Assume that ${\mathbb{R}}^s$ ($s=\kappa$ or $\eta$) is the $2\le s$-Euclidean space and that ${\mathbb{S}}^{s-1}$ is the unit sphere in ${\mathbb{R}}^s$ equipped with the normalized Lebesgue surface measure $d{\sigma }_s(·)$. Also assume that $w^{\prime }=w/\left|w\right|$ for $w\in {\mathbb{R}}^s\setminus \left\{0\right\}$.

Let ℧ be an integrable over ${\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}$ and satisfy

The singular integral operator $T_{\mho }$ on symmetric spaces ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ is defined, initially for $h\in \mathcal{S}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$, by

The study of the boundedness of the operator ${\mathbf{T}}_{\mho }$ was started in [1] in which the authors proved the $L^p$ boundedness of ${\mathbf{T}}_{\mho }$ for all $p\in (1,\infty )$ if $\Omega$ satisfies certain Lipschitz conditions. Subsequently the boundedness of ${\mathbf{T}}_{\mho }$ and some of its extensions has been investigated by many researchers. For example, Duoandikoetxea improved the above results in [2] by proving that ${\mathbf{T}}_{\mho }$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ under the weaker condition $\mho \in L^q({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Later on, the authors of [3], confirmed that ${\mathbf{T}}_{\mho }$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ ($1<p<\infty$) if $\mho \in L{\left({log}^{+}L\right)}^2({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. In [4] the authors established the $L^p$ boundedness of ${\mathbf{T}}_{\mho }$ for $p\in (1,\infty )$ provided that ℧ in the block space $B_q^{(0,1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ for some $q>1$. Thereafter, the discussion of the mapping properties of ${\mathbf{T}}_{\mho }$ and its extensions under various conditions on ℧ has received a large amount of attention by many authors, the readers are referred to [1,2,3,4,5,6,7,8].

Our focus in this paper will be in studying the boundedness of ${\mathbf{T}}_{\mho }$ whenever ℧ belongs to a certain class of functions related to a class of functions introduced by Walsh in [9] and then developed by Grafakos and Stefanov in [10]. To clarify our purpose we recall some definitions and some pertinent results related to our current study. Let ${\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ (for $\alpha >0$) be the class of all functions ℧ which are integrable over ${\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}$ and satisfy the condition on product spaces

By following the same arguments as that employed in [10], we get the following:

Let us recall the definition of the homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$. For $p,\epsilon \in (1,\infty )$ and $\overrightarrow{\gamma }=({\gamma }_1,{\gamma }_2)\in \mathbb{R}\times \mathbb{R}$, the homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ is the class of all tempered distributions h on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ that satisfy where ${\widehat{\mathcal{A}}}_j\left(u\right)=2^{-j\kappa }\mathcal{A}\left(2^{-j}u\right)$ for $j\in \mathbb{Z}$, ${\widehat{\mathcal{B}}}_k\left(v\right)=2^{-k\eta }\mathcal{B}\left(2^{-k}v\right)$ for $k\in \mathbb{Z}$ and the radial functions $\mathcal{A}\in \mathcal{S}\left({\mathbb{R}}^{\kappa }\right)$, $\mathcal{B}\in \mathcal{S}\left({\mathbb{R}}^{\eta }\right)$ satisfy the following:

(1) $0\le \mathcal{A}\le 1$, $0\le \mathcal{B}\le 1$,

(2) $supp\left(\mathcal{A}\right)\subset \left\{u:{\displaystyle \frac{1}{2}}\le \left|u\right|\le 2\right\}$, $supp\left(\mathcal{B}\right)\subset \left\{v:{\displaystyle \frac{1}{2}}\le \left|v\right|\le 2\right\}$,

(3) There exists $M>0$ such that $\mathcal{A}\left(u\right),\mathcal{B}\left(v\right)\ge M$ for all $\left|u\right|,\left|v\right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}]$ ,

(4) $\sum _{j\in \mathbb{Z}}\mathcal{A}\left(2^{-j}u\right)=1$ with $u\ne 0$ and $\sum _{k\in \mathbb{Z}}\mathcal{B}\left(2^{-k}v\right)=1$ with $v\ne 0$.

The authors of [12] proved the following properties:

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

(ii) ${\stackrel{.}{F}}_p^{2,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })=L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for $1<p<\infty$,

(iii) ${\stackrel{.}{F}}_p^{{\epsilon }_1,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })\subseteq {\stackrel{.}{F}}_p^{{\epsilon }_2,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ if ${\epsilon }_1\le {\epsilon }_2$.

In [11], Ying showed that if $\mho \in {\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ for some $\alpha >0$, then ${\mathbf{T}}_{\mho }$ is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for all $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$.

In the one parameter setting, the singular operator related to ${\mathbf{T}}_{\mho }$ is given by

For $\alpha >0$, the class ${\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\right)$ is the collection of all functions $\mho \in L^1\left({\mathbb{S}}^{\kappa -1}\right)$ which satisfy the Grafakos-Stefanov condition

In [13], the authors proved that the integral operator ${\mathbf{H}}_{\mho }$ is bounded on ${\stackrel{.}{F}}_p^{\epsilon ,{\gamma }_1}\left({\mathbb{R}}^{\kappa }\right)$ for $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$, $\epsilon \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$ and ${\gamma }_1\in \mathbb{R}$.

It is worth mentioning that the Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,{\gamma }_1}\left({\mathbb{R}}^{\kappa }\right)$ covers several classes of many well-known function spaces including Lebesgue spaces $L^p\left({\mathbb{R}}^{\kappa }\right)$ , the Hardy spaces $H^p\left({\mathbb{R}}^{\kappa }\right)$ and the Sobolev spaces $L_p^{\alpha }\left({\mathbb{R}}^{\kappa }\right)$. So it is tacitly that the work on these spaces is more intricate than $L^p\left({\mathbb{R}}^{\kappa }\right)$. This clearly has instigated many authors to investigate the boundedness of ${\mathbf{H}}_{\mho }$ and some of its extensions, see for instance [14,15,16,17,18,19,20,21,22,23,24,25,26].

In light of the results obtained in [13] regarding the ${\stackrel{.}{F}}_p^{\epsilon ,{\gamma }_1}$ boundedness of the singular integral ${\mathbf{H}}_{\mho }$ in the one parameter setting whenever $\mho \in {\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\right)$, and the work done in [11] regarding the $L^p$ boundedness of the singular integral ${\mathbf{T}}_{\mho }$ in the product domains whenever $\mho \in {\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$, we are motivated to investigate the boundedness of ${\mathbf{T}}_{\mho }$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ whenever ℧ satisfies the Grafakos-Stefanov condition.

The main result of this paper is the following:

We devote this section to establishing some preliminary lemmas. For $\mho \in L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$, we consider the sequence of measures $\{{{\rm Y}}_{t,r}:t,r\in \mathbb{R}\}$ and its corresponding maximal operator ${{\rm Y}}^{*}$ on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ by and where $I_{t,r}=$$\left\{\left(u,v\right)\in {\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }:2^t\le \left|u\right|<2^{t+1},2^r\le \left|v\right|<2^{r+1}\right\}$.

By adapting the same argument used in [10] to the product case, it is easy to obtain the following:

The following lemma can be found in [4] (see also [2,3,8]).

Let $\mathcal{A}\in \mathcal{S}\left({\mathbb{R}}^{\kappa }\right)$ and $\mathcal{B}\in \mathcal{S}\left({\mathbb{R}}^{\eta }\right)$ be radial functions satisfying the following:

(1) $0\le \mathcal{A},\mathcal{B}\le 1$,

(2) $supp\left(\mathcal{A}\right)\subset \left\{u:{\displaystyle \frac{1}{2}}\le \left|u\right|\le 2\right\}$, $supp\left(\mathcal{B}\right)\subset \left\{v:{\displaystyle \frac{1}{2}}\le \left|v\right|\le 2\right\}$,

(3) There is a constant $M>0$ such that $\mathcal{A}\left(u\right),\mathcal{B}\left(v\right)\ge M$ for all $\left|u\right|,\left|v\right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}]$ ,

(4) ${\int }_{\mathbb{R}}{\left|\widehat{\mathcal{A}}\left(2^{t}u\right)\right|}^2=1$ with $u\ne 0$ and ${\int }_{\mathbb{R}}{\left|\widehat{\mathcal{B}}\left(2^{r}v\right)\right|}^2=1$ with $v\ne 0$.

For simplicity, we denote $\widehat{\mathcal{A}}\left(tu\right)$ by ${\widehat{\mathcal{A}}}_t\left(u\right)$ and $\widehat{\mathcal{B}}\left(rv\right)$ by ${\widehat{\mathcal{B}}}_r\left(v\right)$. Then it is clear that ${\mathcal{A}}_{2^t}\left(u\right)=2^{-t\kappa }\mathcal{A}(u/2^t)$ and ${\mathcal{B}}_{2^r}\left(v\right)=2^{-r\eta }\mathcal{B}(v/2^r)$. Let ${\mathcal{W}}_{2^t,2^r}\left(h\right)(u,v)=({\mathcal{A}}_{2^t}\otimes {\mathcal{B}}_{2^r})*h(u,v)$. Hence, for any $h\in \mathcal{S}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$, we have

Let us give the following result regarding the boundedness of the measures $\left|{{\rm Y}}_{t,r}\right|*\left|h\right|$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$.

Let $\mho \in {\mathbf{G}}_{\alpha }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ for some $\alpha >0$. By the translation invariance of ${\mathbf{T}}_{\mho }$, it is enough to prove the boundedness of ${\mathbf{T}}_{\mho }$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ only whenever $\overrightarrow{\gamma }=\overrightarrow{0}$. It is clear that where

Let us estimate the ${∥{\mathcal{Q}}_{n,m}∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}$. By following the same steps in proving (15), we get that

We need now to consider three cases:

Case 1. $p=2=\epsilon$. In this case, we have ${\stackrel{.}{F}}_2^{2,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })=L^2({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$. So, by invoking Plancherel’s theorem, we obtain where ${\Delta }_{t,r}=\left\{(\xi ,\zeta )\in {\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }:{\displaystyle \frac{1}{2}}\le \mathcal{A}\left(2^t\xi \right)\le 2and{\displaystyle \frac{1}{2}}\le \mathcal{B}\left(2^r\zeta \right)\le 2\right\}$.

Case 2. $p=\epsilon$. By (15), we get that where which is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for $1<p<\infty$. Therefore,

Case 3. $p>\epsilon$. By duality, there is a non-negative function $\varphi$ lies in the space $L^{{(p/\epsilon )}^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ such that ${∥\varphi ∥}_{L^{{(p/\epsilon )}^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}=1$ and

Thus, by the last inequality and (22), we obtain that for all $p\ge \epsilon$. Therefor, by employing the duality along with the interpolation, we conclude that the inequality (23) holds for all $1<p<\infty$ and $1<\epsilon <\infty$, which is when interpolated with (20), we get for all $\theta \in (0,1)$, ${\displaystyle \frac{\theta }{2}}<{\displaystyle \frac{1}{p}}<1-{\displaystyle \frac{\theta }{2}}$ and ${\displaystyle \frac{\theta }{2}}<{\displaystyle \frac{1}{\epsilon }}<1-{\displaystyle \frac{\theta }{2}}$. Since then by invoking (24) and choosing $\theta >{\displaystyle \frac{1}{\alpha +1}}$, we end with for all $p,\epsilon \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$.