Throughout this article, s ($s=\kappa$ or $\eta$) is assumed to be an integer greater than or equal 2. Also, ${\mathbb{S}}^{s-1}$ is assumed to be the unit sphere in the Euclidean space ${\mathbb{R}}^s$ which is equipped with the normalized Lebesgue surface measure $d{\sigma }_{{}_s}(·)\equiv d\sigma$.

For fixed ${\beta }_{s,k}\ge 1$$(k\in \{1,2,\cdots ,s\left\}\right)$, we define the mapping $\Theta :{\mathbb{R}}^{+}\times {\mathbb{R}}^s\to \mathbb{R}$ by $\Theta ({\tau }_s,v)=\sum _{k=1}^s{v}_k^2{\tau }_s^{-2{\beta }_{s,k}}$ with $v=(v_1,v_2,\dots ,v_s)\in {\mathbb{R}}^s$. For a fixed $v\in {\mathbb{R}}^s$, the unique solution to the equation $\Theta ({\tau }_s,v)=1$ is denoted by ${\tau }_s\equiv {\tau }_s\left(v\right)$. The metric space $({\mathbb{R}}^s,{\tau }_s)$ is known by the mixed homogeneity space associated to ${\left\{{\beta }_{s,k}\right\}}_{k=1}^s$. Let $D_{{\tau }_s}$ be the diagonal $s\times s$ matrix

The following transformation presents the change of variables concerning the space $({\mathbb{R}}^s,{\tau }_s)$:

Hence, $dv={\tau }_s^{{\beta }_s-1}{J}_s\left(v^{\prime }\right)d{\tau }_{s}d\sigma \left(v^{\prime }\right)$, where and ${\tau }_s^{{\beta }_s-1}{J}_s\left(v^{\prime }\right)$ is the Jacobian of the transformation.

Fabes and Riviére showed in [1] that $J_s\in {\mathcal{C}}^{\infty }\left({\mathbb{S}}^{s-1}\right)$ and that there is a constant $A\ge 1$ satisfying

For ${\rho }_1=a_1+i{a}_2,{\rho }_2=b_1+i{b}_{2}with{a}_1,b_1\in (0,\infty )and{a}_2,b_2\in (-\infty ,\infty )$, we assume that where h is a measurable function defined on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ and ℧ is a measurable function defined on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ which is integrable over ${\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}$ and satisfies the following properties: and

For $g\in \mathcal{S}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$, we define the generalized parabolic Marcinkiewicz integral ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ on product domains by where and $1<\mu <\infty$.

We notice that if ${\beta }_{\kappa ,1}={\beta }_{\kappa ,2}=\cdots ={\beta }_{\kappa ,\kappa }=1$ and ${\beta }_{\eta ,1}={\beta }_{\eta ,2}=\cdots ={\beta }_{\eta ,\eta }=1$, then we have ${\beta }_{\kappa }=\kappa$, ${\tau }_{\kappa }\left(v\right)=\left|v\right|$, ${\beta }_{\eta }=\eta$, ${\tau }_{\eta }\left(\omega \right)=\left|\omega \right|$, and $({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta },{\tau }_{\kappa },{\tau }_{\eta })=({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta },|·|,|·|)$. In this case, we denote the operator ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ by ${\mathcal{M}}_{\mho ,h}^{\left(\mu \right)}$. In addition, when $\mu =2$, $h\equiv 1$ and ${\rho }_1=1={\rho }_2$, we denote ${\mathcal{M}}_{\mho ,h}^{\left(\mu \right)}$ by ${\mathcal{M}}_{\mho }$ which is the classical Marcinkiewicz integral on product domains. The investigation of the boundedness of ${\mathcal{M}}_{\mho }$ began in [2] in which the author proved the $L^2$ boundedness of ${\mathcal{M}}_{\mho }$ under the condition $\mho \in L{(logL)}^2({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Subsequently, the investigation of the $L^p$ boundedness of ${\mathcal{M}}_{\mho }$ was considered by many authors (see for instance [3,4,5,6]).

On the other hand, the investigation of the $L^p$ boundedness of the operator ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ was considered by many authors. For example, Al-Salman introduced ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ in [7] in which he proved that ${\mathcal{G}}_{\mho ,1}^{\left(2\right)}$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for all $p\in (1,\infty )$ provided that $\mho \in L(logL)({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Later on, the authors of [8] improved the results in [7]. In fact, they proved the $L^p$ boundedness of ${\mathcal{G}}_{\mho ,h}^{\left(2\right)}$ for all $\left|1/2-1/p\right|<min\{1/2,1/{\ell }^{\prime }\}$ whenever ℧ in $B_q^{(0,0)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$ or ℧ in $L(logL)({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ , and $h\in {\Delta }_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ with $\ell >1$. Here, ${\Delta }_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ (for $\ell >1$) refers to the set of all measurable functions h such that

Let us now recall the definition of Triebel-Lizorkin spaces on product domains. Let $1<\mu ,p<\infty$ and $\overrightarrow{c}=(c_1,c_2)\in \mathbb{R}\times \mathbb{R}$. The homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ is defined to be the set of all tempered distributions g on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ satisfying where for $s\in \{\kappa ,\eta$} and $x\in {\mathbb{R}}^s$, $\widehat{{\psi }_{j,s}}\left(x\right)=2^{-js}{D}_s\left(2^{-j}x\right)$ and $D_s\in C_0^{\infty }\left({\mathbb{R}}^s\right)$ is radial function satisfies the following:

(1) $D_s\in [0,1]$,

(2) $supp\left(D_s\right)\subset \left\{x\in {\mathbb{R}}^s:\left|x\right|\in [\frac{1}{2},2]\right\}$,

(3) $D_s\left(x\right)\ge A>0$ if $\left|x\right|\in [\frac{3}{5},\frac{5}{3}]$ for some constant A,

(4) ${\displaystyle \sum _{j\in \mathbb{Z}}}{D}_s\left(2^{-j}x\right)=1$ with $x\ne 0$.

The authors of [9] proved that the space ${\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ satisfies the following properties:

(i) For $p\in (1,\infty )$, we have ${\stackrel{.}{F}}_p^{0,\overrightarrow{2}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })=L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$,

(ii) If ${\mu }_1\le {\mu }_2$, then ${\stackrel{.}{F}}_p^{\overrightarrow{c},{\mu }_1}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })\subseteq {\stackrel{.}{F}}_p^{\overrightarrow{c},{\mu }_2}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$,

(iii) ${\stackrel{.}{F}}_{p^{\prime }}^{-\overrightarrow{c},{\mu }^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })={\left({\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })\right)}^{*}$, where $p^{\prime }$ is the exponent conjugate to p,

(iv) The Schwartz space $\mathcal{S}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ is dense in ${\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$.

Recently, the authors of [10] employed the extrapolation argument of Yano [11] to prove that whenever $\Omega$ lies in the space $L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ or in the space $B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$, then for all $p\in (1,\infty )$, where $B_q^{(0,\alpha )}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$$(\alpha >-1,q>1)$ refers to a special class of block spaces introduced in [12]. Very recently, the result in [10] was improved in [13] in which the authors proved that if $\mho \in L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\cup B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$ and $h\in {\Delta }_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$, then ${\mathcal{M}}_{\mho ,h}^{\left(\mu \right)}$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for $p\in ({\ell }^{\prime },\infty )$ with $\mu \ge {\ell }^{\prime }$ and for $p\in (1,\mu )$ with $\mu \le {\ell }^{\prime }$ if $2<\ell <\infty$; and also for ${\ell }^{\prime }<p<\infty$ with $\mu \ge {\ell }^{\prime }$ and for $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ with $\mu \le {\ell }^{\prime }$ if $1<\ell \le 2$.

In the view of the results in [8] regarding the boundedness of the parabolic Marcinkiewicz operator ${\mathcal{G}}_{\mho ,h}^{\left(2\right)}$ and the results in [13] regarding the boundedness of the generalized parametric Marcinkiewicz operator ${\mathcal{M}}_{\mho ,h}^{\left(\mu \right)}$, we have the following natural question: Is the integral operator ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ bounded under the same conditions on h and ℧ that was assumed in [13] ?

In this article, we shall answer the above question in the affirmative. In fact, we prove the following:

By using the extrapolation argument in [11,14]) and the estimates in Theorems 1 - 2, we obtain the following results.

(i) For the special case $h\equiv 1$ and $\mu =2$, the authors of [5] showed that ${\mathcal{M}}_{\mho ,1}^{\left(2\right)}$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for all $p\in (1,\infty )$ under the condition $\Omega \in L(logL)({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Also, they found that this condition is the weakest possible condition so that the boundedness of ${\mathcal{M}}_{\mho ,1}^{\left(2\right)}$ holds. On the other hand, the $L^p$ ($1<p<\infty$) boundedness of ${\mathcal{M}}_{\mho ,1}^{\left(2\right)}$ was proved in [6] if $\Omega \in B_q^{(0,0)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$. Also, the optimality of the condition $\Omega \in B_q^{(0,0)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ is established. Therefore, the conditions on ℧ in Theorem 3 as well as Theorem 4 are the weakest known conditions in their respective classes for the case $\mu =2$ and $h\equiv 1$.

(ii) In Theorem 4, when we consider the special case $h\equiv 1$, we get that ${\mathcal{G}}_{\mho ,1}^{\left(\mu \right)}$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for all $p\in (1,\infty )$ if $\mho \in L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\cup B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$, which improves the results in [7,10].

(iii) When $\mu ={\ell }^{\prime }$ with $2<\ell <\infty$, Theorem 4 gives the boundedness of ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ for all $p\in (1,\infty )$.

(iv) For the case $\mu =2$ and $\ell \in (1,2]$, the range of p in Theorem 3 is better than the range obtained in Theorem 1.2 in [8] in which the authors proved the $L^p$ boundedness of ${\mathcal{G}}_{\mho ,h}^{\left(2\right)}$ only for $p\in (\frac{2{\ell }^{\prime }}{{\ell }^{\prime }-2},\frac{2\ell }{2-\ell })$.

(v) For $s\in \{\kappa ,\eta \}$ with ${\beta }_{s,1}={\beta }_{s,2}=\cdots ={\beta }_{s,s}=1$, our results are the same as that obtained in [13].

Throughout the rest of the paper, the letter A stands for a positive constant which is independent of the essential variables and its value not necessary the same at each occurrence.

In this section we need to introduce some notations and establish some lemmas. For $\gamma \ge 2$, consider the family of measures $\{{\sigma }_{{\mathcal{K}}_{\mho ,h},s,r}:={\sigma }_{s,r}:s,r\in {\mathbb{R}}_{+}\}$ and its concerning maximal operators ${\sigma }_h^{*}$ and $M_{h,\gamma }$ on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ given by and where $|{\sigma }_{s,r}|$ is defined in the same way as ${\sigma }_{s,r}$ except that $h\mho$ is replaced by $|h\mho |$ .

We shall need the following two lemmas from [8].

By using Lemma 2, it is easy to show that for all $p\in ({\ell }^{\prime },\infty )$.

Now we need to prove the following result: