu, \vec{b}}$ on certain classical Hardy and Herz-Hardy spaces. We refer to [4-6] for an extensive study of multilinear operators.

In recent years, following the fundamental work of Kováčik and Rákosník [7], function spaces with variable exponent, such as variable exponent Lebesgue and Herz-type Hardy spaces etc., have attracted a great attention in connection with problems of the boundedness of classical operators on those spaces, which in turn were motivated by the treatment of recent problems in fluid dynamics, image restoration and differential equations with $p(x)$-growth, see [8-16] and the references therein.

Karlovich and Lerner in [17] showed that $[b, T]$, the commutator of a standard Calderón-Zygmund singular integral operator $T$ and a ${\rm BMO}$ function $b$, is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$, which improved a celebrated result by Coifman et al. in [18]. Recently, Xu [19] made a futher step and proved that the multilinear commutators $T_{\vec{b}}$, a generalization of the commutator $[b, T]$, enjoy the same $L^{p(\cdot)}(\mathbb{R}^n)$ estimates when $b_i\in {\rm BMO}(\mathbb{R}^n)$, $i=1, 2, \cdots, m$. These results inspire us to ask whether the multilinear commutators $g^\ast_{\mu, \vec{b}}$ have the similar mapping properties in variable exponent spaces $L^{p(\cdot)}(\mathbb{R}^n)$? Our first result (see Theorem 3.1 below) will give an affirmative answer to this question.

The variable exponent Herz spaces $\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)$ and $K_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)$ were first studied by Izuki [20, 21]. Simultaneously, he gave some basic lemmas on generalization of the BMO norms to get the boundedness of classical operators on such spaces. On the other hand, the variable exponent Herz-type Hardy spaces, as well as their atomic decomposition characterizations, were intensively studied by a significant number of authors [22, 23]. Using these decompositions, they also established the boundedness results for some singular integral operators. Motivated by the results mentioned above, another purpose of this article is to study the boundedness of $g^\ast_{\mu, \vec{b}}$ in variable exponent Herz-type Hardy spaces, which improves the corresponding main result in classical case (see [3, Theorem 2]).

In general, we denote cubes in $\mathbb{R}^n$ by $Q$. If $E$ is a subset of $\mathbb{R}^n$, $|E|$ denotes its Lebesgue measure and $\chi_E$ denotes its characteristic function. For $l\in \mathbb{Z}$, we define $B_l = \{x \in {\mathbb R} ^n: |x| \leqslant2^l\}$. $p'(\cdot)$ denotes the conjugate exponent defined by $1/p(\cdot)+1/p'(\cdot)=1$. By $\mathcal{S'}(\mathbb{R}^n), $ we denote the space of tempered distributions. We use $x\thickapprox y$ if there exist constants $c_1$, $c_2$ such that $c_1x\leq y\leq c_2x$, $C$ stands for a positive constant, which may vary from line to line.

We begin with a brief and necessarily incomplete review of the variable exponent Lebesgue spaces $L^{p(\cdot)}({\Bbb{ R}}^n)$, see [24, 25] for more information.

Let $p(\cdot): {\Bbb{ R}}^n\rightarrow [1, \infty)$ be a measurable function. We assume that

$ \begin{equation*}\begin{split} 1\leq {p_-}\leq p(x)\leq {p_+}<\infty, \end{split}\end{equation*} $

where and in the sequel

$p_-:=\mbox{ess} \inf\limits_{x\in{\Bbb{ R}}^n}p(x) ~\mbox{and}~ p_+:=\mbox{ess} \sup\limits_{x\in {\Bbb{ R}}^n}p(x). $

By $L^{p(\cdot)}({\Bbb{ R}}^n)$ we denote the set of all measurable functions $f$ on ${\Bbb{ R}}^n$ such that

$ \begin{equation*}\begin{split} \varrho_{p(\cdot)}(f):=\displaystyle\int_{{\Bbb{ R}}^n}|f(x)|^{p(x)}dx<\infty. \end{split}\end{equation*} $

This is a Banach space with the norm (the Luxemburg-Nakano norm)

$ \begin{equation*}\begin{split} \|f\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}=\mbox{inf}\{\mu>0: \varrho_{p(\cdot)}(f/\mu)\leq 1 \}. \end{split}\end{equation*} $

Given an open set $\Omega\subset {\Bbb{ R}}^n$, the space $L^{p(\cdot)}_{\rm loc}(\Omega)$ is defined by

$ \begin{equation*}\begin{split} L^{p(\cdot)}_{\rm loc}(\Omega)=\{f: f\in L^{p(\cdot)}(K)\ \mbox{for all compact subsets $K\subset \Omega$}\}. \end{split}\end{equation*} $

For the sake of simplicity, we use the notation

$ \begin{eqnarray*} &&\mathcal{P}({\Bbb{ R}}^n):=\{p(\cdot): \ p_->1 \ \mbox{and}\ p_+<\infty\}, \\ && \mathcal{B}({\Bbb{ R}}^n):=\{p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n): \ \ M \ \mbox{is bounded on $L^{p(\cdot)}({\Bbb{ R}}^n)$}\}, \end{eqnarray*} $

where $M$ is the Hardy-Littlewood maximal operator defined by

$ \begin{equation*}\begin{split} Mf(x)=\sup\limits_{Q\ni x}\frac{1}{|Q|}\displaystyle\int_Q|f(y)|dy. \end{split}\end{equation*} $

When $p(\cdot)\in\mathcal{P}({\Bbb{ R}}^n)$, the generalized H$\ddot{\rm o}$lder inequality holds in the form

$ \begin{equation}\begin{split}\label{holder} \displaystyle\int_{{\Bbb R}^n}|f(x)g(x)|dx\leq r_p\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\|g\|_{L^{p'(\cdot)}({\Bbb R}^n)} \end{split}\end{equation} $

(2.1)

with $r_p=1+1/p_--1/p_+$, see [7, Theorem 2.1].

We say a measurable function $\phi:{\Bbb{ R}}^n \rightarrow [1, \infty)$ is globally log-H$\ddot{\rm o}$lder continuous if it satisfies

$ \begin{eqnarray}&&\label{log1} |\phi(x)-\phi(y)|\leq \frac{-C}{\mbox{log}(|x-y|)}, \quad |x-y|\leq 1/2, \end{eqnarray} $

(2.2)

$ \begin{eqnarray} &&\label{log2} |\phi(x)-\phi(y)|\leq \frac{C}{\mbox{log}(e+|x|)}, \quad |y|\geq |x| \end{eqnarray} $

(2.3)

for any $x, y\in {{\Bbb{ R}}^n}$. The set of $p(\cdot)$ satisfying (2.2) and (2.3) is denoted by $LH({\Bbb{ R}}^n)$. It is well-known that if $p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n)\bigcap LH({\Bbb{ R}}^n)$, then the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}({\Bbb{ R}}^n)$, thus we have $p(\cdot)\in \mathcal{B}({\Bbb{ R}}^n)$, see [24].

Lemma 2.1 (see [21]) Suppose $p(\cdot)\in\mathcal{B}({\Bbb{R}}^n)$, then we have

$ \begin{equation*}\begin{split} \frac{1}{|B|}\|\chi_B\|_{L^{p(\cdot)}({\Bbb R}^n)}\|\chi_B\|_{L^{p'(\cdot)}({\Bbb R}^n)}\leq C. \end{split}\end{equation*} $

Lemma 2.2 (see [21]) Suppose $p(\cdot)\in\mathcal{B}({\Bbb{R}}^n)$, then we have for all measurable subsets $E\subset B$,

$ \begin{equation*}\begin{split} \frac{\|\chi_E\|_{L^{p(\cdot)}({\Bbb R}^n)}}{\|\chi_B\|_{L^{p(\cdot)}({\Bbb R}^n)}} \leq C\biggl(\frac{|E|}{|B|}\biggr)^{\delta_1}, \quad \frac{\|\chi_E\|_{L^{p'(\cdot)}({\Bbb R}^n)}}{\|\chi_B\|_{L^{p'(\cdot)}({\Bbb R}^n)}}\leq C\biggl(\frac{|E|}{|B|}\biggr)^{\delta_2}, \end{split}\end{equation*} $

where $\delta_1, \delta_2$ are constants with $0 < \delta_1, \delta_2 < 1$.

Remark 2.1 We would like to stress that everywhere below the constants $\delta_1$ and $\delta_2$ are always the same as in Lemma 2.2.

Lemma 2.3 (see [24]) Suppose $p_i(\cdot), p(\cdot)\in\mathcal{P}({\Bbb{ R}}^n)$, $i=1, 2, \cdots, m$, so that

$ \begin{equation*}\begin{split} \frac{1}{p(x)}=\sum\limits_{i=1}^m\frac{1}{p_i(x)}, \end{split}\end{equation*} $

where $m\in\mathbb{N}$. Then for all $f_i\in L^{p_i(\cdot)}({\Bbb{ R}}^n)$, we have

$ \begin{equation*}\begin{split} \bigg\|\prod\limits_{i=1}^mf_i\bigg\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\leq C\prod\limits_{i=1}^m\|f_i\|_{L^{p_i(\cdot)}({\Bbb{ R}}^n)}. \end{split}\end{equation*} $

Lemma 2.4 (see [25]) Suppose $p(\cdot)\in LH({\Bbb{ R}}^n)$ and $0 < {p_-}\leq p(x)\leq {p_+} < \infty$.

(ⅰ) For all balls (or cubes) $|B|\leq 2^n$ and any $x\in B$, we have

$ \begin{equation*}\begin{split} \|\chi_B\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\approx |B|^{1/p(x)}. \end{split}\end{equation*} $

(ⅱ) For all balls (or cubes) $|B|\geq 1$, we have

$ \begin{equation*}\begin{split} \|\chi_B\|_{L^{p(\cdot)}({\Bbb{ R}}^n)}\approx |B|^{1/{p_\infty}}, \end{split}\end{equation*} $

where $p_\infty:=\lim\limits_{x\rightarrow\infty}p(x).$

Combining Lemma 2.3, Lemma 2.4 and Lemma 3 in [21, page 464], a simple computation shows that

Lemma 2.5 Suppose $p(\cdot)\in\mathcal{P}({\Bbb{ R}}^n)\bigcap LH({\Bbb{ R}}^n)$, $b_i\in {\rm {BMO}}({\Bbb{ R}}^n)$, $i=1, 2, \cdots, m$, $k > j\ (k, j\in\mathbb{N})$, then we have

$ \begin{equation*}\begin{split} \sup\limits_{B\subset{\Bbb{R}}^n} \frac{1}{\|\chi_B\|_{L^{p(\cdot)}({\Bbb R}^n)}} \bigg\|\prod\limits_{i=1}^m(b_i-(b_i)_B)\chi_B\bigg\|_{L^{p(\cdot)}({\Bbb R}^n)}\thickapprox \prod\limits_{i=1}^m\|b_i\|_{\ast} \end{split}\end{equation*} $

and

$ \begin{equation*}\begin{split} \bigg\|\prod\limits_{i=1}^m(b_i-(b_i)_{B_j})\chi_{B_k}\bigg\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq C(k-j)^m\prod\limits_{i=1}^m\|b_i\|_{\ast}\|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation*} $

Remark 2.2 We note that Lemma 2.5 generalizes the well known properties for ${\rm BMO}({\Bbb R}^n)$ spaces (see [26]), and is also a generalization of Lemma 3 in [21].

We first recall some pointwise estimates for sharp maximal functions, the duality and density in variable exponent Lebesgue spaces $L^{p(\cdot)}({\Bbb R}^n)$.

For $p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n)$, the spaces $L^{p(\cdot)}({\Bbb R}^n)$ can be endowed with the Orlicz type norm

$ \begin{equation*}\begin{split} \|f\|_{L^{p(\cdot)}({\Bbb R}^n)}^0:= \sup\bigg\{\displaystyle\int_{{\Bbb{ R}}^n}|f(x)g(x)|dx: g\in L^{p'(\cdot)}({\Bbb R}^n), \quad \|g\|_{L^{p'(\cdot)}({\Bbb R}^n)}\leq 1\bigg\}. \end{split}\end{equation*} $

This norm, as pointed out in [7], is equivalent to the Luxemburg-Nakano norm, that is

$ \begin{equation}\begin{split}\label{equivalence} \|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq \|f\|_{L^{p(\cdot)}({\Bbb R}^n)}^0\leq r_p\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}, \end{split}\end{equation} $

(3.1)

where $r_p=1+1/{p_-}-1/{p_+}$.

By $L_c^{\infty}$ we denote the set of all bounded functions $f$ with compact support. From [7, Theorem 2.11] (see also [17, Lemma 2.2]), we get the following result.

Proposition 3.1 Suppose $p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n)$, then $L_c^{\infty}$ is dense in $L^{p(\cdot)}({\Bbb R}^n)$ and in $L^{p'(\cdot)}({\Bbb R}^n)$. For $\delta > 0$ and $f\in L_{\rm loc}^1({\Bbb{ R}}^n)$, we define

$ \begin{equation*}\begin{split} M_\delta(f)(x)=M(|f|^\delta)^{1/\delta}(x)=\bigg(\sup\limits_{Q\ni x}\frac{1}{|Q|}\displaystyle\int_Q|f(y)|^\delta dy\bigg)^{1/\delta}. \end{split}\end{equation*} $

Given a function $f\in L_{\rm loc}^\delta({\Bbb{ R}}^n)$, set also

$ \begin{equation*}\begin{split} f_\delta^\sharp(x)=\sup\limits_{Q\ni x}\inf\limits_{c\in\mathbb{R}}\bigg(\frac{1}{|Q|}\displaystyle\int_Q|f(y)-c|^\delta dy\bigg)^{1/\delta}, \end{split}\end{equation*} $

where the supremums are taken over all cubes $Q\subset {\Bbb{ R}}^n$ containing $x$.

The non-increasing rearrangement of a measurable function $f$ on ${\Bbb{ R}}^n$ is defined as

$ \begin{equation*}\begin{split} f^\ast(t):=\inf\bigg\{a>0: |\{s\in{\Bbb{ R}}^n: |f(s)|>a\}|\leq t\bigg\}, \quad t>0, \end{split}\end{equation*} $

and for a fixed $\lambda\in (0, 1)$, the local sharp maximal function $M_\lambda^\sharp f$ is given by

$ \begin{equation*}\begin{split} M_\lambda^\sharp (f)(x)=\sup\limits_{Q\ni x}\inf\limits_{c\in\mathbb{R}}((f-c)\chi_Q)^\ast(\lambda |Q|). \end{split}\end{equation*} $

The next lemma is due to Karlovich and Lerner [17, Proposition 2.3].

Lemma 3.1 Suppose $\lambda\in (0, 1)$, $\delta > 0$ and $f\in L_{\rm loc}^\delta({\Bbb{ R}}^n)$, then we have

$ \begin{equation*}\begin{split} M_\lambda^\sharp (f)(x)\leq (1/\lambda)^{1/\delta}f_\delta^\sharp(x), \quad x\in {\Bbb{ R}}^n. \end{split}\end{equation*} $

A function $\Phi$ defined on $[0, \infty)$ is said to be a Young function, if $\Phi$ is a continuous, nonnegative, strictly increasing and convex function with $\lim\limits_{t\rightarrow 0^+}\Phi(t)/t=\lim\limits_{t\rightarrow 0^+}t/\Phi(t)=0$. We define the $\Phi$-average of a function $f$ over a cube $Q$ by

$ \begin{equation*}\begin{split} \|f\|_{\Phi, Q}=\inf\bigg\{\eta>0: \frac{1}{|Q|}\displaystyle\int_Q\Phi\bigg(\frac{|f(y)|}{\eta}\bigg)dy\leq 1\bigg\}. \end{split}\end{equation*} $

Associated to this $\Phi$-average, we define the maximal operator $M_\Phi$ by

$ \begin{equation*}\begin{split} M_\Phi(f)(x):=\sup\limits_{Q\ni x}\|f\|_{\Phi, Q}. \end{split}\end{equation*} $

When $\Phi(t)=t{\rm log}^r(e+t)$ $(r\geq 1)$, we denote $M_\Phi$ by $M_{L({\rm log}L)^r}$. It is well-known that if $m\in\mathbb{N}$, then $M_{L({\rm log}L)^m}\approx M^{m+1}$, the $m+1$ iterations of the Hardy-Littlewood maximal operator $M$, see [1].

Lemma 3.2 (see [2]) Let $0 < \delta < 1$. Then there exsits a positive constant $C$, independet of $f$ and $x$, such that $(g^\ast_\mu(f))_\delta^\sharp(x)\leq CMf(x), \quad x\in {\Bbb{ R}}^n$ holds for all bounded function $f$ with compact support.

In fact, there holds a similar piontwise estimate for the multilinear commutators $g^\ast_{\mu, \vec{b}}$. To state it, we first introduce some notations.

As in [1], given any positive integer $m$, for all $1\leq j\leq m$, we denote by $C_j^m$ the family of all finite subset $\sigma=\{\sigma(1), \sigma(2), \cdots, \sigma(j)\}$ of $\{1, 2, \cdots, m\}$ of $j$ different elements. For any $\sigma\in C_j^m$, we associate the complementary sequence $\sigma'$ given by $\sigma'=\{1, 2, \cdots, m\}\backslash\sigma$.

Suppose $\vec{b}=(b_1, b_2, \cdots, b_m)$ and $\sigma=\{\sigma(1), \sigma(2), \cdots, \sigma(j)\}\in C_j^m$. Denote

$ \vec{b}_\sigma=\{b_{\sigma(1)}, b_{\sigma(2)}, \cdots, b_{\sigma(j)}\}, ~~ b_\sigma=b_{\sigma(1)}b_{\sigma(2)}\cdots b_{\sigma(j)} $

and $\|b_\sigma\|=\prod\limits_{j\in\sigma}\|b\|_\ast$. If $\sigma=\{1, 2, \cdots, m\}$, then we denote $\|b_\sigma\|$ by $\|\vec{b}\|$.

For any $\sigma=\{\sigma(1), \sigma(2), \cdots, \sigma(j)\}\in C_j^m$, we define

$ \begin{equation*}\begin{split} &g^\ast_{\mu, \vec{b}_\sigma}(f)(x)\\ =&\bigg(\displaystyle\int\displaystyle\int_{{\mathbb {R}}^{n+1}_+}\bigg(\frac{t}{t+|x-y|}\bigg)^{n\mu} \bigg|\displaystyle\int_{{\mathbb {R}}^n}\psi_t(y-z)\prod^j_{i=1}(b_{\sigma(i)}(x)-b_{\sigma(i)}(z))f(z)dz\bigg|^2\frac{dydt}{t^{n+1}}\bigg)^{1/2}. \end{split}\end{equation*} $

In the case $\sigma=\{1, 2, \cdots, m\}$, we understand $g^\ast_{\mu, \vec{b}_\sigma}=g^\ast_{\mu, \vec{b}}$ and $g^\ast_{\mu, \vec{b}_{\sigma'}}=g^\ast_\mu$.

We now mention an immediate consequence of Proposition 2.4 in [2].

Lemma 3.3 Suppose $\mu > 2$ and $0 < \delta < \varepsilon < 1$. Then for any $f\in L_c^\infty$, there exists a constant $C > 0$, depending only on $\delta$ and $\varepsilon$, such that

$ \begin{equation*}\begin{split} (g^\ast_{\mu, \vec{b}}(f))_\delta^\sharp(x)\leq C \bigg\{\|\vec{b}\|M_{L({\rm log}L)^m}f(x)+\sum\limits_{j=1}^m\sum\limits_{\sigma\in C_j^m} \|b_\sigma\|M_{\varepsilon}(g^\ast_{\mu, \vec{b}_{\sigma'}}f)(x)\bigg\}. \end{split}\end{equation*} $

We also need the following result from Lerner [27, Theorem 1].

Lemma 3.4 Suppose $g\in L_{\rm loc}^1({\Bbb R}^n)$ and let $f$ be a measurable function with $f^\ast(+\infty)=0$, then

$ \begin{equation*}\begin{split} \displaystyle\int_{{\Bbb R}^n}|f(x)g(x)|dx\leq c_n\displaystyle\int_{{\Bbb R}^n}M_{\lambda_n}^\sharp f(x)Mg(x)dx, \end{split}\end{equation*} $

where constants $0 < \lambda_n < 1$ and $c_n$ depend only on dimension $n$.

To prove Theorem 3.1, we first prove the following result which has its independent role.

Lemma 3.5 Suppose $\mu > 2$ and $0 < \gamma < {\rm min}\{(\mu-2)n/2, 1\}$. If $p(\cdot)\in \mathcal{B}({\Bbb{ R}}^n)$, then $g^\ast_\mu$ is bounded from $L^{p(\cdot)}({\Bbb R}^n)$ to itself.

Proof Let $f\in L_c^\infty$ and $g\in L^{p'(\cdot)}({\Bbb R}^n)\subset L_{\rm loc}^1({\Bbb R}^n)$. Since $g^\ast_\mu$ is of weak type $(1, 1)$ provided that $\mu > 2$ and $0 < \gamma < {\rm min}\{(\mu-2)n/2, 1\}$ (see [2, Theorem 1.1]), from Lemmas 3.4, 3.1, 3.2 and the generalized H$\ddot{\rm o}$lder inequality (2.1), we get that

$ \begin{equation}\begin{split}\label{gf} &\displaystyle\int_{{\Bbb R}^n}|g^\ast_\mu(f)(x)g(x)|dx \leq C_n\displaystyle\int_{{\Bbb R}^n}M_{\lambda_n}^\sharp (g^\ast_\mu(f))(x)Mg(x)dx\\ \leq& C_n\displaystyle\int_{{\Bbb R}^n}(1/\lambda_n)^{1/\delta} (g^\ast_\mu(f))^\sharp_\delta(x)Mg(x)dx \leq C_n\displaystyle\int_{{\Bbb R}^n}Mf(x)Mg(x)dx\\ \leq& C_n r_p\|Mf\|_{L^{p(\cdot)}({\Bbb R}^n)}\|Mg\|_{L^{p'(\cdot)}({\Bbb R}^n)} \leq C_p\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\|g\|_{L^{p'(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation} $

(3.2)

Here for the last inequality we have used the fact that if $p(\cdot)\in \mathcal{B}({\Bbb{ R}}^n)$, then $p'(\cdot)\in \mathcal{B}({\Bbb{ R}}^n)$, see [21, Proposition 2]. Thus we have

$ \begin{equation*}\begin{split} \|g^\ast_\mu(f)\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq \|g^\ast_\mu(f)\|_{L^{p(\cdot)}({\Bbb R}^n)}^0\leq C\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation*} $

By Proposition 3.1, this concludes the proof of Lemma 3.5.

We now state the main result of this section.

Theorem 3.1 Suppose $\mu > 2$, $0 < \gamma < {\rm min}\{(\mu-2)n/2, 1\}$ and $b_i\in {\rm BMO}(\mathbb R^n)$, $i=1, 2, \cdots, m$. If $p(\cdot)\in \mathcal{B}({\Bbb{ R}}^n)$, then $g^\ast_{\mu, \vec{b}}$ are bounded from $L^{p(\cdot)}({\Bbb R}^n)$ to itself.

Proof Let $f\in L_c^\infty$ and $g\in L^{p'(\cdot)}({\Bbb R}^n)\subset L_{\rm loc}^1({\Bbb R}^n)$. We show Theorem 3.1 by induction on $m$. For $m=1$, by Theorem 1.2 in [2, page 1850], $[b, g^\ast_\mu]f$ satisfies the assumptions of Lemma 3.4. Thus, as argued in (3.2), we deduce that

$ \begin{equation}\begin{split} \displaystyle\int_{{\Bbb R}^n}|[b, g^\ast_\mu](f)(x)g(x)|dx &\leq C_n\displaystyle\int_{{\Bbb R}^n}M_{\lambda_n}^\sharp ([b, g^\ast_\mu]f)(x)Mg(x)dx\\ &\leq C_n\displaystyle\int_{{\Bbb R}^n}(1/\lambda_n)^{1/\delta} ([b, g^\ast_\mu]f)^\sharp_\delta(x)Mg(x)dx\\ &\leq C_n\|b\|_\ast\displaystyle\int_{{\Bbb{ R}}^n}\bigg(M_{L({\rm log}L)}f(x)+M_\varepsilon(g^\ast_\mu(f))(x)\bigg)Mg(x)dx\\ &\leq C_n r_p(\|M^2f\|_{L^{p(\cdot)}}+\|M_\varepsilon(g^\ast_\mu(f))\|_{L^{p(\cdot)}})\|Mg\|_{L^{p'(\cdot)}({\Bbb R}^n)}\\ &\leq C_p\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\|g\|_{L^{p'(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation} $

(3.3)

This together with (3.1) yields

$ \begin{equation*}\begin{split} \|[b, g^\ast_\mu](f)\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq \|[b, g^\ast_\mu](f)\|_{L^{p(\cdot)}({\Bbb R}^n)}^0\leq C\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation*} $

Suppose now that the Theorem 3.1 is true for $m-1$. We will show that it is true for $m$. Once again by Theorem 1.2 in [2], according to Lemmas 3.4, 3.1, 3.2 and the generalized H$\ddot{\rm o}$lder inequality (2.1), we have

$ \begin{equation}\begin{split}\label{m1} &\displaystyle\int_{{\Bbb{ R}}^n}|(g^\ast_{\mu, \vec{b}}f)(x)g(x)|dx\\ \leq& C\displaystyle\int_{{\Bbb{ R}}^n}M_{\lambda_n}^\sharp(g^\ast_{\mu, \vec{b}}f)(x)Mg(x)dx\\ \leq& C\displaystyle\int_{{\Bbb{ R}}^n}(1/\lambda_n)^{1/\delta}(g^\ast_{\mu, \vec{b}}f)_\delta^\sharp(x)Mg(x)dx\\ \leq& C\displaystyle\int_{{\Bbb{ R}}^n}\bigg\{\|\vec{b}\|M_{L({\rm log}L)^m}f(x)+\sum\limits_{j=1}^m\sum\limits_{\sigma\in C_j^m} \|b_\sigma\|M_{\varepsilon}(g^\ast_{\mu, \vec{b}_{\sigma'}}f)(x)\bigg\}Mg(x)dx\\ \leq& C\displaystyle\int_{{\Bbb{ R}}^n}\bigg\{\|\vec{b}\|M_{L({\rm log}L)^m}f(x)+\sum\limits_{j=1}^m\sum\limits_{\sigma\in C_j^m} \|b_\sigma\|M_{L({\rm log}L)^{m-j}}f(x)\bigg\}Mg(x)dx\\ \leq &C\prod\limits_{j=1}^m\|b\|_\ast\displaystyle\int_{{\Bbb{ R}}^n}\sum\limits_{j=1}^mM_{L({\rm log}L)^{m-j}}f(x)Mg(x)dx\\ \leq& C\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}\|g\|_{L^{p'(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation} $

(3.4)

Now we obtain from (3.4) and (3.1) that

$ \begin{equation*}\begin{split} \|g^\ast_{\mu, \vec{b}}f\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq \|g^\ast_{\mu, \vec{b}}f\|_{L^{p(\cdot)}({\Bbb R}^n)}^0\leq C\|f\|_{L^{p(\cdot)}({\Bbb R}^n)}. \end{split}\end{equation*} $

By Proposition 3.1, this concludes the proof of Theorem 3.1.

The main purpose of this section is to further study the mapping properties of the multilinear commutators $g^\ast_{\mu, \vec{b}}$ in variable exponent Herz-type Hardy spaces. Before stating the main result, we give some definitions.

Let

$ B_k = \{x \in {\mathbb R} ^n: |x| \leqslant2^k\}, ~~ R_k = B_k \backslash B_{k-1} $

and $\chi_k = \chi_{R_k}$ be the characteristic function of the set $R_k$ for $k\in \mathbb{Z}$.

Definition 4.1 Let $p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n)$, $0 < q\leq\infty$ and $\alpha\in\Bbb R$. The homogeneous variable exponent Herz space $\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)$ consists of all $f\in L_{\rm loc}^{p(\cdot)}({{\Bbb{ R}}^n}\backslash\{0\})$ satisfying

$ \begin{equation*}\begin{split} \|f\|_{{\dot{K}_{p(\cdot)}^{\alpha, q}}({\Bbb{ R}}^n)}:= \bigg(\sum\limits_{k\in\mathbb{Z}}2^{\alpha kq}\| f\chi_k\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\bigg)^{1/q}<\infty \end{split}\end{equation*} $

with the usual modification when $q=\infty$.

For $x\in\mathbb{R}$, we denote by $[x]$ the largest integer less than or equal to $x$.

Definition 4.2 Suppose $\alpha\geq n\delta_2$, $p(\cdot)\in\mathcal{P}({\Bbb{R}}^n)$ and non-negative integer $s\geq [\alpha-n\delta_2]$. Let $b_i~(i=1, 2, \cdots, m)$ be alocally integrable function and $\vec{b}=(b_1, b_2, \cdots, b_m)$. A function $a(x)$ on ${\Bbb{R}}^n$ is said to be a central $(\alpha, p(\cdot), s; \vec{b})$-atom, if it satisfies

(ⅰ) ${\rm supp} a \subset \widetilde{B}:=\{x\in\Bbb{R}^n: |x| < r\}.$

(ⅱ) $\|a\|_{L^{p(\cdot)}({\Bbb R}^n)}\leq |\widetilde{B}|^{-\frac{\alpha}{n}}.$

(ⅲ) $\displaystyle\int_{\widetilde{B}}x^{\beta}a(x)\prod_{i\in\sigma}b_i(x)dx=0 \quad \mbox{for}\ |\beta|\leq s, \sigma\in C^m_j, j=0, 1, \cdots, m.$

Remark 4.1 It is easy to see that if $p(x)\equiv p$ is constant, then taking $\delta_2=1-1/p$, we can get the classical case, see [28].

A temperate distribution $f$ is said to belong to $H\dot{K}_{p(\cdot), \vec{b}}^{\alpha, q, s}({\Bbb{R}}^n)$, if it can be written as

$ \begin{equation*}\begin{split} f=\sum\limits_{j=-\infty}^{\infty}\lambda_j a_j, \ \mbox{in the sense of}\ \mathcal{S'}({\Bbb{ R}}^n), \end{split}\end{equation*} $

where $a_j$ is a central $(\alpha, p(\cdot), s; \vec{b})$-atom with support contained in $B_j$, $\lambda_j\in\mathbb{R}$ and

$ \sum |\lambda_j|^q<\infty. $

Moreover,

$ \begin{equation*}\begin{split} \|f\|_{H\dot{K}_{p(\cdot), \vec{b}}^{\alpha, q, s}({\Bbb{R}}^n)} \approx \inf\bigg(\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q \bigg)^{1/q}, \end{split}\end{equation*} $

where the infimum is taken over all above decompositions of $f$.

Our main result in this section can be stated as follows.

Theorem 4.1 Suppose $0 < \gamma < 1$ and $\mu > 3+2/n+2\gamma/n$. If $p(\cdot)\in \mathcal{P}({\Bbb{ R}}^n)\bigcap LH({\Bbb{ R}}^n)$, $0 < q < \infty$ and $n\delta_2\leq\alpha < n\delta_2+\gamma$, where $\delta_2$ is the constant appearing in Lemma 2.2. Then the multilinear commutators $g^\ast_{\mu, \vec{b}}$ map $H\dot{K}_{p(\cdot), \vec{b}}^{\alpha, q, 0}({\Bbb{R}}^n)$ into $\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)$.

Proof Let $a_j$ be a central $(\alpha, p(\cdot), 0; \vec{b})$-atom with support contained in $B_j$. We first restrict $0 < q\leq 1$. In this case, it suffices to show that

$ \|g^\ast_{\mu, \vec{b}}a_j\|_{\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)}\leq C. $

We write

$ \begin{equation*}\begin{split} \|g^\ast_{\mu, \vec{b}}a_j\|^q_{\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)} &=\sum\limits_{k=-\infty}^{j+2}2^{k\alpha q}\|\chi_kg^\ast_{\mu, \vec{b}}a_j\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)} +\sum\limits_{k=j+3}^{\infty}2^{k\alpha q}\|\chi_kg^\ast_{\mu, \vec{b}}a_j\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)}\\ &:=I+J. \end{split}\end{equation*} $

For $I$, by the boundedness of $g^\ast_{\mu, \vec{b}}$ in $L^{p(\cdot)}({\Bbb{ R}}^n)$, we obtain

$ \begin{equation*} I\leq C\sum\limits_{k=-\infty}^{j+2}2^{k\alpha q}\|a_j\|^q_{L^{p(\cdot)}({\Bbb{ R}}^n)} \leq C\sum\limits_{k=-\infty}^{j+2}2^{(k-j)\alpha q}\leq C. \end{equation*} $

We proceed now to estimate $J$. If $x\in R_k$, $y\in B_j$ and $k\geq j+3$, then $2|y| < |x|$. By the vanishing condition of $a_j$, we get that

$ \begin{align} & g_{\mu ,\vec{b}}^{*}({{a}_{j}})(x)={{(\int{{}}\int_{\mathbb{R}_{+}^{n+1}}{(}\frac{t}{t+|x-y|})}^{n\mu }}|\int_{{{B}_{j}}}{{{\psi }_{t}}}(y-z)\prod\limits_{i=1}^{m}{({{b}_{i}}(}x)-{{b}_{i}}(z)) \\ & {{a}_{j}}(z)dz{{|}^{2}}\frac{dydt}{{{t}^{n+1}}}{{)}^{1/2}} \\ & \le {{(\int{{}}\int_{\mathbb{R}_{+}^{n+1}}{(}\frac{t}{t+|x-y|})}^{n\mu }}{{(\int_{{{B}_{j}}}{|}{{\psi }_{t}}(y-z)-{{\psi }_{t}}(y)|\prod\limits_{i=1}^{m}{|}{{b}_{i}}(x)-{{b}_{i}}(z)||{{a}_{j}}(z)|dz)}^{2}} \\ & \frac{dydt}{{{t}^{n+1}}}{{)}^{1/2}} \\ & \le {{(\int{{}}\int_{\mathbb{R}_{+}^{n+1}}{(}\frac{t}{t+|x-y|})}^{n\mu }}\frac{{{t}^{1-n}}dydt}{{{(t+|y|)}^{2(n+\gamma +1)}}}{{)}^{1/2}}\int_{{{B}_{j}}}{|}z{{|}^{\gamma }}|{{a}_{j}}(z)| \\ & \prod\limits_{i=1}^{m}{|}{{b}_{i}}(x)-{{b}_{i}}(z)|dz. \\ \end{align} $

(4.1)

If $\mu > 3+2/n+2\gamma/n$, using the same estimates in [3, page 5], then we have

$ \begin{equation}\begin{split}\label{4.2} &\displaystyle\int_{{\mathbb {R}}^n}\bigg(\frac{t}{t+|x-y|}\bigg)^{n\mu}\frac{t^{1-n}dy}{(t+|y|)^{2(n+\gamma+1)}}\\ \leq& C\sum\limits_{k=1}^\infty2^{k(3n+2+2\gamma-n\mu)}\frac{t}{(t+|x|)^{2(n+\gamma+1)}}\\ \leq& C\frac{t}{(t+|x|)^{2(n+\gamma+1)}}. \end{split}\end{equation} $

(4.2)

Combining (4.1) and (4.2), we arrive at the estimate

$ \begin{equation}\begin{split}\label{4.3} g^\ast_{\mu, \vec{b}}(a_j)(x)&\leq C\bigg(\displaystyle\int_0^\infty\frac{tdt}{(t+|x|)^{2(n+\gamma+1)}}\bigg)^{1/2} \displaystyle\int_{B_j}|z|^\gamma|a_j(z)|\prod^m_{i=1}|b_i(x)-b_i(z)|dz\\ &\leq C|x|^{-n-\gamma}\displaystyle\int_{B_j}|z|^\gamma|a_j(z)|\prod^m_{i=1}|b_i(x)-b_i(z)|dz\\ &\leq C2^{(j-k)\gamma-kn}\displaystyle\int_{B_j}|a_j(z)|\prod^m_{i=1}|b_i(x)-b_i(z)|dz. \end{split}\end{equation} $

(4.3)

Let $\lambda_i=(b_i)_{B_j}$. An application of (4.3), (2.1), Lemmas 2.5, 2.1 and 2.2 give

$ \begin{equation}\begin{split}\label{4.4} &\|\chi_kg^\ast_{\mu, \vec{b}}a_j\|_{L^{p(\cdot)}({\Bbb R}^n)}\\ \leq& C2^{(j-k)\gamma-kn}\|a_j\|_{L^{p(\cdot)}({\Bbb R}^n)} \sum\limits_{i=0}^m\sum\limits_{\sigma\in C_i^m} \|(b(\cdot)-{\vec \lambda})_\sigma\chi_k\|_{L^{p(\cdot)}({\Bbb R}^n)} \|(b(\cdot)-\vec{\lambda})_{\sigma'}\chi_j\|_{L^{p'(\cdot)}({\Bbb R}^n)}\\ \leq& C(k-j)^m\prod^m_{i=1}\|b_i\|_{\ast}2^{(j-k)\gamma-j\alpha}2^{-kn}\|\chi_{B_k}\|_{L^{p(\cdot)}({\Bbb R}^n)} \|\chi_{B_j}\|_{L^{p'(\cdot)}({\Bbb R}^n)}\\ \leq& C(k-j)^m2^{(j-k)\gamma-j\alpha}\frac{\|\chi_{B_j}\|_{L^{p'(\cdot)}({\Bbb R}^n)}}{\|\chi_{B_k}\|_{L^{p'(\cdot)}({\Bbb R}^n)}}\\ \leq& C(k-j)^m2^{(j-k)(\gamma+n\delta_2)-j\alpha}. \end{split}\end{equation} $

(4.4)

Consequently, by the condition $\gamma+n\delta_2-\alpha > 0$, we have

$ \begin{equation*}\begin{split} J\leq C\sum\limits_{k=j+3}^{\infty}(k-j)^{mq}2^{(j-k)(\gamma+n\delta_2-\alpha)q}\leq C. \end{split}\end{equation*} $

Now let $1 < q < \infty$ and $f=\sum\limits_{j=-\infty}^{\infty}\lambda_ja_j$. For convenience below we put $\sigma=\gamma+n\delta_2-\alpha$, then we have $\sigma > 0$. From (4.4) and the $L^{p(\cdot)}({\Bbb{ R}}^n)$-boundedness of the multilinear commutators $g^\ast_{\mu, \vec{b}}$, it follows that

$ \begin{equation*}\begin{split} \|g^\ast_{\mu, \vec{b}}f\|_{\dot{K}_{p(\cdot)}^{\alpha, q}({\Bbb{ R}}^n)} &\leq \bigg\{\sum\limits_{k=-\infty}^{\infty}2^{\alpha kq}\bigg(\sum\limits_{j=-\infty}^{k-3} |\lambda_j|\|\chi_kg^\ast_{\mu, \vec{b}}a_j\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\bigg\}^{1/q}\\ &\quad+\bigg\{\sum\limits_{k=-\infty}^{\infty}2^{\alpha kq}\bigg(\sum\limits_{j=k-2}^{\infty} |\lambda_j|\|\chi_kg^\ast_{\mu, \vec{b}}a_j\|_{L^{p(\cdot)}({\Bbb R}^n)}\bigg)^q\bigg\}^{1/q}\\ &\leq C\bigg\{\sum\limits_{k=-\infty}^{\infty}\bigg(\sum\limits_{j=-\infty}^{k-3} |\lambda_j|(k-j)^{m}2^{(j-k)\sigma}\bigg)^q\bigg\}^{1/q}\\ &\quad+C\bigg\{\sum\limits_{k=-\infty}^{\infty}\bigg(\sum\limits_{j=k-2}^{\infty} |\lambda_j|2^{(k-j)\alpha}\bigg)^q\bigg\}^{1/q}\\ &\leq C\bigg\{\sum\limits_{k=-\infty}^{\infty}\bigg(\sum\limits_{j=-\infty}^{k-3} |\lambda_j|^q2^{(j-k)\sigma q/2}\bigg) \bigg(\sum\limits_{j=-\infty}^{k-3}(k-j)^{mq'} 2^{(j-k)\sigma q'/2}\bigg)^{q/q'}\bigg\}^{1/q}\\ &\quad+C\bigg\{\sum\limits_{k=-\infty}^{\infty}\bigg(\sum\limits_{j=k-2}^{\infty} |\lambda_j|^q2^{(k-j)\alpha q/2}\bigg)\bigg(\sum\limits_{j=k-2}^{\infty} 2^{(j-k)\alpha q'/2}\bigg)^{q/q'}\bigg\}^{1/q}\\ & \le C{{\{\sum\limits_{j=-\infty }^{\infty }{|}{{\lambda }_{j}}{{|}^{q}}(\sum\limits_{k=j+3}^{\infty }{{{2}^{(j-k)\sigma q/2}}})\}}^{1/q}}+ \\ & C{{\{\sum\limits_{j=-\infty }^{\infty }{|}{{\lambda }_{j}}{{|}^{q}}(\sum\limits_{k=-\infty }^{j+2}{{{2}^{(k-j)\alpha q/2}}})\}}^{1/q}}\\ &\leq C\bigg(\sum\limits_{j=-\infty}^{\infty}|\lambda_j|^q \bigg)^{1/q}. \end{split}\end{equation*} $

This completes the proof of Theorem 4.1.