Estimates for Certain Rough Multiple Singular Integrals on Triebel-Lizorkin Space

1. Introduction and Main Results

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

$$\mho (tu,rv)=\mho (u,v),\forall t,r>0,$$

(1)

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{S}}^{\kappa -1}}\mho (u^{\prime },v^{\prime })d{\sigma }_{\kappa }\left(u^{\prime }\right)& =& {\int }_{{\mathbb{S}}^{\eta -1}}\mho (u^{\prime },v^{\prime })d{\sigma }_{\eta }\left(v^{\prime }\right)=0.\hfill \end{array}$$

(2)

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

$${\mathbf{T}}_{\mho }h(x,y)=p.v.\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }h(x-u,y-v){\displaystyle \frac{\mho (u^{\prime },v^{\prime })}{{\left|u\right|}^{\kappa }{\left|v\right|}^{\eta }}}dudv.$$

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

$$\begin{array}{ccc}& & \underset{(\xi ,\zeta )\in {\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{sup}\underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }{log}^{\alpha +1}\left({\left|\xi ·u^{\prime }\right|}^{-1}\right){log}^{\alpha +1}\left({\left|\zeta ·v^{\prime }\right|}^{-1}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & \times \left|\mho \left(u^{\prime },v^{\prime }\right)\right|d{\sigma }_{\kappa }\left(u^{\prime }\right)d{\sigma }_{\eta }\left(v^{\prime }\right)<\infty .\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill \bigcup _{q>1}{L}^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)& ⊈& {\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)forany\alpha >0,\hfill \\ \hfill \bigcap _{\alpha >0}{\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)& ⊈& L{\left({log}^{+}L\right)}^2({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})⊈\bigcup _{\alpha >0}{\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right),\hfill \\ \hfill \bigcap _{\alpha >0}{\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)& ⊈& B_q^{(0,1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})⊈\bigcup _{\alpha >0}{\mathbf{G}}_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right).\hfill \end{array}$$

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

$$\begin{array}{c}\hfill {∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}={∥{\left(\sum _{j,k\in \mathbb{Z}}{2}^{j{\gamma }_1\epsilon }{2}^{k{\gamma }_2\epsilon }{\left|({\mathcal{A}}_j\otimes {\mathcal{B}}_k)*h\right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}<\infty ,\end{array}$$

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

$${\mathbf{H}}_{\mho }h\left(x\right)=p.v.{\int }_{{\mathbb{R}}^{\kappa }}h(x-u){\displaystyle \frac{\mho \left(u^{\prime }\right)}{{\left|u\right|}^{\kappa }}}du.$$

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

$$\begin{array}{c}\hfill \underset{\xi \in {\mathbb{S}}^{\kappa -1}}{sup}{\int }_{{\mathbb{S}}^{\kappa -1}}\left|\mho \left(u^{\prime }\right)\right|{log}^{\alpha +1}\left({\left|\xi ·u^{\prime }\right|}^{-1}\right)d{\sigma }_{\kappa }\left(u^{\prime }\right)<\infty .\end{array}$$

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:

Theorem 1. 

Suppose that $\mho \in {\mathbf{G}}_{\alpha }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ for some $\alpha >0$. Then ${\mathbf{T}}_{\mho }$ is bounded on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ 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 $\overrightarrow{\gamma }\in \mathbb{R}\times \mathbb{R}$.

2. Auxiliary Lemmas

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

$$\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }h d{{\rm Y}}_{t,r}=\underset{I_{t,r}}{\int \int }h\left(u,v\right){\displaystyle \frac{\mho (u^{\prime },v^{\prime })}{{\left|u\right|}^{\kappa }{\left|v\right|}^{\eta }}}dudv$$

and

$${{\rm Y}}^{*}\left(h\right)=\underset{t,r\in \mathbb{R}}{sup}\left|{{\rm Y}}_{t,r}\right|*\left|h\right|,$$

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:

Lemma 1. 

Let $\mho \in G_{\alpha }\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ for some $\alpha >0$ and satisfy the conditions $\left(\right)$-$\left(\right)$. Then there is a constant $C>0$ such that the estimates

$$\begin{array}{ccc}\hfill \left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|& \le & C,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|& \le & Cmin\left\{\left|2^t\xi \right|,{\left({log}^{+}\left|2^t\xi \right|\right)}^{-(\alpha +1)}\right\},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|& \le & Cmin\left\{\left|2^r\zeta \right|,{\left({log}^{+}\left|2^r\zeta \right|\right)}^{-(\alpha +1)}\right\}\hfill \end{array}$$

(5)

hold for all $t,r\in \mathbb{R}$ and $(\xi ,\zeta )\in {\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$.

Proof. 

By the definition of ${\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta ),$ it is easy to see that

$$\left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le {(log2)}^2{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})},$$

(6)

which proves (3). By a change of variable, we deduce that

$$\left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le \underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }\left|\mho (u,v)\right|{\int }_{2^r}^{2^{r+1}}\left|J_t(\xi ,u,l)\right|{\displaystyle \frac{d\tau }{\tau }}d{\sigma }_{\kappa }\left(u\right)d{\sigma }_{\eta }\left(v\right),$$

(7)

where

$$J_t(\xi ,u,l)={\int }_1^2{e}^{-i(l{2}^t\xi ·u)}{\displaystyle \frac{dl}{l}}$$

which leads to

$$J_t(\xi ,u,l)\le C{\left|2^t\xi \left|u·{\xi }^{\prime }\right|\right|}^{-1/2}.$$

Hence, by the last estimate and the trivial estimate $\left|J_t(\xi ,u,l)\right|\le$$(log2)$ along with the fact that $t/{(logt)}^{\alpha }$ is increasing on $(2^{\alpha },\infty )$, we get that

$$\left|J_t(\xi ,u,l)\right|\le C{\displaystyle \frac{{\left(log2{\left|{\xi }^{\prime }·u\right|}^{-1}\right)}^{\alpha +1}}{{\left(log\left|2^t\xi \right|\right)}^{\alpha +1}}}if\left|2^t\xi \right|>2^{\alpha }.$$

(8)

Thus, the inequalities (7) and (8) give that

$$\begin{array}{ccc}& & \left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\hfill \\ & \le & C{\left(log\left|2^t\xi \right|\right)}^{-(\alpha +1)}\underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }{\left(log\left(2{\left|{\xi }^{\prime }·u\right|}^{-1}\right)\right)}^{\alpha +1}\left|\mho (u,v)\right|d{\sigma }_{\kappa }\left(u\right)d{\sigma }_{\eta }\left(v\right),\hfill \end{array}$$

which in turn implies that

$$\left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le C{\left(log\left|2^t\xi \right|\right)}^{-(\alpha +1)} if\left|2^t\xi \right|>2^{\alpha }.$$

(9)

Similarly, we derive that

$$\left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le C{\left(log\left|2^r\zeta \right|\right)}^{-(\alpha +1)} if\left|2^r\zeta \right|>2^{\alpha }.$$

(10)

Now, by the cancellation property (1), we have

$$\begin{array}{ccc}\hfill \left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le & C& \underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }\left|\mho (u,v)\right|{\int }_{2^r}^{2^{r+1}}{\int }_1^2\left|e^{-il{2}^t\xi ·u}-1\right|{\displaystyle \frac{dld\tau }{l\tau }}d{\sigma }_{\kappa }\left(u\right)d{\sigma }_{\eta }\left(v\right)\hfill \\ & \le & C\left|2^t\xi \right|.\hfill \end{array}$$

(11)

In the same manner, we obtain that

$$\left|{\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\right|\le C\left|2^r\zeta \right|.$$

(12)

Therefore, by combining (9) with (11) we get (), and by combining (10) with (12), we get (). The lemma is proved. □

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

Lemma 2. 

Let $\mho \in L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Then there exists a constant $C_p>0$ such that

$${∥{{\rm Y}}^{*}\left(f\right)∥}_{L^p\left({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }\right)}\le C_p{∥h∥}_{L^p\left({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }\right)}{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}$$

(13)

for all $1<p<\infty$ and $h\in L^p\left({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }\right)$.

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

$$\begin{array}{ccc}\hfill {∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \sim & {∥{\left(\underset{{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}}{\int \int }{\left|({\mathcal{A}}_t\otimes {\mathcal{B}}_r)*h\right|}^{\epsilon }{\displaystyle \frac{dtdr}{tr}}\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \\ & \sim & {∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{\mathcal{W}}_{2^t,2^r}\left(h\right)\right|}^{\epsilon }dtdr\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\hfill \end{array}$$

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

Lemma 3. 

Let $\mho \in L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Then, the estimate

$${∥\left|{{\rm Y}}_{t,r}\right|*\left|h\right|∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le C_p{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}$$

(14)

holds for all $1<p,\epsilon <\infty$.

Proof. 

Let $h\in {\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$. Then for any function $f\in {\stackrel{.}{F}}_{p^{\prime }}^{{\epsilon }^{\prime },\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ with ${∥f∥}_{{\stackrel{.}{F}}_{p^{\prime }}^{{\epsilon }^{\prime },\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le 1$, by Hölder’s inequality we get

$$\begin{array}{ccc}& & \left|\langle \left|{{\rm Y}}_{t,r}\right|*\left|h\right|,f\rangle \right|\hfill \\ & \le & \left|\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right){\mathcal{W}}_{2^{t+n},2^{r+m}}^{*}\left(f\right)(u,v)dndmdudv\right|\hfill \\ & \le & {∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|}^{\epsilon }dndm\right)}^{1/\epsilon }∥}_p\hfill \\ & \times & {∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{\mathcal{W}}_{2^{t+n},2^{r+m}}^{*}\left(f\right)\right|}^{{\epsilon }^{\prime }}dndm\right)}^{1/{\epsilon }^{\prime }}∥}_{p^{\prime }}\hfill \end{array}$$

which in turn implies

$$\begin{array}{c}\hfill {∥\left|{{\rm Y}}_{t,r}\right|*\left|h\right|∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le C{∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|}^{\epsilon }dndm\right)}^{1/\epsilon }∥}_p.\end{array}$$

(15)

Let us now estimate the $L^p$-norm of ${\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|}^{\epsilon }dndm\right)}^{1/\epsilon }$. Since $p>1$, by duality there exits a function $g\in L^{p^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ such that ${∥g∥}_{L^{p^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}=1$ and

$${∥\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|dndm∥}_p=\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\langle \left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|,g\rangle dndm$$

$$\begin{array}{ccc}\hfill & \le & \underset{\mathbb{R}\times \mathbb{R}}{\int \int }\langle \left|{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)(u,v)\right|,{{\rm Y}}^{*}\left(\overline{g}\right)(u,v)\rangle dndm\hfill \\ & \le & {∥\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)dndm∥}_p{∥{{\rm Y}}^{*}\left(\overline{g}\right)∥}_{p^{\prime }}\hfill \\ & \le & {∥\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)dndm∥}_p{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})},\hfill \end{array}$$

(16)

where $\overline{g}(u,v)=g(-u,-v)$ and the last inequality is obtained by Lemma 2.

By following similar arguments as that employed in the proof of Lemma 2 in [4] we get

$${∥\underset{t,r\in \mathbb{R}}{sup}\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|∥}_p\le C_p{∥\underset{t,r\in \mathbb{R}}{sup}\left|{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|∥}_p{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}.$$

(17)

By interpolating between (16) and (17) we obtain that

$$\begin{array}{c}\hfill {∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|\left|{{\rm Y}}_{t,r}\right|*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(\left|h\right|\right)\right|}^{\epsilon }dndm\right)}^{1/\epsilon }∥}_p\le {∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\end{array}$$

Consequently, by the last inequality and (15), we get (14). □

3. Proof of Theorem 1

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

$$\begin{array}{ccc}\hfill {\mathbf{T}}_{\mho }h(x,y)& =& \underset{\mathbb{R}\times \mathbb{R}}{\int \int }{{\rm Y}}_{t,r}*h(x,y)dtdr\hfill \\ & =& \underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\mathcal{Q}}_{n,m}\left(h\right)dndm,\hfill \end{array}$$

(18)

where

$${\mathcal{Q}}_{n,m}\left(h\right)=\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\mathcal{W}}_{2^{t+n},2^{r+m}}\left({{\rm Y}}_{t,r}*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\right)dtdr.$$

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

$$\begin{array}{c}\hfill {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le C{∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{{\rm Y}}_{t,r}*{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\right|}^{\epsilon }dtdr\right)}^{1/\epsilon }∥}_p.\end{array}$$

(19)

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

$$\begin{array}{c}\hfill {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_2^{2,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^2\end{array}$$

$$\begin{array}{ccc}\hfill & \le & C\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }{\left|\left(\widehat{\mathcal{A}}\left(2^{t+n}\xi \right)\otimes \widehat{\mathcal{B}}\left(2^{r+m}\zeta \right)\right){\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\widehat{f}(\xi ,\zeta )\right|}^{2}d\xi d\zeta drdt\hfill \\ & \le & C\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\underset{{\Delta }_{t+n,r+m}}{\int \int }{\left|\left(\widehat{\mathcal{A}}\left(2^{t+n}\xi \right)\otimes \widehat{\mathcal{B}}\left(2^{r+m}\zeta \right)\right){\widehat{{\rm Y}}}_{t,r}(\xi ,\zeta )\widehat{h}(\xi ,\zeta )\right|}^{2}d\xi d\zeta drdt\hfill \\ & \le & C{\left((1+\left|n\right|)(1+\left|m\right|)\right)}^{-\alpha -1}{∥h∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })},\hfill \end{array}$$

(20)

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

$$\begin{array}{ccc}& & {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& \le & C\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }\left(\underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }{\mathrm{M}}_{u,v}\left({\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)(u,v)\right)\right.\right.\hfill \\ & \times & {\left.{\left.\left|\mho (u,v)\right|{\sigma }_{\kappa }\left(u\right)d{\sigma }_{\eta }\left(v\right)\right)}^{\epsilon }dxdydtdr\right)}^{1/\epsilon }\hfill \\ & \le & C\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\left(\underset{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\int \int }\left|\mho (u,v)\right|\right.\right.\hfill \\ & \times & {\left.{\left.{∥{\mathrm{M}}_{u,v}({\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)∥}_p{\sigma }_{\kappa }\left(u\right)d{\sigma }_{\eta }\left(v\right)\right)}^{p}dtdr\right)}^{1/p},\hfill \end{array}$$

where

$${\mathrm{M}}_{u,v}\left(h\right)(u,v)=\underset{k_1{k}_2\in \mathbb{R}}{sup}{\displaystyle \frac{1}{k_1,k_2}}{\int }_0^{k_2}{\int }_0^{k_1}h(x-tu,y-rv)dtdr$$

which is bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for $1<p<\infty$. Therefore,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}{∥h∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\hfill \end{array}$$

(22)

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

$$\begin{array}{c}\hfill {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^{\epsilon }\end{array}$$

$$\begin{array}{ccc}& \le & C\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }{\left|\underset{I_{t,r}}{\int \int }{\displaystyle \frac{\mho (u^{\prime },v^{\prime })}{{\left|u\right|}^{\kappa }{\left|v\right|}^{\eta }}}{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\left(x-u,y-v\right)dudv\right|}^{\epsilon }\varphi (x,y)dxdydtdr\hfill \\ & \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{\epsilon /{\epsilon }^{\prime }}\underset{\mathbb{R}\times \mathbb{R}}{\int \int }\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }\underset{I_{t,r}}{\int \int }{\displaystyle \frac{\left|\mho (u^{\prime },v^{\prime })\right|}{{\left|u\right|}^{\kappa }{\left|v\right|}^{\eta }}}\hfill \\ & \times & {\left|{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\left(x-u,y-v\right)\right|}^{\epsilon }dudv\varphi (x,y)dxdydtdr\hfill \\ & \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{\epsilon /{\epsilon }^{\prime }}\underset{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int \int }{{\rm Y}}^{*}\left(\overline{\varphi }\right)(x,y)\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\left(x,y\right)\right|}^{\epsilon }dtdr\right)dxdy\hfill \\ & \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{\epsilon /{\epsilon }^{\prime }}{∥\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{\mathcal{W}}_{2^{t+n},2^{r+m}}\left(h\right)\left(x,y\right)\right|}^{\epsilon }dtdr∥}_{(p/q)}{∥{{\rm Y}}^{*}\left(\overline{\varphi }\right)∥}_{{(p/q)}^{\prime }}\hfill \\ & \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{\epsilon /{\epsilon }^{\prime }+1}{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^{\epsilon }.\hfill \end{array}$$

Thus, by the last inequality and (22), we obtain that

$$\begin{array}{ccc}\hfill {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\hfill \end{array}$$

(23)

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

$$\begin{array}{ccc}\hfill {∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \le & C{\left((1+\left|n\right|)(1+\left|m\right|)\right)}^{-\theta (\alpha +1)}{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \end{array}$$

(24)

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

$$\begin{array}{ccc}\hfill {∥{\mathbf{T}}_{\mho }\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \le & C\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{∥{\mathcal{Q}}_{n,m}\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}dndm,\hfill \end{array}$$

then by invoking (24) and choosing $\theta >{\displaystyle \frac{1}{\alpha +1}}$, we end with

$$\begin{array}{ccc}\hfill {∥{\mathbf{T}}_{\mho }\left(h\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \le & C{∥h∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \end{array}$$

(25)

for all $p,\epsilon \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$.

4. Conclusions

In this work, we proved the boundedness of the singular integrals ${\mathbf{T}}_{\mho }$ on Triebel-Lizorkin spaces ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for all $p,\epsilon \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2+2\alpha )$ whenever the kernel function ℧ in ${\mathbf{G}}_{{ }_{\alpha }}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ for some $\alpha >0$. The main result in this paper generalizes and improves the main results proved in [1,2,11]. In future work, we aim to prove the boundedness of ${\mathbf{T}}_{\mho }$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{\gamma }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for a wider range of p provided that $\mho \in {\mathbf{G}}_{{ }_{\alpha }}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$.