Boundedness of Rough Multiple Oscillatory Singular Integral Operators on Triebel–Lizorkin Space

Abstract

In this article, we investigate oscillatory singular integral operators on product domains. We establish the boundedness of these operators on Triebel–Lizorkin spaces under weak assumptions on the rough kernel functions. Our results extend several known results from the one-parameter setting to the two-parameter setting.

1. Introduction and Main Result

Let ${\mathbb{R}}^q$ ($q=\tau$ or $\kappa$) be the $2\le q$-Euclidean space and ${\mathbb{S}}^{q-1}$ be the unit sphere in ${\mathbb{R}}^q$ equipped with the normalized Lebesgue surface measure $d{\sigma }_q(·)$. For $u\in {\mathbb{R}}^q\setminus \left\{0\right\}$, let $u^{\prime }=u/\left|u\right|$.

Let $\mathcal{P}(q,d)$ be the set of all polynomials on ${\mathbb{R}}^q$ with real coefficients and degrees less than or equal to d, and let $\mathcal{H}(q,d)$ be the set of all polynomials in $\mathcal{P}(q,d)$ which are homogeneous of degree d. Let $\mathcal{P}(q,d,0)$ denote the set of all $P\in \mathcal{P}(q,d)$ with $\nabla P\left(0\right)=0$. For $P\left(u\right)={\displaystyle \sum _{\gamma \le d}}{c}_{\gamma }{u}^{\gamma }$, set $∥P∥={\displaystyle \sum _{\gamma \le d}}\left|c_{\gamma }\right|$. For $d_1,d_2\in \mathbb{N}$, and $\alpha >0$, let $\mathcal{A}(\tau ,d_1,\kappa ,d_2,\alpha )$ be the class of all functions ℧ which are integrable over ${\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}$ and satisfy the condition

$$\underset{\begin{array}{c}{P}_1\in \mathcal{H}(\tau ,d_1),∥P_1∥=1,\\ P_2\in \mathcal{H}(\kappa ,d_2),∥P_2∥=1\end{array}}{sup}{\int \int }_{{\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}}{\left(log(|P_1\left(v\right){|}^{-1})log(|P_2\left(w\right){|}^{-1})\right)}^{\alpha +1}|\mho (v^{\prime },w^{\prime })|d{\sigma }_{\tau }\left(v\right)d{\sigma }_{\kappa }\left(w\right)<\infty .$$

For $P_1\in \mathcal{P}(\tau ,d_1)$ and $P_2\in \mathcal{P}(\kappa ,d_2)$, we consider the oscillatory singular integral operator $T_{\mho ,P_1,P_2}$ on product spaces ${\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }$ defined, initially for $U\in \mathcal{S}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$, by

$$T_{\mho ,P_1,P_2}U(x,y)=\mathrm{p}.\mathrm{v}.\underset{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}{\int \int }{e}^{i{P}_1\left(v\right)+i{P}_2\left(w\right)}U(x-v,y-w){\displaystyle \frac{\mho (v^{\prime },w^{\prime })}{{\left|v\right|}^{\tau }{\left|w\right|}^{\kappa }}}dvdw,$$

where $\mho \in L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$ such that

$$\mho (sv,tw)=\mho (v,w),\forall s,t>0,$$

(1)

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

(2)

It is well known that the oscillatory singular integral, $T_{\mho ,P_1,P_2}$, on product spaces naturally generalizes the oscillatory singular in one parameter setting which was introduced by Stein in [1]. The singularity of $T_{\mho ,P_1,P_2}$ is along the diagonals $\{x=v\}$ and $\{y=w\}$. The study of oscillatory singular integrals on product spaces has attracted the attention of many authors in recent years. One of the principal motivations for the study of such operators is the requirement of several complex variables and large classes of “subelliptic” equations. For more background information, readers may refer to [1,2,3].

When $P_1\equiv 0$ and $P_2\equiv 0$, the operator $T_{\mho ,P_1,P_2}$ reduces to the classical singular integral operator on product domains, denoted by $T_{\mho }$. The boundedness of $T_{\mho }$ and its various extensions has been extensively investigated by many authors over the years. Historically, the study of such operators was initiated by Fefferman and Stein in [4], where they established the $L^p$ boundedness of $T_{\mho }$ for all $p\in (1,\infty )$ whenever ℧ satisfies certain Lipschitz conditions. Subsequently, in [5], the author improved the above result by establishing the $L^p$ boundedness of $T_{\mho }$ under the weaker assumption $\mho \in L^q({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$. In [6], the authors proved the $L^p$ boundedness of $T_{\mho }$ for all $1<p<\infty$, provided that $\mho \in L{\left({log}^{+}L\right)}^2({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$. Later, in [7], the authors proved the $L^p$ ($1<p<\infty$) boundedness of $T_{\mho }$ when ℧ belongs to the block space $B_q^{(0,1)}({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$ for some $q>1$. Since then, the boundedness of $T_{\mho }$ and its various extensions under different conditions on ℧ has attracted considerable attention from many mathematicians. For further developments, we refer the readers to [8,9,10,11] and the references therein.

On the other hand, Ying [12] obtained that $T_{\mho }$ is bounded on $L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ provided that $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)$ with $\alpha >0$, where ${\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)$ is the collection of all functions $\mho \in L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$ which satisfy the condition:

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

We note that the above condition in the one-parameter case was introduced by Walsh in [13] and then developed by Grafakos and Stefanov in [14].

By following similar arguments as in [14], we get the following:

$$\begin{array}{ccc}\hfill \bigcup _{q>1}{L}^q\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)& ⊊& {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)\mathrm{for}\mathrm{any}\alpha >0,\hfill \\ \hfill \bigcap _{\alpha >0}{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)& ⊈& B_q^{(0,1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})⊈\bigcup _{\alpha >0}{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right),\hfill \\ \hfill \bigcap _{\alpha >0}{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)& ⊈& L{\left({log}^{+}L\right)}^2({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})⊈\bigcup _{\alpha >0}{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right).\hfill \end{array}$$

Let us now recall the definition of the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$. For $\overrightarrow{a}=(a_1,a_2)\in \mathbb{R}\times \mathbb{R}$ and $\epsilon ,p\in (1,\infty )$, the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ is the collection of all tempered distributions U on ${\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }$ such that

$$\begin{array}{c}\hfill {∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}={∥{\left(\sum _{k,j\in \mathbb{Z}}{2}^{(k{a}_1+j{a}_2)\epsilon }{\left|({\mathsf{\Phi }}_k^{\left(1\right)}\otimes {\mathsf{\Phi }}_j^{\left(2\right)})\ast U\right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}<\infty ,\end{array}$$

where $\widehat{{\mathsf{\Phi }}_k^{\left(1\right)}}\left(v\right)=2^{-k\tau }\mathrm{I}\left(2^{-k}v\right)$, $\widehat{{\mathsf{\Phi }}_j^{\left(2\right)}}\left(w\right)=2^{-j\kappa }\mathrm{J}\left(2^{-j}w\right)$ for $k,j\in \mathbb{Z}$, and the radial mappings $\mathrm{I}\in \mathcal{S}\left({\mathbb{R}}^{\tau }\right)$, $\mathrm{J}\in \mathcal{S}\left({\mathbb{R}}^{\kappa }\right)$ satisfying the following:

(1)

$0\le \mathrm{I},\mathrm{J}\le 1$,

(2)

$supp\left(\mathrm{I}\right)\subset \left\{v:{\displaystyle \frac{1}{2}}\le \left|v\right|\le 2\right\}$, $supp\left(\mathrm{J}\right)\subset \left\{w:{\displaystyle \frac{1}{2}}\le \left|w\right|\le 2\right\}$,

(3)

There is a constant $C>0$ such that

$$\mathrm{I}\left(v\right)\ge C\forall \left|v\right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}],and\mathrm{J}\left(w\right)\ge C\forall \left|w\right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}],$$

(4)

${\displaystyle \sum _{k\in \mathbb{Z}}}\mathrm{I}\left(2^{-k}v\right)=1$ and ${\displaystyle \sum _{j\in \mathbb{Z}}}\mathrm{J}\left(2^{-j}w\right)=1$ with $v\ne 0$ and $w\ne 0$.

The following properties are proved in [15]:

(1)

The space of Schwartz functions $\mathcal{S}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ is dense in ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$,

(2)

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

(3)

${\stackrel{.}{F}}_p^{{\epsilon }_1,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })\subseteq {\stackrel{.}{F}}_p^{{\epsilon }_2,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ for $1<{\epsilon }_1\le {\epsilon }_2$.

We remark here that the Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ is an extension of many well-known important spaces as Lebesgue spaces $L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$, the Sobolev spaces $L_p^{\alpha }({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$, and the Hardy spaces $H^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$. So, obtaining the boundedness of $T_{\mho ,P_1,P_2}$ on the space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ is more intricate than establishing the boundedness on the space $L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$.

Very recently, the authors of [16] improved the result in [12] as described in the following result:

Theorem 1.

Let $\mho \in {\mathcal{F}}_{\alpha }({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$ with $\alpha >0$ satisfy (1) and (2). Then $T_{\mho }$ is bounded on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$, $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$.

In the one parameter setting, the oscillatory singular operator related to $T_{\mho }$ is given by

$${\mathcal{W}}_{\mho ,P}U\left(x\right)=p.v.{\int }_{{\mathbb{R}}^{\tau }}{e}^{iP\left(v\right)}U(x-v){\displaystyle \frac{\mho \left(v^{\prime }\right)}{{\left|v\right|}^{\tau }}}dv.$$

We point out that the boundedness of the integral ${\mathcal{W}}_{\mho }$ on ${\stackrel{.}{F}}_p^{\epsilon ,r}\left({\mathbb{R}}^{\tau }\right)$ was proved in [17] under certain conditions as described in the following theorem:

Theorem 2.

Let $d\in \mathbb{N}$ be given and $\mho \in {\displaystyle \bigcap _{1\le n\le d}}\mathcal{O}(\tau ,n,\alpha )$ with $\alpha >0$. Then the estimate

$$\begin{array}{ccc}\hfill \underset{P\in \mathcal{P}(\tau ,d,0)}{sup}{∥{\mathcal{W}}_{\mho ,P}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,r}\left({\mathbb{R}}^{\tau }\right)}& \le & C(logd+1){∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,r}\left({\mathbb{R}}^{\tau }\right)}\hfill \end{array}$$

holds for all $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$, $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ and $r\in \mathbb{R}$, where $\mathcal{O}(\tau ,n,\alpha )$ is the set of all functions $\mho \in L^1\left({\mathbb{S}}^{\tau -1}\right)$ satisfying:

$$\begin{array}{cc}\hfill \underset{P\in \mathcal{H}(\tau ,n)),∥P∥=1}{sup}& {\int }_{{\mathbb{S}}^{\tau -1}}\left|\mho \left(v^{\prime }\right)\right|{\left(log{\left|P\left(v\right)\right|}^{-1}\right)}^{\alpha +1}d{\sigma }_{\tau }\left(v\right)<\infty .\hfill \end{array}$$

It is worth mentioning that the space $\mathcal{O}(\tau ,1,\alpha )={\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\right)$, and in the case $\tau =2$, ${\displaystyle \bigcap _{n=1}^{\infty }}\mathcal{O}(2,n,\alpha )={\mathcal{F}}_{\alpha }\left({\mathbb{S}}^1\right)$, see [18]. For more information concerning the operator ${\mathcal{W}}_{\mho ,P}$, readers should consult [19,20,21,22,23] and their references.

In view of the results in [16] regarding the boundedness of $T_{\mho }$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ whenever $\mho \in {\mathcal{F}}_{\alpha }({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$ and of the results in [17] regarding the boundedness of ${\mathcal{W}}_{\mho ,P}$ on ${\stackrel{.}{F}}_p^{\epsilon ,r}\left({\mathbb{R}}^{\tau }\right)$ whenever $\mho \in {\displaystyle \bigcap _{1\le n\le d}}\mathcal{O}(\tau ,n,\alpha )$, it is natural to ask the following question:

Question. 

Is the operator $T_{\mho ,P_1,P_2}$ bounded on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ under the condition $\mho \in {\displaystyle \bigcap _{1\le n\le d_1,1\le m\le d_2}}\mathcal{A}(\tau ,n,\kappa ,m,\alpha )$?

In this work, we shall answer the above question in the affirmative, as described in the following result:

Theorem 3.

Let $d_1,d_2\in \mathbb{N}$, and let $\mho \in {\displaystyle \bigcap _{1\le n\le d_1,1\le m\le d_2}}\mathcal{A}(\tau ,n,\kappa ,m,\alpha )$ for some $\alpha >0$ satisfy the conditions (1) and (2). Then there exists $C>0$ such that

$$\begin{array}{ccc}\hfill \underset{P_1\in \mathcal{P}(\tau ,d_1,0),P_2\in \mathcal{P}(\kappa ,d_2,0)}{sup}{∥T_{\mho ,P_1,P_2}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C(log{d}_1+1)(log{d}_2+1){∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$, $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ and $\overrightarrow{a}\in \mathbb{R}\times \mathbb{R}$.

Remark 1.

(1) 

For the special cases $P_1\equiv 0$, $P_2\equiv 0$, and if ℧ satisfies certain Lipschitz conditions, the $L^p$ boundedness of $T_{\mho ,P_1,P_2}$ was established in [4]. So, our result extends and improves the result in [4].

(2) 

In [5] the author proved the boundedness of $T_{\mho ,0,0}$ on $L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ provided that $\mho \in L^q({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})⊊{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}\right)$. Hence, our result represents an extension and improvement of the aforementioned result.

(3) 

In Theorem 3, if we take $P_1\equiv 0$, $P_1\equiv 0$, the operator $T_{\mho ,P_1,P_2}$ is reduced to $T_{\mho }$ and hence our result is a natural generalization of the result obtained in [16].

2. Some Notations

For $P_1\in \mathcal{P}(\tau ,d_1,0)$, $P_2\in \mathcal{P}(\kappa ,d_2,0)$ and $\mho \in L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})$, we consider the family of measures $\{{{\rm Y}}_{P_1,P_2,s,t}:s,t\in \mathbb{R}\}$ and its corresponding maximal operator ${{\rm Y}}^{*}$ on ${\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }$ given by

$${\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}U d{{\rm Y}}_{P_1,P_2,s,t}={\int \int }_{{\mathcal{D}}_{s,t}}{e}^{i{P}_1\left(v\right)+i{P}_2\left(w\right)}U\left(v,w\right){\displaystyle \frac{\mho (v^{\prime },w^{\prime })}{{\left|v\right|}^{\tau }{\left|w\right|}^{\kappa }}}dvdw$$

and

$${{\rm Y}}^{*}\left(U\right)=\underset{s,t\in \mathbb{R}}{sup}\left|{{\rm Y}}_{P_1,P_2,s,t}\right|\ast \left|U\right|,$$

where ${\mathcal{D}}_{s,t}=$$\left\{\left(v,w\right)\in {\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }:2^s\le \left|v\right|<2^{s+1},2^t\le \left|w\right|<2^{t+1}\right\}$.

Let $\mathrm{I}\in \mathcal{S}\left({\mathbb{R}}^{\tau }\right)$ and $\mathrm{J}\in \mathcal{S}\left({\mathbb{R}}^{\kappa }\right)$ be radial mappings satisfying the following:

(1)

$0\le \mathrm{I},\mathrm{J}\le 1$,

(2)

$supp\left(\mathrm{I}\right)\subset \left\{v:{\displaystyle \frac{1}{2}}\le \left|v\right|\le 2\right\}$, $supp\left(\mathrm{J}\right)\subset \left\{w:{\displaystyle \frac{1}{2}}\le \left|w\right|\le 2\right\}$,

(3)

A constant $C>0$ exists such that $\mathrm{I}\left(v\right),\mathrm{I}\left(w\right)\ge C$ for all $\left|v\right|,\left|w\right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}]$,

(4)

${\int }_{\mathbb{R}}{\left|\widehat{\mathrm{I}}\left(2^{s}v\right)\right|}^2=1$ with $v\ne 0$ and ${\int }_{\mathbb{R}}{\left|\widehat{\mathrm{J}}\left(2^{t}w\right)\right|}^2=1$ with $w\ne 0$.

For simplicity, we let ${\widehat{\mathrm{I}}}_s\left(v\right)$ indicate to $\widehat{\mathrm{I}}\left(sv\right)$ and ${\widehat{\mathrm{J}}}_t\left(w\right)$ indicate to $\widehat{\mathrm{J}}\left(tw\right)$. Hence, we have ${\mathrm{I}}_{2^s}\left(v\right)=2^{-s\tau }\mathrm{I}(v/2^s)$ and ${\mathrm{J}}_{2^t}\left(w\right)=2^{-t\kappa }\mathrm{J}(w/2^t)$. Define ${\mathcal{H}}_{2^s,2^t}\left(U\right)(v,w)=({\mathrm{I}}_{2^s}\otimes {\mathrm{J}}_{2^t})\ast U(v,w)$. Therefore, for any $U\in \mathcal{S}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$, we have

$$\begin{array}{cc}\hfill {∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\sim & {∥{\left({\int \int }_{{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}}{\left|({\mathrm{I}}_s\otimes {\mathrm{J}}_t)\ast U\right|}^{\epsilon }{\displaystyle \frac{dsdt}{st}}\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \\ \hfill \sim & {∥{\left({\int \int }_{\mathbb{R}\times \mathbb{R}}{\left|{\mathcal{H}}_{2^s,2^t}\left(U\right)\right|}^{\epsilon }dsdt\right)}^{1/\epsilon }∥}_{L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\hfill \end{array}$$

3. Proof of Theorem 3

Let $P_1\in \mathcal{P}(\tau ,d_1,0)$ and $P_2\in \mathcal{P}(\kappa ,d_2,0)$. So we have $P_1\left(v\right)={\displaystyle \sum _{{\gamma }_1\le d_1}}{a}_{{\gamma }_1}{v}^{{\gamma }_1}$ and $P_2\left(w\right)={\displaystyle \sum _{{\gamma }_2\le d_2}}{b}_{{\gamma }_2}{w}^{{\gamma }_2}$ with $\nabla P_1\left(0\right)=\nabla P_2\left(0\right)=0$. Suppose that $d_1=2^n$ and $d_2=2^m$ for some natural numbers $n,m\ge 1$ (the general case can be easily obtained from this special case). Without loss of generality, we may assume that $P_1\left(l_{1}v\right)$ and $P_2\left(l_{2}w\right)$ do not have constant terms. Hence, we write $P_1\left(l_{1}v\right)={\displaystyle \sum _{j=2}^{d_1}}{P}_{1,j}\left(v\right)l_1^j$ and $P_2\left(l_{2}w\right)={\displaystyle \sum _{k=2}^{d_2}}{P}_{2,k}\left(w\right)l_2^k$, where $P_{1,j}\left(v\right)={\displaystyle \sum _{\left|{\gamma }_1\right|=j}}{a}_{{\gamma }_1}{v}^{{\gamma }_1}$ and $P_{2,k}\left(w\right)={\displaystyle \sum _{\left|{\gamma }_2\right|=k}}{b}_{{\gamma }_2}{w}^{{\gamma }_2}$. Let ${\mathbf{A}}_j=∥P_{1,j}∥$ and ${\mathbf{B}}_k=∥P_{2,k}∥$. Take $Q_1\left(l_{1}v\right)={\displaystyle \sum _{j=2}^{d_1/2}}{P}_{1,j}\left(v\right)l_1^j$ and $Q_2\left(l_{2}w\right)={\displaystyle \sum _{k=2}^{d_2/2}}{P}_{2,k}\left(w\right)l_2^k$ with $\underset{{\displaystyle \frac{d_1}{2}}<j\le d_1}{max}{\mathbf{A}}_j=1$ and $\underset{{\displaystyle \frac{d_2}{2}}<k\le d_2}{max}{\mathbf{B}}_k$. Hence, there are ${\displaystyle \frac{d_1}{2}}<j_0\le d_1$ and ${\displaystyle \frac{d_2}{2}}<k_0\le d_2$ such that ${\mathbf{A}}_{j_0}=1={\mathbf{B}}_{k_0}$.

By the translation invariance of $T_{\mho ,P_1,P_2}$, it is enough to prove the boundedness of $T_{\mho ,P_1,P_2}$ on ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ only for the case $\overrightarrow{a}=\overrightarrow{0}$. It is clear that

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}U(x,y)={\int \int }_{\mathbb{R}\times \mathbb{R}}{{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)dsdt\end{array}$$

$$\begin{array}{ccc}\hfill & =& T_{\mho ,P_1,P_2}^{0,0}U(x,y)+T_{\mho ,P_1,P_2}^{0,\infty }U(x,y)+T_{\mho ,P_1,P_2}^{\infty ,0}U(x,y)+T_{\mho ,P_1,P_2}^{\infty ,\infty }U(x,y),\hfill \end{array}$$

(3)

where

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{0,0}U(x,y)={\int }_{t\le t_0}{\int }_{s\le s_0}{{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)dsdt,\end{array}$$

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{\infty ,\infty }U(x,y)={\int }_{t>t_0}{\int }_{s>s_0}{{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)dsdt,\end{array}$$

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{0,\infty }U(x,y)={\int }_{t>t_0}{\int }_{s\le s_0}{{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)dsdt,\end{array}$$

and

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{\infty ,0}U(x,y)={\int }_{t\le t_0}{\int }_{s>s_0}{{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)dsdt\end{array}$$

for any $s_0,t_0\in \mathbb{R}$. Let us first prove that for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ and $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$,

$$\begin{array}{cc}\hfill E(d_1,d_2)\cong & \underset{P_1\in \mathcal{P}(\tau ,d_1,0),P_2\in \mathcal{P}(\kappa ,d_2,0)}{sup}{∥T_{\mho ,P_1,P_2}^{0,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \\ \hfill \le & C(log{d}_1+1)(log{d}_2+1){∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\hfill \end{array}$$

(4)

We will argue by induction on the degrees of the polynomials $P_1$ and $P_2$. If $n=1=m$, then $d_1=2=d_2$. In this case we have $P_1\left(v\right)={\displaystyle \sum _{\left|{\gamma }_1\right|=2}}{a}_{{\gamma }_1}{v}^{{\gamma }_1}$ and $P_2\left(w\right)={\displaystyle \sum _{\left|{\gamma }_2\right|=2}}{b}_{{\gamma }_2}{w}^{{\gamma }_2}$. Without loss of generality, we may assume that ${\displaystyle \sum _{\left|{\gamma }_1\right|=2}}\left|a_{{\gamma }_1}\right|=1={\displaystyle \sum _{\left|{\gamma }_2\right|=2}}\left|b_{{\gamma }_2}\right|$. Write

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{0,0}U(x,y)={\mathbf{J}}_{\mho }^{0,0}U(x,y)+{\mathbf{J}}_{\mho ,P_1,P_2}^{0,0}U(x,y),\end{array}$$

(5)

where

$$\begin{array}{c}\hfill {\mathbf{J}}_{\mho }^{0,0}U(x,y)={\int }_{t\le t_0}{\int }_{s\le s_0}{{\rm Y}}_{0,0,s,t}\ast U(x,y)dsdt\end{array}$$

and

$$\begin{array}{c}\hfill {\mathbf{J}}_{\mho ,P_1,P_2}^{0,0}U(x,y)={\int }_{t\le t_0}{\int }_{s\le s_0}\left({{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)-{{\rm Y}}_{0,0,s,t}\ast U(x,y)\right)dsdt.\end{array}$$

By employing Theorem 1 in [16], we get that

$$\begin{array}{c}\hfill {∥{\mathbf{J}}_{\mho }^{0,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le C{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\end{array}$$

(6)

for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ and $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$. As $P_1$ and $P_2$ do not have constant terms, $\nabla P_1\left(0\right)=0=\nabla P_2\left(0\right)$ and ${\displaystyle \sum _{\left|{\gamma }_1\right|=2}}\left|a_{{\gamma }_1}\right|=1={\displaystyle \sum _{\left|{\gamma }_2\right|=2}}\left|b_{{\gamma }_2}\right|$ together with the fact that $|e^{it}|\le |t|$, we deduce that

$$\left|e^{i{P}_1\left(v\right)+i{P}_2\left(w\right)}-1\right|\le \sum _{\left|{\gamma }_1\right|=2}\left|a_{{\gamma }_1}\right|{\left|v\right|}^2+\sum _{\left|{\gamma }_2\right|=2}\left|b_{{\gamma }_2}\right|{\left|w\right|}^2.$$

This leads to

$$\begin{array}{c}\hfill \left|{\mathbf{J}}_{\mho ,P_1,P_2}^{0,0}U(x,y)\right|\le {\int }_{t\le t_0}{\int }_{s\le s_0}\left(2^{s+1}+2^{t+1}\right)\left(\left|{{\rm Y}}_{s,t}\right|\ast \left|U\right|(x,y)\right)dsdt\end{array}$$

(7)

for all $\left|v\right|\le 2^{s+1}\le 1$ and $\left|w\right|\le 2^{t+1}\le 1$ with $s\le s_0$ and $t\le t_0$. Hence, by (7) and Lemma 3 in [16], we obtain that

$$\begin{array}{c}\hfill {∥{\mathbf{J}}_{\mho ,P_1,P_2}^{0,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le C{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\end{array}$$

(8)

for fixed $s_0,t_0\in \mathbb{R}$. Therefore, by (5), (6) and (8), we deduce that

$$\begin{array}{c}\hfill E(2,2)\le C{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\end{array}$$

(9)

for $p\in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$ and $\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$. We note that the above argument, together with the results obtained in [17], ensures that Equation (4) holds whenever $d_1=2$ or $d_2=2$.

Now, assume that (4) holds for all polynomials $P_1$ and $P_2$ with $deg\left(P_1\right)\le {\displaystyle \frac{d_1}{2}}=2^{n-1}$ and for any $P_2$ of degree $d_2$, and also holds whenever $deg\left(P_1\right)=d_1$ and $deg\left(P_2\right)={\displaystyle \frac{d_2}{2}}$. It remains to show that condition (4) still holds when $deg\left(P_1\right)=d_1$ and $deg\left(P_2\right)=d_2$. Write

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2}^{0,0}U(x,y)=T_{\mho ,Q_1,Q_2}^{0,0}U(x,y)+T_{\mho ,P_1,P_2,Q_1,Q_2}^{0,0}U(x,y),\end{array}$$

(10)

where

$$\begin{array}{c}\hfill T_{\mho ,Q_1,Q_2}^{0,0}U(x,y)={\int }_{t\le t_0}{\int }_{s\le s_0}{{\rm Y}}_{Q_1,Q_2,s,t}\ast U(x,y)dsdt\end{array}$$

and

$$\begin{array}{c}\hfill T_{\mho ,P_1,P_2,Q_1,Q_2}^{0,0}U(x,y)={\int }_{t\le t_0-1}{\int }_{s\le s_0-1}\left({{\rm Y}}_{P_1,P_2,s,t}\ast U(x,y)-{{\rm Y}}_{Q_1,Q_2,s,t}\ast U(x,y)\right)dsdt.\end{array}$$

Since $\nabla P_i\left(0\right)=0$ ($i=1,2$), we have $\nabla Q_i\left(0\right)=0$; and since $deg\left(Q_1\right)\le {\displaystyle \frac{d_1}{2}}$, we conclude by induction that

$$\begin{array}{c}\hfill {∥T_{\mho ,Q_1,Q_2}^{0,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le E({\displaystyle \frac{d_1}{2}},{\displaystyle \frac{d_2}{2}})=E(2^{n-1},2^{m-1}).\end{array}$$

(11)

Choose $s_0$ and $t_0$ so that for all $s\le s_0$ and $t\le t_0$, we have $2^{(s+1)({\displaystyle \frac{d_1}{2}}-1)}\le {\displaystyle \frac{2}{d_1}}$ and $2^{(t+1)({\displaystyle \frac{d_2}{2}}-1)}\le {\displaystyle \frac{2}{d_2}}$. This leads to

$$\left|e^{i{P}_1\left(l_{1}v\right)+i{P}_2\left(l_{2}w\right)}-e^{i{Q}_1\left(l_{1}v\right)+i{Q}_2\left(l_{2}w\right)}\right|\le \sum _{{\displaystyle \frac{d_1}{2}}<j\le d_1}{\mathbf{A}}_j{l}_1^j+\sum _{{\displaystyle \frac{d_2}{2}}<k\le d_2}{\mathbf{B}}_k{l}_2^k\le 2^{s+1}+2^{t+1}$$

for all $2^s\le l_1\le 2^{s+1}$ with $s\le s_0$ and $2^t\le l_2\le 2^{t+1}$ with $t\le t_0$. Thus, we obtain

$$\begin{array}{c}\hfill \left|{\mathbf{J}}_{\mho ,P_1,P_2,Q_1,Q_2}^{0,0}U(x,y)\right|\le {\int }_{t\le t_0}{\int }_{s\le s_0}\left(2^{s+1}+2^{t+1}\right)\left(\left|{{\rm Y}}_{s,t}\right|\ast \left|U\right|(x,y)\right)dsdt,\end{array}$$

which, by Lemma 3 in [16], yields

$$\begin{array}{c}\hfill {∥T_{\mho ,P_1,P_2,Q_1,Q_2}^{0,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\end{array}$$

(12)

By the inequalities (9)–(12), we deduce that

$$\begin{array}{cc}\hfill E(d_1,d_2)=& E(2^n,2^m)\le E(2^{n-1},2^{m-1})+C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \\ \hfill \le & E(2,2)+C\left(n\right)\left(m\right){∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \\ \hfill \le & Clog(d_1+1)log(d_2+1){∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\hfill \end{array}$$

Thus, (4) is proved for all polynomials $P_1$ and $P_2$ with $deg\left(P_1\right)=d_1=2^n$ and $deg\left(P_2\right)=d_2=2^m$. For the general cases of $d_1$ and $d_2$, we have $2^{n-1}<d_1\le 2^n$ and $2^{m-1}<d_2\le 2^m$. So

$$\begin{array}{cc}\hfill E(d_1,d_2)\le & E(2^n,2^m)\le E(2^{n-1},2^{m-1})+C(n+1)(m+1){∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \\ \hfill \le & Clog(d_1+1)log(d_2+1){∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\hfill \end{array}$$

(13)

Therefore, the proof of (4) is complete.

Now, let us estimate ${∥T_{\mho ,P_1,P_2}^{\infty ,\infty }\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}$. It is clear that

$$\begin{array}{ccc}\hfill {∥T_{\mho ,P_1,P_2}^{\infty ,\infty }\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & {\int }_{t>t_0}{\int }_{s>s_0}{∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}dsdt.\hfill \end{array}$$

(14)

Notice that for any function $V\in {\stackrel{.}{F}}_{p^{\prime }}^{{\epsilon }^{\prime },\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ with ${∥V∥}_{{\stackrel{.}{F}}_{p^{\prime }}^{{\epsilon }^{\prime },\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le 1$, Hölder’s inequality leads to

$$\begin{array}{cc} & \left|\langle {{\rm Y}}_{P_1,P_2,s,t}\ast U,V\rangle \right|\hfill \\ \le & \left|{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}{\int \int }_{\mathbb{R}\times \mathbb{R}}{{\rm Y}}_{P_1,P_2,s,t}\ast {\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right){\mathcal{H}}_{2^{s+\mu },2^{s+\nu }}^{*}\left(V\right)(v,w)d\mu d\nu dvdw\right|\hfill \\ \le & {∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{{\rm Y}}_{P_1,P_2,s,t}\ast {\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\right|}^{\epsilon }d\mu d\nu \right)}^{1/\epsilon }∥}_p{∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}^{*}\left(V\right)\right|}^{{\epsilon }^{\prime }}d\mu d\nu \right)}^{1/{\epsilon }^{\prime }}∥}_{p^{\prime }},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill {∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le C{∥{\left(\underset{\mathbb{R}\times \mathbb{R}}{\int \int }{\left|{{\rm Y}}_{P_1,P_2,s,t}\ast {\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\right|}^{\epsilon }d\mu d\nu \right)}^{1/\epsilon }∥}_p.\end{array}$$

(15)

To estimate ${∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}$, we need now to consider three cases:

Case 1. $p=\epsilon =2$. Let us first estimate $\left|{\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\right|$. By the definition of ${\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta ),$ we directly deduce that

$$\left|{\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\right|\le C.$$

(16)

By a simple change of variables, we get that

$$\left|{\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\right|\le \left|{\int \int }_{{\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}}\mho (v,w){\mathcal{A}}_s(\xi ,v){\mathcal{B}}_t(\zeta ,w)d{\sigma }_{\tau }\left(v\right)d{\sigma }_{\kappa }\left(w\right)\right|,$$

(17)

where

$${\mathcal{A}}_s(\xi ,v)={\int }_{1/2}^1{e}^{i{P}_1\left(2^{s+1}v{l}_1\right)-i(l_1{2}^{s+1}\xi ·v)}{\displaystyle \frac{d{l}_1}{l_1}}$$

and

$${\mathcal{B}}_t(\zeta ,w)={\int }_{1/2}^1{e}^{i{P}_2\left(2^{t+1}w{l}_2\right)-i(l_2{2}^{t+1}\zeta ·w)}{\displaystyle \frac{d{l}_2}{l_2}}.$$

Notice that the authors of [17] proved that

$$\left|{\mathcal{A}}_s(\xi ,v)\right|\le C{(\left|s\right|+1)}^{-(\alpha +1)}\left\{1+{\left(log{\displaystyle \frac{1}{{\mathbf{A}}_{j_0}}}\right)}^{\alpha +1}\right\}$$

and

$$\left|{\mathcal{B}}_t(\zeta ,w)\right|\le C{(\left|t\right|+1)}^{-(\alpha +1)}\left\{1+{\left(log{\displaystyle \frac{1}{{\mathbf{B}}_{k_0}}}\right)}^{\alpha +1}\right\}.$$

As ${\mathbf{A}}_{j_0}=1$ and ${\mathbf{B}}_{k_0}=1$, then by the last two inequalities and (17), we obtain that

$$\begin{array}{c}\hfill \left|{\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\right|\le C{(\left|s\right|+1)}^{-(\alpha +1)}{(\left|t\right|+1)}^{-(\alpha +1)}.\end{array}$$

(18)

Let us back to estimate the norm ${∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_2^{2,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}$. By Plancherel’s theorem and (18), we get

$$\begin{array}{c}\hfill {∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_2^{2,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^2={∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{L^2({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^2\end{array}$$

$$\begin{array}{cc}\le & C{\int \int }_{\mathbb{R}\times \mathbb{R}}{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}{\left|\left(\widehat{\mathrm{I}}\left(2^{s+\mu }\xi \right)\otimes \widehat{\mathrm{J}}\left(2^{t+\nu }\zeta \right)\right){\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\widehat{U}(\xi ,\zeta )\right|}^{2}d\xi d\zeta dtds\hfill \\ \le & C{\int \int }_{\mathbb{R}\times \mathbb{R}}{\int \int }_{{\Omega }_{s+\mu ,t+\nu }}{\left|\left(\widehat{\mathrm{I}}\left(2^{s+\mu }\xi \right)\otimes \widehat{\mathrm{J}}\left(2^{t+\nu }\zeta \right)\right){\widehat{{\rm Y}}}_{P_1,P_2,s,t}(\xi ,\zeta )\widehat{U}(\xi ,\zeta )\right|}^{2}d\xi d\zeta dtds\hfill \\ \le & C{\left((\left|s\right|+1)(\left|t\right|+1)\right)}^{-2\alpha -2}{∥U∥}_{L^2({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^2\hfill \\ =& C{\left((\left|s\right|+1)(\left|t\right|+1)\right)}^{-2\alpha -2}{∥U∥}_{{\stackrel{.}{F}}_2^{2,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^2.\hfill \end{array}$$

(19)

where ${\Omega }_{s,t}=\left\{(\xi ,\zeta )\in {\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }:\mathrm{I}\left(2^s\xi \right),\mathrm{J}\left(2^t\zeta \right)\in [{\displaystyle \frac{1}{2}},2]\right\}$.

Case 2. $p=\epsilon$. By (15), we get

$$\begin{array}{c}{∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\le & C\left({\int \int }_{\mathbb{R}\times \mathbb{R}}{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}\left({\int \int }_{{\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}}{\mathrm{M}}_{u,v}\left({\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)(v,w)\right)\right.\right.\hfill \\ \times & {\left.{\left.\left|\mho (v,w)\right|{\sigma }_{\tau }\left(v\right)d{\sigma }_{\kappa }\left(w\right)\right)}^{\epsilon }dxdydsdt\right)}^{1/\epsilon }\hfill \\ \le & C{\left({\int \int }_{\mathbb{R}\times \mathbb{R}}{\left({\int \int }_{{\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1}}\left|\mho (v,w)\right|{∥{\mathrm{M}}_{u,v}({\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)∥}_p{\sigma }_{\tau }\left(v\right)d{\sigma }_{\kappa }\left(w\right)\right)}^{p}dsdt\right)}^{1/p},\hfill \end{array}$$

where

$${\mathrm{M}}_{u,v}\left(U\right)(x,y)=\underset{h_1,h_2\in \mathbb{R}}{sup}{\displaystyle \frac{1}{h_1,h_2}}{\int }_0^{h_2}{\int }_0^{h_1}U(x-P_1\left(l_{1}v\right),y-P_2\left(l_{2}w\right))d{l}_{1}d{l}_2$$

which is bounded on $L^p({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ for $p\in (1,\infty )$. Thus,

$$\begin{array}{c}{∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\le C{∥U∥}_{{\stackrel{.}{F}}_p^{p,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}.\hfill \end{array}$$

(21)

Case 3. $p>\epsilon$. By the duality, we deduce that a non-negative function $\psi \in L^{{(p/\epsilon )}^{\prime }}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ exists such that ${∥\psi ∥}_{L^{{(p/\epsilon )}^{\prime }}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}=1$ and

$$\begin{array}{cc} & {∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^{\epsilon }\le C{\int \int }_{\mathbb{R}\times \mathbb{R}}{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}\left|{\int \int }_{{\mathcal{D}}_{s,t}}{e}^{i{P}_1\left(v\right)+i{P}_2\left(w\right)}\right.\hfill \\ \times & {\left.{\displaystyle \frac{\mho (v^{\prime },w^{\prime })}{{\left|v\right|}^{\tau }{\left|w\right|}^{\kappa }}}{\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\left(x-v,y-w\right)dvdw\right|}^{\epsilon }\psi (x,y)dxdydsdt\hfill \\ \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}^{\epsilon /{\epsilon }^{\prime }}{\int \int }_{\mathbb{R}\times \mathbb{R}}{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}{\int \int }_{{\mathcal{D}}_{s,t}}{\displaystyle \frac{\left|\mho (v^{\prime },w^{\prime })\right|}{{\left|v\right|}^{\tau }{\left|w\right|}^{\kappa }}}\hfill \\ \times & {\left|{\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\left(x-v,y-w\right)\right|}^{\epsilon }dvdw\psi (x,y)dxdydsdt\hfill \\ \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}^{\epsilon /{\epsilon }^{\prime }}{\int \int }_{{\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }}{{\rm Y}}^{*}\left(\overline{\psi }\right)(x,y)\left({\int \int }_{\mathbb{R}\times \mathbb{R}}{\left|{\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\left(x,y\right)\right|}^{\epsilon }dsdt\right)dxdy\hfill \\ \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}^{\epsilon /{\epsilon }^{\prime }}{∥{\int \int }_{\mathbb{R}\times \mathbb{R}}{\left|{\mathcal{H}}_{2^{s+\mu },2^{t+\nu }}\left(U\right)\left(x,y\right)\right|}^{\epsilon }dsdt∥}_{(p/q)}{∥{{\rm Y}}^{*}\left(\overline{\psi }\right)∥}_{{(p/q)}^{\prime }}\hfill \\ \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}^{\epsilon /{\epsilon }^{\prime }+1}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}^{\epsilon }.\hfill \end{array}$$

(22)

Hence, (21) and (22) lead to

$$\begin{array}{ccc}\hfill {∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C{∥\mho ∥}_{L^1({\mathbb{S}}^{\tau -1}\times {\mathbb{S}}^{\kappa -1})}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

(23)

for all $p\ge \epsilon$. Consequently, by applying duality and the interpolation argument used in ([24], p. 302), we conclude that (23) holds for all $p\in (1,\infty )$ and $\epsilon \in (1,\infty )$, which when interpolated with (19), yields that

$$\begin{array}{ccc}\hfill {∥{{\rm Y}}_{P_1,P_2,s,t}\ast U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C{\left((1+\left|s\right|)(1+\left|t\right|)\right)}^{-\vartheta (\alpha +1)}{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

(24)

for all $\vartheta \in (0,1)$, ${\displaystyle \frac{\vartheta }{2}}<{\displaystyle \frac{1}{p}}<1-{\displaystyle \frac{\vartheta }{2}}$ and ${\displaystyle \frac{\vartheta }{2}}<{\displaystyle \frac{1}{\epsilon }}<1-{\displaystyle \frac{\vartheta }{2}}$. Therefore, by (14) and (24), and choose $\vartheta >{\displaystyle \frac{1}{\alpha +1}}$, we get that

$$\begin{array}{ccc}\hfill {∥T_{\mho ,P_1,P_2}^{\infty ,\infty }\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C{∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

(25)

for all $p,\epsilon \in ({\displaystyle \frac{2\alpha +2}{2\alpha +1}},2\alpha +2)$. In the same manner, we derive that

$$\begin{array}{ccc}\hfill \underset{P_1\in \mathcal{P}(\tau ,d_1,0),P_2\in \mathcal{P}(\kappa ,d_2,0)}{sup}{∥T_{\mho ,P_1,P_2}^{0,\infty }\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C(log{d}_1+1){∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}\hfill \end{array}$$

(26)

and

$$\begin{array}{ccc}\hfill \underset{P_1\in \mathcal{P}(\tau ,d_1,0),P_2\in \mathcal{P}(\kappa ,d_2,0)}{sup}{∥T_{\mho ,P_1,P_2}^{\infty ,0}\left(U\right)∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}& \le & C(log{d}_2+1){∥U∥}_{{\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{0}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })}.\hfill \end{array}$$

(27)

Consequently, by (3) and (4) and (25)–(27), we finish the proof of Theorem 3.

4. Conclusions

In this work, we introduced the oscillatory singular operator $T_{\mho ,P_1,P_2}$ on product spaces ${\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa }$. We established the boundedness of this operator on the Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\epsilon ,\overrightarrow{a}}({\mathbb{R}}^{\tau }\times {\mathbb{R}}^{\kappa })$ provided that the rough kernel function ℧ belongs to the space ${\displaystyle \bigcap _{\begin{array}{c}1\le n\le d_1,1\le m\le d_2\end{array}}}\mathcal{A}(\tau ,n,\kappa ,m,\alpha )$ for some $\alpha >0$ and the polynomials $P_1$ on ${\mathbb{R}}^{\tau }$, $P_2$ on ${\mathbb{R}}^{\kappa }$ satisfy some certain conditions. The results presented in this paper extend, improve, and generalize several known results on oscillatory integrals, including those obtained in [4,7,16].