1 Introduction

The study of Hardy spaces began in the early 1900s in the context of Fourier series and complex analysis in one variable. It was not until 1960 when the groundbreaking work in Hardy space theory in \({\mathbb {R}}^n\) came from Coifman [1], Coifman and Weiss [2] and Fefferman and Stein [9]. The classical Hardy space can be characterized by the Littlewood–Paley–Stein square functions, maximal functions and atomic decompositions. Especially, atomic decomposition is a significant tool in harmonic analysis and wavelet analysis for the study of function spaces and the operators acting on these spaces. Atomic decomposition was first introduced by Coifman [1] in one dimension in 1974 and later was extended to higher dimensions by Latter [18]. As we all know, atomic decompositions of Hardy spaces play an important role in the boundedness of operators on Hardy spaces and it is commonly sufficient to check that atoms are mapped into bounded elements of quasi-Banach spaces.

Another stage in the progress of the theory of Hardy spaces was done by Nakai and Sawano [21] and Cruz-Uribe and Wang [7] when they independently considered Hardy spaces with variable exponents. It is quiet different to obtain the boundedness of operators on Hardy spaces with variable exponents. It is not sufficient to show the \(H^{p(\cdot )}\)-boundedness merely by checking the action of the operators on \(H^{p(\cdot )}\)-atoms. In the linear theory, the boundedness of some operators on variable Hardy spaces have been established in [7, 13, 21, 25, 33] as applications of the corresponding atomic decompositions theories.

In more recent years, the study of multilinear operators on Hardy space theory has received increasing attention by many authors, see for example [12, 14, 15]. While the multilinear operators worked well on the product of Hardy spaces, it is surprising that these similar results in the setting of variable exponents were unknown for a long time. The boundedness of some multilinear operators on products of classical Hardy spaces was investigated by Grafakos and Kalton [12] and Li et al. [19]. In Tan et al. [29] studied some multilinear operators are bounded on variable Lebesgue spaces \(L^{p(\cdot )}\). However, there are some subtle difficulties in proving the boundedness results when we deal with the \(H^{p(\cdot )}\)-norm. The main goal of this article is to show that multilinear fractional type operators are bounded from product of Hardy spaces with variable exponents into Lebesgue spaces with variable exponents by using the atomic decompositions theory. We also remark that some boundedness of many types of multilinear operators on some variable Hardy spaces have established in [5, 27, 28, 30]. After we were completing this paper we learned that via using Rubio de Francia extrapolation the similar result had been also established independently by Cruz-Uribe et al. [6] independently though our approaches are very different.

First we recall the definition of Lebesgue spaces with variable exponents. Note that the variable exponent spaces, such as the variable Lebesgue spaces and the variable Sobolev spaces, were studied by a substantial number of researchers (see, for instance, [4, 17]). For any Lebesgue measurable function \(p(\cdot ): {\mathbb {R}}^n\rightarrow (0,\infty ]\) and for any measurable subset \(E\subset {\mathbb {R}}^n\), we denote \(p^-(E)= \inf _{x\in E}p(x)\) and \(p^+(E)= \sup _{x\in E}p(x).\) Especially, we denote \(p^-=p^{-}({\mathbb {R}}^n)\) and \(p^+=p^{+}({\mathbb {R}}^n)\). Let \(p(\cdot )\): \({\mathbb {R}}^n\rightarrow (0,\infty )\) be a measurable function with \(0<p^-\le p^+ <\infty \) and \({\mathcal {P}}^0\) be the set of all these \(p(\cdot )\). Let \({\mathcal {P}}\) denote the set of all measurable functions \(p(\cdot ):{\mathbb {R}}^n \rightarrow [1,\infty ) \) such that \(1<p^-\le p^+ <\infty .\)

Definition 1.1

Let \(p(\cdot ):{\mathbb {R}}^n\rightarrow (0,\infty ]\) be a Lebesgue measurable function. The variable Lebesgue space \(L^{p(\cdot )}\) consisits of all Lebesgue measurable functions f, for which the quantity \(\int _{{\mathbb {R}}^n}|\varepsilon f(x)|^{p(x)}dx\) is finite for some \(\varepsilon >0\) and

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}}=\inf {\left\{ \lambda >0: \int _{{\mathbb {R}}^n}\left( \frac{|f(x)|}{\lambda }\right) ^{p(x)}dx\le 1 \right\} }. \end{aligned}$$

The variable Lebesgue spaces were first established by Orlicz [23]. Two decades later, Nakano [22] first systematically studied modular function spaces which include the variable Lebesgue spaces as specific examples. However, the modern development started with the paper Kováčik and Rákosník [17]. We also refer to [10] for some basic theory of these spaces. As a special case of the theory of Nakano and Luxemberg, we see that \(L^{p(\cdot )}\) is a quasi-normed space. Especially, when \(p^-\ge 1\), \(L^{p(\cdot )}\) is a Banach space.

We also recall the following class of exponent function, which can be found in [8]. Let \({\mathcal {B}}\) be the set of \(p(\cdot )\in {\mathcal {P}}\) such that the Hardy–littlewood maximal operator M is bounded on \(L^{p(\cdot )}\). An important subset of \({\mathcal {B}}\) is the LH condition. In the study of variable exponent function spaces it is common to assume that the exponent function \(p(\cdot )\) satisfies LH condition. We say that \(p(\cdot )\in LH\), if \(p(\cdot )\) satisfies

$$\begin{aligned} |p(x)-p(y)|\le \frac{C}{-\log (|x-y|)} ,\quad |x-y| \le 1/2 \end{aligned}$$

and

$$\begin{aligned} |p(x)-p(y)|\le \frac{C}{\log |x|+e} ,\quad |y|\ge |x|. \end{aligned}$$

It is well known that \(p(\cdot )\in {\mathcal {B}}\) if \(p(\cdot )\in {\mathcal {P}}\cap LH.\) Moreover, example shows that the above LH conditions are necessary in certain sense, see Pick and R\(\mathring{u}\)z̆ic̆ka [24] for more details. We now recall the definition of variable Hardy spaces \({H}^{p(\cdot )}\) as follows.

Definition 1.2

 ([7, 21]) Let \(f\in \mathcal {S'}\), \(\psi \in {\mathcal {S}}\), \(p(\cdot )\in {{\mathcal {P}}^0}\) and \(\psi _t(x)=t^{-n}\psi (t^{-1}x)\), \(x\in {\mathbb {R}}^n\). Denote by \({\mathcal {M}}\) the grand maximal operator given by \({\mathcal {M}}f(x)= \sup \{|\psi _t*f(x)|: t>0,\psi \in {\mathcal {F}}_N\}\) for any fixed large integer N, where \({\mathcal {F}}_N=\{\varphi \in {\mathcal {S}}:\int \varphi (x)dx=1,\sum _{|\alpha |\le N}\sup (1+|x|)^N|\partial ^\alpha \varphi (x)|\le 1\}\). The variable Hardy space \({H}^{p(\cdot )}\) is the set of all \(f\in {\mathcal {S}}^\prime \), for which the quantity

$$\begin{aligned} \Vert f\Vert _{{H}^{p(\cdot )}}=\Vert {\mathcal {M}}f\Vert _{{L}^{p(\cdot )}}<\infty . \end{aligned}$$

Throughout this paper, C or c will denote a positive constant that may vary at each occurrence but is independent to the essential variables, and \(A\sim B\) means that there are constants \(C_1>0\) and \(C_2>0\) independent of the essential variables such that \(C_1B\le A\le C_2B\). Given a measurable set \(S\subset {\mathbb {R}}^n\), |S| denotes the Lebesgue measure and \(\chi _S\) means the characteristic function. For a cube Q, let \(Q^*\) denote with the same center and \(2\sqrt{n}\) its side length, i.e. \(l(Q^*)=100\sqrt{n}l(Q)\). The symbols \({\mathcal {S}}\) and \({\mathcal {S}}'\) denote the class of Schwartz functions and tempered functions, respectively. As usual, for a function \(\psi \) on \({\mathbb {R}}^n\), \(\psi _t(x)=t^{-n}\psi (t^{-1}x)\). We also use the notations \(j\wedge j'=\min \{j,j'\}\) and \(j\vee j'=\max \{j,j'\}\).

In what follows, we recall the new atoms for Hardy spaces with variable exponents \({H}^{p(\cdot )}\), which is introduced in [25]. Define

$$\begin{aligned} p_-=p^-\wedge 1,\quad d_{p(\cdot )}\equiv [n/{p_-}-n]\vee -1 \end{aligned}$$

for \(p\in (0,\infty ).\) Let \(p(\cdot ): {\mathbb {R}}^n\rightarrow (0,\infty )\), \(0< p^{-}\le p^+\le \infty \). Fix an integer \(d\ge d_{p(\cdot )}\) and \(1<q\le \infty \). A function a on \({\mathbb {R}}^n\) is called a \((p(\cdot ),q)\)-atom, if there exists a cube Q such that \(\mathrm{supp}\,a\subset Q\); \(\Vert a\Vert _{L^q}\le {|Q|^{1/q}}\); \(\int _{{\mathbb {R}}^n} a(x)x^\alpha dx=0\; \mathrm{for}\;|\alpha | \le d\). Especially, the first two conditions can be replaced by \(|a|\le \chi _Q\) when \(q=\infty \). The atomic decomposition of Hardy spaces with variable exponents was first established independently in [7, 21]. Moreover, Yang et al. [31, 32, 34] established some equivalent characterizations of Hardy spaces with variable exponents. Recently, the author revisited the atomic decomposition for \(H^{p(\cdot )}\) via the Littlewood–Paley–Stein theory in [26]. In this paper we will use the following decomposition results obtained by Sawano [25], which extends and sharp the ones of above papers.

Theorem 1.1

[25] Let \(p(\cdot )\in LH\cap {\mathcal {P}}^0\) and \(q>(p^+\vee 1)\). Suppose that \(d\ge d_{p(\cdot )}\) and \(s\in (0,\infty )\). If \(f\in H^{p(\cdot )}\), there exists sequences of \((p(\cdot ),\infty )-\)atoms \(\{a_j\}\) and scalars \(\{\lambda _j\}\) such that \(f=\sum _{j=1}^\infty \lambda _ja_j\) in \(H^{p(\cdot )}\cap L^q\) and that

$$\begin{aligned} \left\| \left\{ \sum _{j=1}^\infty \left( {\lambda _j \chi _{Q_j}}\right) ^{s} \right\} ^{\frac{1}{s}}\right\| _{L^{p(\cdot )}} \le C\Vert f\Vert _{{H}^{p(\cdot )}}. \end{aligned}$$

The multilinear fractional type operators are natural generalization of linear ones. Their earliest version was originated on the work of Grafakos [11], in which he studied the multilinear fractional integral defined by

$$\begin{aligned} {\bar{I}}_\alpha (\mathbf {f})(x)=\int _{{\mathbb {R}}^n} \frac{1}{|y|^{n-\alpha }}\prod _{i=1}^mf_i(x-\theta _iy)dy, \end{aligned}$$

where \(\theta _i(i=1,\ldots ,m)\) are fixed distinct and nonzero real numbers and \(0<\alpha <n.\) Later on, Kenig and Stein [16] established the boundedness of another type of multilinear fractional integral \({I}_{\alpha ,A}\) on product of Lebesgue spaces. \({I}_{\alpha ,A}(\mathbf {f})\) is defined by

$$\begin{aligned} {I}_{\alpha ,A}(\mathbf {f})(x)=\int _{({\mathbb {R}}^n)^m} \frac{1}{|(y_1,\ldots ,y_m)|^{mn-\alpha }}\prod _{i=1}^mf_i( \ell _i(y_1,\ldots ,y_m,x))dy_i, \end{aligned}$$

where \(\ell _i\) is a linear combination of \(y_j\)’s and x depending on the matrix A. In [20], Lin and Lu obtained \(I_{\alpha ,A}\) is bounded from product of Hardy spaces to Lebegue spaces when \(\ell _i(y_1, \ldots , y_m, x)=x-y_i\). We denote this multilinear fractional type integral operators by \(I_\alpha \), namely,

$$\begin{aligned} {I}_{\alpha }(\mathbf {f})(x)=\int _{({\mathbb {R}}^n)^m} \frac{1}{|(y_1,\ldots ,y_m)|^{mn-\alpha }}\prod _{i=1}^mf_i (x-y_i)dy_i. \end{aligned}$$

For convenience, we also denote \(K_\alpha (y_1,\ldots ,y_m)=\frac{1}{|(y_1,\ldots ,y_m)|^{mn-\alpha }}\).

The main goal of this paper is to prove the following result:

Theorem 1.2

Let \(0<\alpha <n\). Suppose that \(p_1(\cdot ),\ldots ,p_m(\cdot ) \in LH\cap {\mathcal {P}}^0\) and \(q(\cdot )\in {\mathcal {P}}^0\) be Lebesgue measure functions satisfying

$$\begin{aligned} \frac{1}{p_1(x)}+\cdots +\frac{1}{p_m(x)}-\frac{\alpha }{n}=\frac{1}{q(x)}, \quad \quad x\in {\mathbb {R}}^n. \end{aligned}$$
(1.1)

Then \(I_\alpha \) can be extended to a bounded operator from \(\prod _{j=1}^m H^{p_j(\cdot )}\) into \(L^{q(\cdot )}\).

2 Proof of Theorem 1.2

In this section, we will discuss the boundedness of multilinear fractional type operators on product of Hardy spaces with variable exponents. First we introduce some necessary notations and requisite lemmas. The following generalized Hölder inequality on variable Lebesgue spaces can be found in [3] or [29].

Lemma 2.1

Given exponent function \(p_i(\cdot )\in {\mathcal {P}}^0,\) define \(p(\cdot )\in {\mathcal {P}}^0\) by

$$\begin{aligned} \frac{1}{p(x)}=\sum _{i=1}^m\frac{1}{p_i(x)}, \end{aligned}$$

where \(i=1,\ldots ,m.\) Then for all \(f_i\in L^{p_i(\cdot )}\) and \(f_1\cdots f_m\in L^{p(\cdot )}\) and

$$\begin{aligned} \Vert \prod _{i=1}^mf_i\Vert _{p(\cdot )}\le C\prod _{i=1}^m\Vert f_i\Vert _{p_i(\cdot )}. \end{aligned}$$

Lemma 2.2

([29]) Let \(m\in {\mathbb {N}}\),

$$\begin{aligned} \frac{1}{s(x)}=\sum _{i=1}^m\frac{1}{r_i(x)}-\frac{\alpha }{n}, x\in {\mathbb {R}}^n, \end{aligned}$$

with \(0<\alpha <mn\), \(1< r_i\le \infty \). Then

$$\begin{aligned} \Vert I_\alpha (\mathbf {f})\Vert _{s(\cdot )}\le C\prod _{i=1}^m\Vert f_i\Vert _{r_i(\cdot )}. \end{aligned}$$

We also need the following boundedness of the vector-valued fractional maximal operators on variable Lebesgue spaces whose proof can be found in [3]. Let \(0\le \alpha <n\), we define

$$\begin{aligned} M_\alpha f(x)=\sup _{Q\ni x}\frac{1}{|Q|^{1-\alpha /n}}\int _Q|f(y)|dy. \end{aligned}$$

Lemma 2.3

Let \(0\le \alpha <n\), \(p(\cdot ),q(\cdot )\in {\mathcal {B}}\) be such that \(p^+<\frac{n}{\alpha }\) and

$$\begin{aligned} \frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha }{n},\quad x\in {\mathbb {R}}^n. \end{aligned}$$

If \(q(\cdot )(n-\alpha )/n\in {\mathcal {B}}\), then for any \(q>1\), \(f=\{f_i\}_{i\in {\mathbb {Z}}}\), \(f_i\in L_{loc}\), \(i\in {\mathbb {Z}}\)

$$\begin{aligned} \Vert \Vert \mathbb {M}_{\alpha }(f)\Vert _{l^q}\Vert _{L^{q(\cdot )}}\le C\Vert \Vert f||_{l^q}\Vert _{L^{p(\cdot )}}, \end{aligned}$$

where \(\mathbb {M}_{\alpha }(f)=\{M_\alpha (f_i)\}_{i\in {\mathbb {Z}}}\).

Lemma 2.4

[17] Let \(p(\cdot )\in {\mathcal {P}}\), \(f\in L^{p(\cdot )}\) and \(g\in L^{p'(\cdot )}\), then fg is integrable on \({\mathbb {R}}^n\) and

$$\begin{aligned} \int _{{\mathbb {R}}^n}|f(x)g(x)|dx\le r_p\Vert f\Vert _{L^{p(\cdot )}} \Vert g\Vert _{L^{p'(\cdot )}}, \end{aligned}$$

where \(r_p= 1+1/p^--1/p^+.\) Moreover, for all \(g\in L^{p'(\cdot )}\) such that \(\Vert g\Vert _{L^{p'(\cdot )}}\le 1\) we get that

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}}\le \sup _{g}|\int _{{\mathbb {R}}^n}f(x)g(x)dx| \le r_p\Vert f\Vert _{L^{p(\cdot )}}. \end{aligned}$$

We now give the proof of our main result.

Proof of Theorem 1.2

Observe that \(p_j(\cdot )\in LH\cap {\mathcal {P}}^0\) and choose that \(\bar{q}>(p_j^+\vee 1)\), \(j=1,\ldots ,m\). By Theorem 1.1, for each \(f_j\in {H}^{p_j(\cdot )}\cap {L}^{{{\bar{q}}}}\), \(j=1,\ldots ,m\), \(f_j\) admits an atomic decomposition: Suppose that \(d_j\ge d_{p_j(\cdot )}\) and \(s\in (0,\infty )\). If \(f\in H^{p_j(\cdot )}\), there exists sequences of \((p_j(\cdot ),\infty )-\)atoms \(\{a_{j,k_j}\}\) and scalars \(\{\lambda _{j,k_j}\}\)

such that \(f_j=\sum _{k_j=1}^\infty \lambda _{j,k_j}a_{j,k_j}\) in \(H^{p_j(\cdot )}\cap L^{{{\bar{q}}}}\) and that

$$\begin{aligned} \left\| \left\{ \sum _{k_j=1}^\infty \left( {\lambda _{j,k_j} \chi _{Q_{j,k_j}}}\right) ^{s} \right\} ^{\frac{1}{s}}\right\| _{L^{p_j(\cdot )}} \le C\Vert f_j\Vert _{{H}^{p_j(\cdot )}}. \end{aligned}$$

For the decomposition of \(f_j\in {H}^{p_j(\cdot )}\cap {L}^{{{\bar{q}}}}\), \(j=1,\ldots ,m\), so we can write

$$\begin{aligned} I_\alpha (\mathbf {f})(x)=\sum _{k_1}\cdots \sum _{k_m}\lambda _{1,k_1} \cdots \lambda _{m,k_m}I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x) \end{aligned}$$

in the sense of distributions.

Fixed \(k_1,\ldots ,k_m\), there are two cases for \(x\in {\mathbb {R}}^n\).

  1. Case 1:

    \(x\in Q^*_{1,k_1}\cap \cdots \cap Q^*_{m,k_m}\).

  2. Case 2:

    \(x\in Q^{*,c}_{1,k_1}\cup \cdots \cup Q^{*,c}_{m,k_m}\).

Then we have

$$\begin{aligned} |I_\alpha (\mathbf {f})(x)|&\le \sum _{k_1}\cdots \sum _{k_m}|\lambda _{1,k_1}| \cdots |\lambda _{m,k_m}||I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)| \chi _{Q^*_{1,k_1}\cap \cdots \cap Q^*_{m,k_m}}(x)\\&\quad +\sum _{k_1}\cdots \sum _{k_m}|\lambda _{1,k_1}| \cdots |\lambda _{m,k_m}||I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)| \chi _{Q^{*,c}_{1,k_1}\cup \cdots \cup Q^{*,c}_{m,k_m}}(x)\\&=I_1(x)+I_2(x). \end{aligned}$$

First, we consider the estimate of \(I_1(x)\). We will show that

$$\begin{aligned} \begin{aligned} \Vert I_1\Vert _{L^{q(\cdot )}} \le C\prod _{j=1}^m\Vert f_j\Vert _{{H}^{p_j(\cdot )}}. \end{aligned} \end{aligned}$$
(2.1)

For fixed \(k_1,\ldots ,k_m\), assume that \(Q^*_{1,k_1}\cap \cdots \cap Q^*_{m,k_m}\ne 0\), otherwise there is nothing to prove. Without loss of generality, suppose that \(Q_{1,k_1}\) has the smallest size among all these cubes. We can pick a cube \(G_{k_1,\ldots ,k_m}\) such that

$$\begin{aligned} Q^*_{1,k_1}\cap \cdots \cap Q^*_{m,k_m}\subset G_{k_1,\ldots ,k_m}\subset G^*_{k_1,\ldots ,k_m}\subset Q^{**}_{1,k_1}\cap \cdots \cap Q^{**}_{m,k_m} \end{aligned}$$

and \(|G_{k_1,\ldots ,k_m}|\ge C|Q_{1,k_1}|\).

Denote \(H(x):=|I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)| \chi _{Q_{1,k_1}^*\cap \cdots \cap Q_{m,k_m}^*}(x)\). Obviously,

$$\begin{aligned} \text{ supp } H(x)\subset {Q_{1,k_1}^*\cap \cdots \cap Q_{m,k_m}^*}\subset G_{k_1,\ldots ,k_m}. \end{aligned}$$

By Lemma 2.2, since \(I_\alpha \) maps \(L^{r_1}\times L^\infty \times \cdots \times L^\infty \) into \(L^{s}\) (\(s>1\) and \(\frac{1}{s}=\frac{1}{r_1}-\frac{\alpha }{n}\)), we get that

$$\begin{aligned} \begin{aligned} \Vert H\Vert _{L^{s}}&\le \Vert I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})\Vert _{L^{s}}\\&\le C\Vert a_{1,k_1}\Vert _{L^{r_1}}\Vert a_{2,k_2}\Vert _{L^\infty } \cdots \Vert a_{m,k_m}\Vert _{L^\infty }\\&\le C|Q_{1,k_1}|^{\frac{1}{r_1}} \Vert a_{2,k_2}\Vert _{L^\infty }\Vert a_{m,k_m}\Vert _{L^\infty }\\&\le C |Q_{1,k_1}|^{\frac{1}{r_1}}\\&\le C |G_{k_1,\ldots ,k_m}|^{\frac{1}{r_1}}. \end{aligned} \end{aligned}$$
(2.2)

For any \(g\in L^{(q(\cdot )/{q^-})'}\) with \(\Vert g\Vert _{L^{(q(\cdot )/{q^-}})'}\le 1\), by Hölder’s inequality and (2.2) we find that

$$\begin{aligned}&\left| \int _{{\mathbb {R}}^n}H(x)^{q^-}g(x)dx\right| \le \Vert H^{q^-}\Vert _{L^{s/q^-}}\Vert g\Vert _{L^{(s/q)'}}\\&\quad \le C |G_{k_1,\ldots ,k_m}|^{\frac{q^-}{r_1}} \left( \int _{G_{k_1,\ldots ,k_m}}|g(x)|^{(s/q^-)'}dx\right) ^{\frac{1}{(s/q^-)'}}, \end{aligned}$$

where \((s/q^-)'\) is the conjugate of \(s/q^-\). Thus,

$$\begin{aligned}&\left| \int _{{\mathbb {R}}^n}H(x)^{q^-}g(x)dx\right| \\&\quad \le C |G_{k_1,\ldots ,k_m}|^{1+\frac{\alpha q^-}{n}} \left( \frac{1}{|G_{k_1,\ldots ,k_m}|}\int _{G_{k_1,\ldots ,k_m}} |g(x)|^{(s/q^-)'}dx\right) ^{\frac{1}{(s/q^-)'}}\\&\quad \le C |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}} |G_{k_1,\ldots ,k_m}| \inf _{x\in R_{k_1,k_2}} M(|g|^{(s/q^-)'})(x)^{\frac{1}{(s/q^-)'}}\\&\quad \le C |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}} \int _{G_{k_1,\ldots ,k_m}}(M(|g|^{(s/q^-)'}))^{\frac{1}{(s/q^-)'}}(x)dx. \end{aligned}$$

When \(0<q^-\le 1\), using Lemma 2.4 we obtain that

$$\begin{aligned} \Bigg |\int _{{\mathbb {R}}^n}&\left( \sum _{k_1,\ldots ,k_m}| \prod _{j=1}^m\lambda _{j,k_j}|^{q^-} |I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)|^{q^-} \chi _{Q_{1,k_1}^*\cap \cdots \cap Q_{m,k_m}^*}(x)\right) g(x)dx \Bigg |\\ \le&\,\sum _{k_1}\cdots \sum _{k_m}|\prod _{j=1}^m\lambda _{j,k_j}|^{q^-} \left| \int _{G_{k_1,\ldots ,k_m}} H(x)^{q^-}g(x)dx\right| \\ \le&\,C \sum _{k_1}\cdots \sum _{k_m}| \prod _{j=1}^m\lambda _{j,k_j}|^{q^-} |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}}\\&\times \int _{G_{k_1,\ldots ,k_m}}(M(|g|^{(s/q^-)'}))^{\frac{1}{(s/q^-)'}}(x)dx\\ =&\,C \int _{{\mathbb {R}}^n}\sum _{k_1}\cdots \sum _{k_m} \prod _{j=1}^m |\lambda _{j,k_j}|^{q^-}\chi _{G_{k_1,\ldots ,k_m}} |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}}\\&\times (M(|g|^{(s/q^-)'}))^{\frac{1}{(s/q^-)'}}(x)dx\\ \le&\, C \left\| \sum _{k_1}\cdots \sum _{k_m} \prod _{j=1}^m |\lambda _{j,k_j}|^{q^-} |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}} \chi _{G_{k_1,\ldots ,k_m}} \right\| _{L^{q(\cdot )/{q^-}}}\\&\times \Vert (M(|g|^{(s/q^-)'}))^{\frac{1}{(s/q^-)'}}\Vert _{L^{(q(\cdot )/{q^-}})'}\\ \le&\, C \left\| \sum _{k_1}\cdots \sum _{k_m} \prod _{j=1}^m |\lambda _{j,k_j}|^{q^-} |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}} \chi _{G_{k_1,\ldots ,k_m}} \right\| _{L^{q(\cdot )/{q^-}}}\\&\times \Vert (M(|g|^{(s/q^-)'}))\Vert ^{\frac{1}{(s/q^-)'}}_{L^{(q(\cdot )/{q^-})'/{(s/q^-)'}}}. \end{aligned}$$

Choose s large enough such that \((q(\cdot )/{q^-})'/{(s/q^-)'}>1\). Then by the fact that Hardy–Littlewood operator M is bounded on \(L^{q'(\cdot )/{(s/q^-)'q^-}}\) and \(\Vert g\Vert _{L^{(q(\cdot )/{q^-})'}}\le 1\), we know that

$$\begin{aligned} \Vert (M(|g|^{(s/q^-)'}))\Vert ^{\frac{1}{(s/q^-)'}}_{L^{(q(\cdot )/{q^-})'/{(s/q^-)'}}} \le C. \end{aligned}$$

Applying Lemma 2.4 again, we get that

$$\begin{aligned}&\Bigg \Vert \sum _{k_1}\cdots \sum _{k_m}| \lambda _{1,k_1}|^{q^-}\cdots |\lambda _{m,k_m}|^{q^-} |I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)|^{q^-} \chi _{Q_{1,k_1}^*\cap \cdots \cap Q_{m,k_m}^*}\Bigg \Vert _{L^{q(\cdot )/{q^-}}}\\&\le C \left\| \sum _{k_1}\cdots \sum _{k_m} \prod _{j=1}^m |\lambda _{j,k_j}|^{q^-} |G_{k_1,\ldots ,k_m}|^{\frac{\alpha q^-}{n}} \chi _{G_{k_1,\ldots ,k_m}} \right\| _{L^{q(\cdot )/{q^-}}}\\&\le C\left\| \prod _{j=1}^m \sum _{k_j} |\lambda _{j,k_j}|^{q^-} |Q_{j,k_j}|^{\frac{\alpha q^-}{mn}} \chi _{Q^{**}_{j,k_j}} \right\| _{L^{q(\cdot )/{q^-}}}. \end{aligned}$$

Denote \(\frac{1}{q_j(x)}=\frac{1}{p_j(x)}-\frac{\alpha }{mn}\) for any \(x\in {\mathbb {R}}^n\), \(j=1,\ldots ,m\). Then \(q_j(\cdot )\in {\mathcal {P}}^0\) and for any \(x\in {\mathbb {R}}^n\)

$$\begin{aligned} \frac{1}{q(x)}=\frac{1}{p_1(x)}+\cdots +\frac{1}{p_m(x)}-\frac{\alpha }{n} =\frac{1}{q_1(x)}+\cdots +\frac{1}{q_m}. \end{aligned}$$

By Lemma 2.1, we obtain

$$\begin{aligned} \begin{aligned} \left\| \prod _{j=1}^m \sum _{k_j} |\lambda _{j,k_j}|^{q^-} |Q_{j,k_j}|^{\frac{\alpha q^-}{mn}} \chi _{Q^{**}_{j,k_j}} \right\| _{L^{q(\cdot )/{q^-}}}^{\frac{1}{q^-}}&\le \prod _{j=1}^m\left\| \sum _{k_j} |\lambda _{j,k_j}|^{q^-} |Q_{j,k_j}|^{\frac{\alpha q^-}{mn}} \chi _{Q^{**}_{j,k_j}} \right\| _{L^{q_j(\cdot )/{q^-}}}^{\frac{1}{q^-}}. \end{aligned} \end{aligned}$$
(2.3)

Furthermore, it is easy to verify that

$$\begin{aligned} |Q_{j,k_j}|^{\frac{\alpha }{mn}}\chi _{Q^{**}_{j,k_j}}(x)\le CM_{{\alpha }{q^-}/2m}(\chi _{Q^{**}_{j,k_j}})^{\frac{2}{q^-}}(x). \end{aligned}$$

Applying Lemma 2.6, then we get that

$$\begin{aligned} \begin{aligned}&\left\| \sum _{k_j} |\lambda _{j,k_j}|^{q^-} |Q_{j,k_j}|^{\frac{\alpha q^-}{mn}} \chi _{Q^{**}_{j,k_j}} \right\| _{L^{q_j(\cdot )/{q^-}}}^{\frac{1}{q^-}} = \left\| \left( \sum _{k_j} \left( |\lambda _{j,k_j}| |Q_{j,k_j}|^{\frac{\alpha }{mn}} \chi _{Q^{**}_{j,k_j}}\right) ^{q^-}\right) ^{\frac{1}{q^-}} \right\| _{L^{q_j(\cdot )}}\\&\le C\left\| \left( \sum _{k_j} \left( |\lambda _{j,k_j}| M_{{\alpha }{q^-}/2m}(\chi _{Q^{**}_{j,k_j}})^{\frac{2}{q^-}} \right) ^{q^-}\right) ^{\frac{1}{q^-}} \right\| _{L^{q_j(\cdot )}} \\&= C\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|^{q^-} M_{{\alpha }{q^-}/2m}(\chi _{Q^{**}_{j,k_j}})^2 \right) ^{\frac{1}{q^-}} \right\| _{L^{q_j(\cdot )}}\\&\le C\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|^{q^-} M_{{\alpha }{q^-}/2m}(\chi _{Q^{**}_{j,k_j}})^2 \right) ^{\frac{1}{2}} \right\| ^{\frac{2}{q^-}}_{L^{2q_j(\cdot )/{q^-}}} \le C\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|^{q^-} \chi _{Q^{**}_{j,k_j}} \right) ^{\frac{1}{2}} \right\| ^{\frac{2}{q^-}}_{L^{2p_j(\cdot )/{q^-}}}. \end{aligned} \end{aligned}$$
(2.4)

Applying Fefferman-Stein vector value inequality on \(L^{2p_j(\cdot )/{q^-}}\), we get that

$$\begin{aligned} \begin{aligned}&\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|^{q^-} \chi _{Q^{**}_{j,k_j}} \right) ^{\frac{1}{2}} \right\| ^{\frac{2}{q^-}}_{L^{2p_j(\cdot )/{q^-}}} \le \left\| \left( \sum _{k_j} |\lambda _{j,k_j}|^{q^-} M{(\chi _{Q_{j,k_j}})^2} \right) ^{\frac{1}{2}} \right\| ^{\frac{2}{q^-}}_{L^{2p_j(\cdot )/{q^-}}}\\&\le C\left\| \left( \sum _{k_j} \left( |\lambda _{j,k_j}| \chi _{Q_{j,k_j}}\right) ^{q^-} \right) ^{\frac{1}{q^-}} \right\| _{L^{p_j(\cdot )}}\le \Vert f\Vert _{H^{p_j(\cdot )}}. \end{aligned} \end{aligned}$$
(2.5)

Therefore, when \(0<q^-\le 1\), by (2.3), (2.4) and (2.5) we have that

$$\begin{aligned} \begin{aligned} \Vert I_1\Vert _{L^{q(\cdot )}}&=\Vert (I_1)^{q^-}\Vert _{L^{q(\cdot )/{q^-}}}^{\frac{1}{q^-}}\\&\le C\left\| \prod _{j=1}^m \sum _{k_j} |\lambda _{j,k_j}|^{q^-} |Q_{j,k_j}|^{\frac{\alpha q^-}{mn}} \chi _{Q^{**}_{j,k_j}} \right\| _{L^{q(\cdot )/{q^-}}}^{\frac{1}{q^-}}\\&\le C\prod _{j=1}^m \Vert f\Vert _{H^{p_j(\cdot )}}. \end{aligned} \end{aligned}$$

When \(q^->1\), repeating similar but more easier argument, we can also get the desired result (2.1). In fact, we only need to replace \(p^-\) by 1 in the above proof. We omit the detail.

Secondly, we consider the estimate of \(I_2\). Let A be a nonempty subset of \(\{1,\ldots ,m\}\), and we denote the cardinality of A by |A|, then \(1\le |A|\le m\). Let \(A^c=\{1,\ldots ,m\}\backslash A.\) If \(A=\{1,\ldots ,m\}\), we define

$$\begin{aligned} (\cap _{j\in A}Q_{j,k_j}^{*,c}) \cap (\cap _{j\in A^c}Q_{j,k_j}^{*,c})= \cap _{j\in A}Q_{j,k_j}^{*,c}, \end{aligned}$$

then

$$\begin{aligned} Q_{1,k_1}^{*,c}\cup \cdots \cup Q_{m,k_m}^{*,c} =\cup _{A\subset \{1,\ldots ,m\}}((\cap _{j\in A}Q_{j,k_j}^{*,c}) \cap (\cap _{j\in A^c}Q_{j,k_j}^{*})). \end{aligned}$$

Set \(E_A=(\cap _{j\in A}Q_{j,k_j}^{*,c}) \cap (\cap _{j\in A^c}Q_{j,k_j}^{*}).\) For fixed A, assume that \(Q_{\tilde{j},k_{{\tilde{j}}}}\) is the smallest cubes in the set of cubes \(Q_{j,k_j}, j\in A\). Let \(z_{{\tilde{j}},k_{{\tilde{j}}}}\) is the center of the cube \(Q_{{\tilde{j}},k_{{\tilde{j}}}}\).

Denote \(K_\alpha (x,y_1,\ldots ,y_m) =|(x-y_1,\ldots ,x-y_m)|^{-mn+\alpha }\). Notice that for all \(|\beta |=d+1,\; \beta =(\beta _1,\ldots ,\beta _m)\)

$$\begin{aligned} \begin{aligned} |\partial _{y_1}^{\beta _1}\cdots \partial _{y_m}^{\beta _m} K_\alpha (x,y_1,\ldots ,y_m)| \le C|(x-y_1,\ldots ,x-y_m)|^{-mn+\alpha -|\beta |}. \end{aligned} \end{aligned}$$
(2.6)

Since \(a_{{\tilde{j}},k_{{\tilde{j}}}}\) has zero vanishing moment up to order \(d_j\), using the Taylor expansion we get

$$\begin{aligned} \begin{aligned}&I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x) \\&\quad = \int _{({\mathbb {R}}^n)^m}K_\alpha (x,y_1,\ldots ,y_m) a_{1,k_1}(y_1)\cdots a_{m,k_m}(y_m)d\mathbf {y}\\&\quad = \int _{({\mathbb {R}}^n)^{m-1}}\prod _{j\ne {\tilde{j}}}a_{j,k_j}(y_j) \int _{{\mathbb {R}}^n}[K_\alpha (x,y_1,\ldots ,y_m) -P_{z_{\tilde{j},k_{{\tilde{j}}}}}^d(x,y_1,\ldots ,y_m)] a_{{\tilde{j}},k_{{\tilde{j}}}}d\mathbf {y}\\&\quad = \int _{({\mathbb {R}}^n)^{m-1}}\prod _{j\ne {\tilde{j}}}a_{j,k_j}(y_j) \int _{{\mathbb {R}}^n}\sum _{|\gamma |=d+1} (\partial _{y_{\tilde{j}}}^\gamma K_\alpha )(x,y_1,\ldots ,\xi ,\ldots ,y_m)\frac{(y_{{\tilde{j}}} -z_{{\tilde{j}},k_{{\tilde{j}}}})^\gamma }{\gamma !} a_{{\tilde{j}}}(y_{\tilde{j}})d\mathbf {y} \end{aligned} \end{aligned}$$

for some \(\xi \) on the line segment joining \(y_{{\tilde{j}}}\) to \(z_{{\tilde{j}},k_{{\tilde{j}}}}\), where \(P_{z_{{\tilde{j}},k_{\tilde{j}}}}^d(x,y_1,\ldots ,y_m)\) is the Taylor polynomial of \(K_\alpha (x,y_1,\ldots ,y_m)\). Since \(x\in (Q_{{\tilde{j}},k_{\tilde{j}}}^*)^c\), we can easily obtain that \(|x-\xi |\ge \frac{1}{2}|x-z_{{\tilde{j}},k_{{\tilde{j}}}}|\). Similarly, \(|x-y_j|\ge \frac{1}{2}|x-z_{j,k_{j}}|\) for \(y_j\in Q_{j,k_j}\), \(j\in A\backslash \{{\tilde{j}}\}\).

Applying the estimate for the kernel \(K_\alpha \) which satisfies (2.6) and the size estimates for the new atoms yield

$$\begin{aligned} \begin{aligned}&\int _{({\mathbb {R}}^n)^{m-1}}\prod _{j\ne {\tilde{j}}}|a_{j,k_j}(y_j)| \int _{{\mathbb {R}}^n}\sum _{|\gamma |=d+1} |(\partial _{y_{\tilde{j}}}^\gamma K_\alpha )(x,y_1,\ldots ,\xi ,\ldots ,y_m)|\frac{|y_{\tilde{j}} -z_{{\tilde{j}},k_{{\tilde{j}}}}|^\gamma }{\gamma !} |a_{{\tilde{j}}}(y_{{\tilde{j}}})|d\mathbf {y}\\&\quad \le C\int _{({\mathbb {R}}^n)^{|A|}}\prod _{j\in A}|a_{j,k_j}(y_j)| \int _{({\mathbb {R}}^n)^{m-|A|}} \frac{|y_{{\tilde{j}}}-z_{\tilde{j},k_{{\tilde{j}}}}|^{d+1}}{(|x-\xi |+\sum _{j\ne \tilde{j}}|x-y_j|)^{mn+d+1-\alpha }} \prod _{j\in A^c}|a_{ j,k_{ j}}(y_{{\tilde{j}}})|d\mathbf {y}\\&\quad \le C\bigg (\prod _{j\in A}\Vert a_{j,k_j}\Vert _{L^1}\bigg ) \bigg (\prod _{j\in A^c}\Vert a_{ j,k_{ j}}\Vert _{L^\infty }\bigg ) \int _{({\mathbb {R}}^n)^{m-|A|}} \frac{|y_{{\tilde{j}}}-z_{\tilde{j},k_{{\tilde{j}}}}|^{d+1}}{(|x-\xi |+\sum _{j\ne \tilde{j}}|x-y_j|)^{mn+d+1-\alpha }} d\mathbf {y}_{A^c}\\&\quad \le C\bigg (\prod _{j\in A}\Vert a_{j,k_j}\Vert _{L^1}\bigg ) \bigg (\prod _{j\in A^c}\Vert a_{ j,k_{ j}}\Vert _{L^\infty }\bigg ) \frac{|y_{{\tilde{j}}}-z_{{\tilde{j}},k_{{\tilde{j}}}}|^{d+1}}{(\sum _{j\in A}|x-z_{j,k_j}|)^{mn+d+1-\alpha -n(m-|A|)}}\\&\quad \le C\bigg (\prod _{j\in A}{|Q_{j,k_j}|} \bigg ) \bigg (\prod _{j\in A^c} {\Vert a_{ j,k_{ j}}\Vert _{L^\infty }}\bigg ) \frac{|Q_{\tilde{j},k_{{\tilde{j}}}}|^{(d+1)/n}}{(\sum _{j\in A}|x-z_{j,k_j}|)^{mn+d+1-\alpha -n(m-|A|)}}. \end{aligned} \end{aligned}$$

Observe that \(x\in \cap _{j\in A}Q_{j,k_j}^{*,c}\), then we can found constant C such that \(|x-z_{j,k_j}|\ge C(|x-z_{j,k_j}|+l{(Q_{j,k_j})})\). On the other hand, using the fact that \(x\in \cap _{j\in A^c}Q_{j,k_j}^*\) yields that there exists a constant C such that \(|x-z_{j,k_j}|\le Cl{(Q_{j,k_j})}\) for \(j\in A^c\). Then we have that

$$\begin{aligned} \frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}} \ge C, \quad \text{ for }\quad j\in A^c. \end{aligned}$$

Moreover, since \(Q_{{\tilde{j}},k_{{\tilde{j}}}}\) is the smallest cube among \(\{Q_{j,l_j}\}_{j\in A}\), we have that

$$\begin{aligned} |Q_{\tilde{j},k_{{\tilde{j}}}}|\le \prod _{j\in A}|Q_{j,l_j}|^{\frac{1}{|A|}}. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned}&|I_\alpha (a_{1,k_1},\ldots ,a_{m,k_m})(x)|\\&\quad \le C\bigg (\prod _{j\in A}\frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}}\bigg ) \bigg (\prod _{j\in A^c} {\Vert a_{ j,k_{ j}}\Vert _{L^\infty }}\bigg )\\&\quad \le C\prod _{j\in A}\frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}} \prod _{j\in A^c}\frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}} \end{aligned} \end{aligned}$$
(2.7)

for all \(x\in E_A\).

Then applying the generalized Hölder inequality in variable Lebesgue spaces and the Fefferman-Stein inequality in Lemma 2.3 we obtain the estimate

$$\begin{aligned} \begin{aligned} \Vert I_2\Vert _{L^{q(\cdot )}}&\le C\Bigg \Vert \sum _{k_1}\cdots \sum _{k_m}\prod _{j=1}^m|\lambda _{j,{k_j}}| \sum _{A\subset \{1,\ldots ,m\}}\prod _{j\in A}\frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}}\\&\quad \times \prod _{j\in A^c}\frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}} \chi _{E_A}\Bigg \Vert _{L^{q(\cdot )}}\\&\quad \le C\sum _{A\subset \{1,\ldots ,m\}}\Bigg \Vert \prod _{j\in A}\sum _{k_j}|\lambda _{j,{k_j}}| \frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}}\\&\quad \times \prod _{j\in A^c}\sum _{k_j}|\lambda _{j,{k_j}}| \frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}} \chi _{E_A}\Bigg \Vert _{L^{q(\cdot )}}. \end{aligned} \end{aligned}$$
(2.8)

For \(x\in {\mathbb {R}}^n\), denote \(\frac{1}{s_j(x)}=\frac{1}{p_j(x)}-\frac{\alpha }{n|A|}\), \(j\in A\). Then \(0<r(x)<\infty \) and

$$\begin{aligned} \frac{1}{q(x)}=\sum _{j}^m\frac{1}{p_j(x)}-\frac{\alpha }{n} =\sum _{j\in A}\frac{1}{s_j(x)}+\sum _{j\in A^c}\frac{1}{p_j(x)}. \end{aligned}$$

For convenience, we denote that

$$\begin{aligned} U_A=\sum _{k_j}|\lambda _{j,{k_j}}| \frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}} \end{aligned}$$

and

$$\begin{aligned} U_{A^c}=\sum _{k_j}|\lambda _{j,{k_j}}| \frac{|Q_{j,k_j}|^{1+\frac{d+1}{n|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}}. \end{aligned}$$

Applying (2.8) and the generalized Hölder inequality with variable exponents \(s_j(\cdot )\;,j\in A\), \(p_j(\cdot )\;,j\in A^c\) and \(q(\cdot )\) yield that

$$\begin{aligned} \begin{aligned} \Vert I_2\Vert _{L^{q(\cdot )}}&\le C\sum _{A\subset \{1,\ldots ,m\}}\Bigg \Vert \prod _{j\in A}U_{A} \prod _{j\in A^c}U_{A^c}\chi _{E_A}\Bigg \Vert _{L^{q(\cdot )}}\\&\le C\sum _{A\subset \{1,\ldots ,m\}} \left( \prod _{j\in A}\Vert U_{A}\chi _{E_A}\Vert _{L^{s_j(\cdot )}}\right) \left( \prod _{j\in A^c}\Vert U_{A^c} \chi _{E_A}\Vert _{L^{p_j(\cdot )}}\right) . \end{aligned} \end{aligned}$$
(2.9)

Denote \(\theta =\frac{n+\frac{d+1}{|A|}}{n}\) and we can choose d large enough such that \(\theta p_j^->1\) and \(\theta s^-_j>1\). Notice that \(\frac{1}{\theta s_j(x)}= \frac{1}{\theta p_j(x)}-\frac{\alpha /\theta |A|}{n}\). By Lemma 2.3, we get that

$$\begin{aligned} \begin{aligned} \prod _{j\in A}\Vert U_{A} \chi _{E_A}\Vert _{L^{s_j(\cdot )}}&=\prod _{j=A}\left\| \sum _{k_j}|\lambda _{j,k_j}| \frac{l(Q_{j,k_j})^{n+\frac{d+1}{|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}-\frac{\alpha }{|A|}}}\right\| _{L^{s_j(\cdot )}}\\&\le C \prod _{j\in A} \left\| \left( \sum _{k_j}|\lambda _{j,k_j} |{(M_{\alpha /\theta |A|}{\chi _{Q_{j,k_j}}})^{\theta }}\right) ^{1/\theta } \right\| ^\theta _{L^{\theta s_i(\cdot )}}\\&\le \prod _{j\in A}\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|{{\chi _{Q_{j,k_j} }}}\right) ^{\frac{1}{\theta }} \right\| ^\theta _{L^{\theta p_j(\cdot )}}\\&\le C\prod _{j\in A}\Vert f_j\Vert _{{H}^{p_j(\cdot )}}, \end{aligned} \end{aligned}$$
(2.10)

where the first inequality follows from the following claim: For any \(x\in {\mathbb {R}}^n\) and \(0\le \alpha <\infty \), there exists a constant C such that

$$\begin{aligned} \frac{r^n}{(r+|x-y|)^{n-\alpha }}\le CM_\alpha \chi _{Q(y,r)}(x), \end{aligned}$$

where Q(y, r) is a cube centered in y and r its side length. To prove the claim, for \(x\in (Q^*)^c\), if P is the smallest cube containing x and Q, then

$$\begin{aligned} M_\alpha \chi _{Q(y,r)}(x)\sim \frac{1}{|P|^{1-\frac{\alpha }{n}}} \int _P\chi _{Q(y,r)}dy\sim \frac{r^n}{|x-y|^{n-\alpha }}; \end{aligned}$$

and for \(x\in Q^*\), we only need to note that

$$\begin{aligned} r^\alpha \le CM_\alpha \chi _{Q(y,r)}(x). \end{aligned}$$

Repeating the similar argument to (2.10) with \(\alpha =0\), we get that

$$\begin{aligned} \begin{aligned} \prod _{j\in A^c}\Vert U_{A^c} \chi _{E_A}\Vert _{L^{p_j(\cdot )}}&=\prod _{j=A^c}\left\| \sum _{k_j}|\lambda _{j,k_j}| \frac{l(Q_{j,k_j})^{n+\frac{d+1}{|A|}}}{(|x-z_{j,k_j}|+l(Q_{j,k_j}))^{n+\frac{d+1}{|A|}}}\right\| _{L^{p_j(\cdot )}}\\&\le C \prod _{j\in A^c} \left\| \sum _{k_j}|\lambda _{j,k_j}|{(M{\chi _{Q_{j,k_j}}})^{\theta }} \right\| _{L^{p_i(\cdot )}}\\&\le \prod _{j\in A^c}\left\| \left( \sum _{k_j} |\lambda _{j,k_j}|{{\chi _{Q_{j,k_j} }}}\right) ^{\frac{1}{\theta }} \right\| ^\theta _{L^{\theta p_j(\cdot )}}\\&\le C\prod _{j\in A^c}\Vert f_j\Vert _{{H}^{p_j(\cdot )}}. \end{aligned} \end{aligned}$$
(2.11)

Therefore, for any \(f_j\in {H}^{p_j(\cdot )}\cap {L}^{p_j^-+1}\) by the estimates (2.1), (2.9), (2.10) and (2.11) then we have

$$\begin{aligned} \Vert I_\alpha (\mathbf {f})\Vert _{L^{q(\cdot )}}\le C\prod _{j=1}^m\Vert f_j\Vert _{{H}^{p_j(\cdot )}}. \end{aligned}$$
(2.12)

From Remark 4.12 in [21], we have that \({H}^{p_j(\cdot )}\cap {L}^{{{\bar{p}}}}\) is dense in \({H}^{p_j(\cdot )}\). Thus, by the density argument we prove Theorem 1.2. \(\square \)