Estimates for Generalized Parabolic Marcinkiewicz Integrals with Rough Kernels on Product Domains

Abstract

We prove $L^p$ estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument, we obtain the boundedness of our operators under very weak conditions on the kernel functions. Our results in this paper improve and extend several known results on both generalized Marcinkiewicz integrals and parabolic Marcinkiewicz integrals on product domains.

1. Introduction

Throughout this article, let $s\ge 2$ ($s=\kappa$ or $\eta$) and ${\mathbb{S}}^{s-1}$ 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 $\mathrm{\Theta }:{\mathbb{R}}^{+}\times {\mathbb{R}}^s\to \mathbb{R}$ by $\mathrm{\Theta }({\tau }_s,v)={\displaystyle \sum _{k=1}^s}{v}_k^2{\tau }_s^{-2{\beta }_{s,k}}$ 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 $\mathrm{\Theta }({\tau }_s,v)=1$ 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

$$D_{{\tau }_s}=\left[\begin{array}{ccc}{\tau }_s^{{\beta }_{s,1}}& & 0\\ & \ddots & \\ 0& & {\tau }_s^{{\beta }_{s,s}}\end{array}\right].$$

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

$$\begin{array}{c}{v}_1={\tau }_s^{{\beta }_{s,1}}cos{x}_1\cdots cos{x}_{s-2}cos{x}_{s-1},\hfill \\ v_2={\tau }_s^{{\beta }_{s,2}}cos{x}_1\cdots cos{x}_{s-2}sin{x}_{s-1},\hfill \\ ⋮\hfill \\ v_{{}_{s-1}}={\tau }_s^{{\beta }_{{}_{s,s-1}}}cos{x}_{1}sin{x}_2,\hfill \\ v_{{}_s}={\tau }_s^{{\beta }_{{}_{s,s}}}sin{x}_1.\hfill \end{array}$$

Hence, $dv={\tau }_s^{{\beta }_s-1}{J}_s\left(v^{\prime }\right)d{\tau }_{s}d\sigma \left(v^{\prime }\right)$, where

$${\beta }_s=\sum _{k=1}^s{\beta }_{s,k},J_s\left(v^{\prime }\right)=\sum _{k=1}^s{\beta }_{s,k}{\left(v_k^{\prime }\right)}^2,v^{\prime }=D_{{\tau }_s^{-1}}v\in {\mathbb{S}}^{s-1},$$

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

$$1\le J_s\left(v^{\prime }\right)\le A.$$

For ${\rho }_1=a_1+i{a}_2,{\rho }_2=b_1+i{b}_2$ $(a_1,a_2,b_1,b_2\in \mathbb{R}$ with $a_1,b_1>0)$, we assume that

$${\mathcal{K}}_{\mho ,h}(\upsilon ,\omega )=\frac{\mho (\upsilon ,\omega )h({\tau }_{\kappa }\left(\upsilon \right),{\tau }_{\eta }\left(\omega \right))}{{\left({\tau }_{\kappa }\left(\upsilon \right)\right)}^{{\beta }_{\kappa }-{\rho }_1}{\left({\tau }_{\eta }\left(\omega \right)\right)}^{{\beta }_{\eta }-{\rho }_2}},$$

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:

$$\begin{array}{ccc}\hfill \mho (D_{{\tau }_{\kappa }}\upsilon ,D_{{\tau }_{\eta }}\omega )& =& \mho (\upsilon ,\omega ),\forall {\tau }_{\kappa },{\tau }_{\eta }>0\hfill \end{array}$$

(1)

and

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

(2)

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

$${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)(x,y)={\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|T_{s,r}\left(g\right)(x,y)\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu },$$

where

$$T_{s,r}\left(g\right)(x,y)=\frac{1}{s^{{\rho }_1}{r}^{{\rho }_2}}{\int }_{{\tau }_{\kappa }\left(\upsilon \right)\le s}{\int }_{{\tau }_{\eta }\left(\omega \right)\le r}g(x-\upsilon ,y-\omega ){\mathcal{K}}_{\mho ,h}(\upsilon ,\omega )d\upsilon d\omega$$

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 [3,4,5,6,7,8,9]).

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 [10] 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 [11] improved the results presented in [10]. 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 $h\in {\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ with $\ell >1$. Here, ${\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ (for $\ell >1$) refers to the set of all measurable functions h such that

$${∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}=\underset{k,j\in \mathbb{Z}}{sup}{\left({\int }_{2^j}^{2^{j+1}}{\int }_{2^k}^{2^{k+1}}{\left|h({\tau }_{\kappa },{\tau }_{\eta })\right|}^{\ell }\frac{d{\tau }_{\kappa }d{\tau }_{\eta }}{{\tau }_{\kappa }{\tau }_{\eta }}\right)}^{1/\ell }<\infty .$$

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

$$\begin{array}{c}\hfill {∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}={∥{\left(\sum _{j,k\in \mathbb{Z}}{2}^{k{c}_1\mu }{2}^{j{c}_2\mu }{\left|({\psi }_{k,\kappa }\otimes {\psi }_{j,\eta })\ast g\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}<\infty ,\end{array}$$

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 [12] 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)

${\stackrel{.}{F}}_{p^{\prime }}^{-\overrightarrow{c},{\mu }^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })={\left({\stackrel{.}{F}}_p^{\overrightarrow{c},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })\right)}^{\ast }$, 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 [13] employed the extrapolation argument of Yano [14] to prove that whenever $\mathrm{\Omega }$ 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 $p\in (1,\infty )$,

$${∥{\mathcal{M}}_{\mho ,1}^{\left(\mu \right)}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_p{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })},$$

(3)

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 [15]. Very recently, the result in [13] was improved in [16] 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 $h\in {\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$, 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 [11] regarding the boundedness of the parabolic Marcinkiewicz operator ${\mathcal{G}}_{\mho ,h}^{\left(2\right)}$ and the results in [16] 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 ℧ as that was assumed in [16]?

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

Theorem 1.

Let $\mho \in L^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ for some $q\in (1,2]$ and $h\in {\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ for some $\ell \in (1,2]$. There then exists a real number $A_p>0$ such that

$${∥{\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{p,\mho ,h}{\left((q-1)(\ell -1)\right)}^{-2/\mu }{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}$$

for $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ if $\mu \le {\ell }^{\prime }$, and for ${\ell }^{\prime }<p<\infty$ if $\mu \ge {\ell }^{\prime }$; where $A_{p,\mho ,h}=A_p{∥\mho ∥}_{L^q({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}{∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}$ and $A_p$ is independent of $\mho ,h,q,\ell$

Theorem 2.

Let $\mho \in L^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ with $q\in (1,2]$ and $h\in {\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ for some $\ell \in (2,\infty )$. Then

$${∥{\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{p,\mho ,h}{\left(\frac{\ell }{q-1}\right)}^{2/\mu }{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}$$

for all $p\in ({\ell }^{\prime },\infty )$ if $\mu \ge {\ell }^{\prime }$ and for all $p\in (1,\mu )$ if $\mu \le {\ell }^{\prime }$.

Now by using the estimates in Theorems 1 and 2 and following the same method as employed in [17] along with the extrapolation argument as in [14,18,19], we obtain the following results.

Theorem 3.

Assume that h is given as in Theorem 1.

(i) 

If $\mho \in B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$, then the inequality

$$\begin{array}{c}\hfill {∥{\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_p\left({∥\mho ∥}_{B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}+1\right){∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

holds for ${\ell }^{\prime }<p<\infty$ if $\mu \ge {\ell }^{\prime }$, and for $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ if $\mu \le {\ell }^{\prime }$.

(ii) 

If $\mho \in L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$, then the inequality

$$\begin{array}{c}\hfill {∥{\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_p\left({∥\mho ∥}_{L{\left(logL\right)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}+1\right){∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

holds for ${\ell }^{\prime }<p<\infty$ if $\mu \ge {\ell }^{\prime }$, and for $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ if $\mu \le {\ell }^{\prime }$.

Theorem 4.

Suppose that $\mho \in L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\cup B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$ and $h\in {\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$ with $2<\ell <\infty$. The integral operator ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ is then bounded on $L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ for $p\in ({\ell }^{\prime },\infty )$ if $\mu \ge {\ell }^{\prime }$, and for $p\in (1,\mu )$ if $\mu \le {\ell }^{\prime }$.

Remark 1.

(i) 

For any $0<\gamma \le 1$, $m>0$ and $q>1$, the following inclusions hold and are proper:

$$C^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset Li{p}_{\gamma }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset L^q({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset L{(logL)}^m({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}),$$

$$\bigcup _{r>1}{L}^r({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset B_q^{(0,\tau )}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})forany\tau >-1,$$

$$L{(logL)}^{m_1}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset L{(logL)}^{m_2}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})for0<m_2<m_1,$$

$$B_q^{(0,{\tau }_1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\subset B_q^{(0,{\tau }_2)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})for-1<{\tau }_2<{\tau }_1.$$

(ii) 

For the special cases $h\equiv 1$ and $\mu =2$, the authors of [7] 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 $\mathrm{\Omega }\in L(logL)({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. In addition, 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 [8] if $\mathrm{\Omega }\in B_q^{(0,0)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ with $q>1$. Furthermore, the optimality of the condition $\mathrm{\Omega }\in B_q^{(0,0)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$ is established. Therefore, our conditions on ℧ in both Theorems 3 and 4 are known to be the best possible in their respective classes in the cases $\mu =2$ and $h\equiv 1$.

(iii) 

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 $\mho \in L{(logL)}^{2/\mu }({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})\cup B_q^{(0,\frac{2}{\mu }-1)}({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$. Hence, The results in Theorem 4 are improvement as well as generalization to the results in [10,13].

(iv) 

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

(v) 

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 [11] in which the authors proved the $L^p$ boundedness of ${\mathcal{G}}_{\mho ,h}^{\left(2\right)}$ only for $p\in (\frac{2{\ell }^{\prime }}{{\ell }^{\prime }-2},\frac{2\ell }{2-\ell })$. Therefore, our results improve the main results in [11].

(vi) 

For the special case $s\in \{\kappa ,\eta \}$ with ${\beta }_{s,1}={\beta }_{s,2}=\cdots ={\beta }_{s,s}=1$, $\mu =2$ and $h\equiv 1$, we extend the results in [4] in which the authors proved the $L^p$ boundedness of ${\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}$ for $p\in (1,\infty )$ under stronger condition $L^q({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})$.

(vii) 

For the special case $s\in \{\kappa ,\eta \}$ with ${\beta }_{s,1}={\beta }_{s,2}=\cdots ={\beta }_{s,s}=1$, our results are the same as that obtained in [16]. Thus, the results in [16] are special cases of our results.

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

2. Auxiliary Lemmas

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 ${\sigma }_h^{\ast }$ and $M_{h,\gamma }$ on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$ given by

$$\begin{array}{ccc}\hfill {\int \int }_{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}gd{\sigma }_{s,r}& =& \frac{1}{s^{{\rho }_1}{r}^{{\rho }_2}}{\int }_{1/2s\le {\tau }_{\kappa }\left(\upsilon \right)\le s}{\int }_{1/2r\le {\tau }_{\eta }\left(\omega \right)\le r}{\mathcal{K}}_{\mho ,h}(\upsilon ,\omega )g(\upsilon ,\omega )d\upsilon d\omega ,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\sigma }_h^{\ast }\left(g\right)(\upsilon ,\omega )& =& \underset{s,r\in {\mathbb{R}}_{+}}{sup}\left|\right|{\sigma }_{s,r}|\ast g(\upsilon ,\omega )|,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill M_{h,\gamma }\left(g\right)(\upsilon ,\omega )& =& \underset{j,k\in \mathbb{Z}}{sup}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}\left|\right|{\sigma }_{s,r}|\ast g(\upsilon ,\omega )|\frac{dsdr}{sr},\hfill \end{array}$$

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 [11].

Lemma 1.

Let $\mho \in L^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ and $h\in {\mathrm{\Delta }}_{\ell }\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)$ for some $q,\ell >1$. There then exists $A_{h,\mho }>0$ such that

$$\begin{array}{ccc}\hfill ∥{\sigma }_{s,r}∥& \le & A_{h,\mho },\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\widehat{\sigma }}_{s,r}(\zeta ,\xi )\right|}^2\frac{dsdr}{sr}& \le & A_{h,\mho }^2{ln}^2\left(\gamma \right){\left|D_{{\gamma }^k}\zeta \right|}^{\pm \frac{2\delta }{n_{1}ln\left(\gamma \right)}}{\left|D_{{\gamma }^j}\xi \right|}^{\pm \frac{2\delta }{n_{2}ln\left(\gamma \right)}},\hfill \end{array}$$

(5)

where $∥{\sigma }_{s,r}∥$ is the total variation of ${\sigma }_{s,r}$, $0<\delta <min\{\frac{1}{2},\frac{n_1}{2q^{\prime }},\frac{n_2}{2q^{\prime }},\frac{n_1}{{\beta }_{\kappa }},\frac{n_2}{{\beta }_{\eta }}\}$ and $n_1$, $n_2$ denote the distinct numbers of $\left\{{\beta }_{\kappa ,k}\right\}$, $\left\{{\beta }_{\eta ,j}\right\}$, respectively.

Lemma 2.

Let $\mho \in L^1\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ and $h\in {\mathrm{\Delta }}_{\ell }\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)$ for some $\ell >1$. Then we have that

$$\parallel {\sigma }_h^{\ast }{\left(g\right)\parallel }_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le {\tilde{A}}_{p,h,\mho }{\parallel g\parallel }_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}$$

(6)

for all $p\in ({\ell }^{\prime },\infty )$, where ${\tilde{A}}_{p,h,\mho }=A_p{∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}$.

By using Lemma 2, it is easy to show that

$$\parallel M_{h,\gamma }{\left(g\right)\parallel }_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le {\tilde{A}}_{p,h,\mho }{ln}^2\left(\gamma \right){\parallel g\parallel }_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}$$

(7)

for all $p\in ({\ell }^{\prime },\infty )$.

Now we need to prove the following result:

Lemma 3.

Let $\mho \in L^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$, $h\in {\mathrm{\Delta }}_{\ell }\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)$ with $1<\ell ,q\le 2$ and $\gamma =2^{{\ell }^{\prime }{q}^{\prime }}$. Then for all $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ with $\mu \in (1,{\ell }^{\prime }]$, we have

$$\begin{array}{c}\hfill {∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{{\gamma }^j}{\overset{{\gamma }^{j+1}}{\int }}\underset{{\gamma }^k}{\overset{{\gamma }^{k+1}}{\int }}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{h,\mho }{\left(\frac{1}{(q-1)(\ell -1)}\right)}^{2/\mu }{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })},\end{array}$$

where $\{{\mathcal{F}}_{j,k}(·,·),j,k\in \mathbb{Z}\}$ is any class of functions defined on ${\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$.

Proof. 

Let us start with the case $p\in (\mu ,\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$. It is clear that

$$\begin{array}{ccc}\hfill & & {\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}(\upsilon ,\omega )\right|}^{\mu }\le A{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{(\mu /{\mu }^{\prime })}{∥h∥}_{{\mathrm{\Delta }}_{\ell }({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}^{(\mu /{\mu }^{\prime })}\underset{r/2}{\overset{r}{\mathit{\int }}}\underset{s/2}{\overset{s}{\mathit{\int }}}{\mathit{\int }\mathit{\int }}_{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}\left|J_{\kappa }\left(\upsilon \right)\right|\left|J_{\eta }\left(\omega \right)\right|\hfill \\ & \times & {\left|{\mathcal{F}}_{j,k}(\upsilon -D_{{\tau }_{\kappa }}x,\omega -D_{{\tau }_{\eta }}y)\right|}^{\mu }\left|\mho (x,y)\right|d\sigma \left(x\right)d\sigma \left(y\right){\left|h({\tau }_{\kappa },{\tau }_{\eta })\right|}^{\mu -\frac{\mu \ell }{{\mu }^{\prime }}}\frac{d{\tau }_{\kappa }d{\tau }_{\eta }}{{\tau }_{\kappa }{\tau }_{\eta }}.\hfill \end{array}$$

(8)

By duality there exists a non-negative function $\phi \in L^{{(p/\mu )}^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$ such that ${∥\phi ∥}_{L^{{(p/\mu )}^{\prime }}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le 1$ and

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^{\mu }\hfill \\ & =& {\int \int }_{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}(\upsilon ,\omega )\right|}^{\mu }\frac{dsdr}{sr}\phi (\upsilon ,\omega )d\upsilon d\omega .\hfill \end{array}$$

(9)

Thus, by the last two inequalities and Hölder’s inequality, we obtain that

$$\begin{array}{ccc}& & {∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^{\mu }\le A{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{(\mu /{\mu }^{\prime })}{∥h∥}_{{\mathrm{\Delta }}_1({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}^{(\mu /{\mu }^{\prime })}\hfill \\ & \times & {\int \int }_{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}(\upsilon ,\omega )\right|}^{\mu }\right)M_{{\left|h\right|}^{\mu -\frac{\mu \ell }{{\mu }^{\prime }}},\gamma }\left(\overline{\phi }\right)(-\upsilon ,-\omega )d\upsilon d\omega \hfill \\ & \le & A{∥h∥}_{{\mathrm{\Delta }}_1({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}^{(\mu /{\mu }^{\prime })}{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{(\mu /{\mu }^{\prime })}{∥\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}\right|}^{\mu }∥}_{L^{(p/\mu )}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}{∥M_{{\left|h\right|}^{\frac{\mu ({\mu }^{\prime }-\ell )}{{\mu }^{\prime }}},\gamma }\left(\overline{\phi }\right)∥}_{L^{{(p/\mu )}^{\prime }}({\mathbb{R}}^n\times {\mathbb{R}}^m)},\hfill \end{array}$$

where $\overline{\phi }(\upsilon ,\omega )=\phi (-\upsilon ,-\omega )$. As ${\left|h\right|}^{\frac{\mu ({\mu }^{\prime }-\ell )}{{\mu }^{\prime }}}$ belongs to the space ${\mathrm{\Delta }}_{\frac{{\mu }^{\prime }\ell }{\mu ({\mu }^{\prime }-\ell )}({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}$, then by employing (7), we obtain that

$$\begin{array}{c}\hfill {∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{\mho ,h}{ln}^{2/\mu }\left(\gamma \right){∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

(10)

for all $p\in (\mu ,\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$.

Let us consider the case $p=\mu$, by Hölder’s inequality and (8), we get

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^{\mu }\le A{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{(\mu /{\mu }^{\prime })}{∥h∥}_{{\mathrm{\Delta }}_1({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}^{(\mu /{\mu }^{\prime })}\hfill \\ & \times & \sum _{j,k\in \mathbb{Z}}{\int }_{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\int }_{r/2}^r{\int }_{s/2}^s{\int }_{{\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}}{\left|{\mathcal{F}}_{j,k}(\upsilon -D_{{\tau }_{\kappa }}x,\omega -D_{{\tau }_{\eta }}y)\right|}^{\mu }\hfill \\ & \times & \left|\mho (x,y)\right|{\left|h({\tau }_{\kappa },{\tau }_{\eta })\right|}^{\frac{\mu ({\mu }^{\prime }-\ell )}{{\mu }^{\prime }}}d\sigma \left(x\right)d\sigma \left(y\right)\frac{d{\tau }_{\kappa }d{\tau }_{\eta }}{{\tau }_{\kappa }{\tau }_{\eta }}\frac{dsdr}{sr}d\upsilon d\omega \hfill \\ & \le & A{\left(\frac{1}{(q-1)(\ell -1)}\right)}^2{∥\mho ∥}_{L^1({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1})}^{(\mu /{\mu }^{\prime })+1}{∥h∥}_{{\mathrm{\Delta }}_1({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})}^{(\mu /{\mu }^{\prime })+1}{\int }_{{\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }}\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}(\upsilon ,\omega )\right|}^{\mu }\right)d\upsilon d\omega .\hfill \end{array}$$

(11)

Finally we prove the lemma for the case $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\mu )$. Let $\mathcal{L}$ be the linear operator defined on any function $\mathcal{F}={\mathcal{F}}_{j,k}(x,y)$ by $\mathcal{L}\left(\mathcal{F}\right)={\sigma }_{{\gamma }^{k}s,{\gamma }^{j}r}\ast {\mathcal{F}}_{j,k}(x,y)$. It is easy to see that

$$\begin{array}{c}\hfill {∥{∥{∥\mathcal{L}\left(\mathcal{F}\right)∥}_{L^1([1,\gamma )\times [1,\gamma )),\frac{dsdr}{sr}}∥}_{l^1(\mathbb{Z}\times \mathbb{Z})}∥}_{L^1({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A{ln}^2\left(\gamma \right){∥\left(\sum _{j,k\in \mathbb{Z}}\left|{\mathcal{F}}_{j,k}\right|\right)∥}_{L^1({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\end{array}$$

(12)

Furthermore, by the inequality (6) we get

$$\begin{array}{ccc}\hfill {∥\underset{j,k\in \mathbb{Z}}{sup}\underset{(s,r)\in [1,\gamma ]\times [1,\gamma ]}{sup}\left|{\sigma }_{{\gamma }^{k}s,{\gamma }^{j}r}\ast {\mathcal{F}}_{j,k}\right|∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}& \le & {∥{\sigma }_h^{\ast }\left(\underset{j,k\in \mathbb{Z}}{sup}\left|{\mathcal{F}}_{j,k}\right|\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \\ & \le & A_{h,\mho }{∥\underset{j,k\in \mathbb{Z}}{sup}\left|{\mathcal{F}}_{j,k}\right|∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \end{array}$$

for all $p\in ({\ell }^{\prime },\infty )$, which in turn implies that

$$\begin{array}{ccc}\hfill {∥{∥\parallel {\sigma }_{{\gamma }^{k}s,{\gamma }^{j}r}\ast {\mathcal{F}}_{j,k}{\parallel }_{L^{\infty }([1,\gamma ]\times [1,\gamma ],\frac{dsdr}{sr})}∥}_{l^{\infty }(\mathbb{Z}\times \mathbb{Z})}∥}_{L^p({\mathbb{R}}^n\times {\mathbb{R}}^m)}& \le & A_{h,\mho }{∥{∥{\mathcal{F}}_{j,k}∥}_{l^{\infty }(\mathbb{Z}\times \mathbb{Z})}∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}.\hfill \end{array}$$

(13)

Consequently, the proof of the lemma is finished in the case $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\mu )$ if we interpolate (12) with (13). □

Lemma 4.

Let ℧ and $\{{\mathcal{F}}_{j,k}(·,·),j,k\in \mathbb{Z}\}$ be given as in Lemma 3. Suppose that $h\in {\mathrm{\Delta }}_{\ell }\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)$ for some $\ell \in (1,\infty )$. Then there exists a positive constant $A_{h,\mho }$ such that

$$\begin{array}{c}\hfill {∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{{\gamma }^j}{\overset{{\gamma }^{j+1}}{\int }}\underset{{\gamma }^k}{\overset{{\gamma }^{k+1}}{\int }}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{h,\mho }{\left(\frac{\ell }{q-1}\right)}^{2/\mu }{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

(14)

for all $p\in (1,\mu )$ if $\mu \le {\ell }^{\prime }$ and $\gamma \ge 2$; and

$$\begin{array}{c}\hfill {∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{{\gamma }^j}{\overset{{\gamma }^{j+1}}{\int }}\underset{{\gamma }^k}{\overset{{\gamma }^{k+1}}{\int }}{\left|{\sigma }_{s,r}\ast {\mathcal{F}}_{j,k}\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{h,\mho }{\left(\frac{\ell }{(q-1)(\ell -1)}\right)}^{2/\mu }{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{F}}_{j,k}\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

(15)

for all $p\in ({\ell }^{\prime },\infty )$ if $\mu \ge {\ell }^{\prime }$.

A proof of this Lemma can be constructed by following a similar argument as that employed in the proof of Lemma 3 and following similar argument as that used in the proofs of Theorems 4–5 in [16] (with minor modifications). We omit the details.

3. Proof of the Main Results

Proof of Theorem 1. Suppose that $\mho \in L^q\left({\mathbb{S}}^{\kappa -1}\times {\mathbb{S}}^{\eta -1}\right)$ and $h\in {\mathrm{\Delta }}_{\ell }\left({\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\right)$ for some $q,\ell \in (1,2]$, and that $\mu >1$. By Minkowski’s inequality we get

$$\begin{array}{ccc}\hfill {\mathcal{G}}_{\mho ,h}^{\left(\mu \right)}\left(g\right)(x,y)& =& \left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}\left|\sum _{j,k=0}^{\infty }\frac{1}{s^{{\rho }_1}{r}^{{\rho }_2}}{\int }_{2^{-j-1}s<{\rho }_{\tau }\left(\upsilon \right)\le 2^{-j}s}{\int }_{2^{-k-1}r<{\rho }_{\kappa }\left(\omega \right)\le 2^{-k}r}{\mathcal{K}}_{\mho ,h}(\upsilon ,\omega )\right.\right.\hfill \\ & \times & {\left.{\left.g(x-\upsilon ,y-\omega )d\upsilon d\omega \right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }\hfill \\ & \le & \sum _{j,k=0}^{\infty }\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}\left|\frac{1}{s^{{\rho }_1}{r}^{{\rho }_2}}{\int }_{2^{-j-1}s<{\rho }_{\tau }\left(\upsilon \right)\le 2^{-j}s}{\int }_{2^{-k-1}r<{\rho }_{\kappa }\left(\omega \right)\le 2^{-k}r}{\mathcal{K}}_{\mho ,h}(\upsilon ,\omega )\right.\right.\hfill \\ & \times & {\left.{\left.g(x-\upsilon ,y-\omega )d\upsilon d\omega \right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }\hfill \\ & \le & \frac{2^{a_1+b_1}}{(2^{a_1}-1)(2^{b_1}-1)}{\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|{\sigma }_{s,r}\ast g(x,y)\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }.\hfill \end{array}$$

(16)

Let $\gamma =2^{{\ell }^{\prime }{q}^{\prime }}$. For $k\in \mathbb{Z}$, choose a collection of smooth functions $\left\{{\psi }_k\right\}$ defined on ${\mathbb{R}}^{+}$ satisfying the following properties:

$$\begin{array}{ccc}\hfill {\psi }_k& \subset & [0,1],\sum _{k\in \mathbb{Z}}{\psi }_k\left(s\right)=1,\hfill \\ \hfill \mathrm{supp}\left({\psi }_k\right)& \subseteq & [{\gamma }^{-1-k},{\gamma }^{1-k}],and\left|\frac{d^t{\psi }_k\left(s\right)}{d{s}^t}\right|\le \frac{C_t}{s^t},\hfill \end{array}$$

where $C_t$ does not depend on $\gamma$. For $(\zeta ,\xi )\in {\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }$, define the operators $\left(\widehat{{\mathrm{\Psi }}_k}\left(\zeta \right)\right)={\psi }_k\left({\tau }_{\kappa }\left(\zeta \right)\right)$ and $\left(\widehat{{\mathrm{\Psi }}_j}\left(\xi \right)\right)={\psi }_j\left({\tau }_{\eta }\left(\xi \right)\right)$. Hence, for any $g\in \mathcal{S}({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })$,

$${\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|{\sigma }_{s,r}\ast g(x,y)\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }\le A\sum _{n,m\in \mathbb{Z}}{\mathcal{H}}_{n,m}\left(g\right)(x,y),$$

(17)

where

$${\mathcal{H}}_{n,m}\left(g\right)(x,y)={\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|{\mathcal{V}}_{n,m}\left(g\right)(x,y,s,r)\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }$$

and

$${\mathcal{V}}_{n,m}\left(g\right)(x,y,s,r)=\sum _{j,k\in \mathbb{Z}}{\sigma }_{s,r}\ast \left({\mathrm{\Psi }}_{k+m}\otimes {\mathrm{\Psi }}_{j+n}\right)\ast g(x,y){\chi }_{{}_{[{\gamma }^k,{\gamma }^{k+1})\times [{\gamma }^j,{\gamma }^{j+1})}}(s,r).$$

Thus, to finish the proof of Theorem 1, it is enough to show that there exists a positive constant $\epsilon$ such that

$$\begin{array}{c}\hfill {∥{\mathcal{H}}_{n,m}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\le A_{p,h,\mho }{\left(\frac{1}{(q-1)(\ell -1)}\right)}^{2/\mu }{2}^{-\frac{\epsilon }{2}\left(\right|n|+\left|m\right|)}{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\end{array}$$

(18)

for all ${\ell }^{\prime }<p<\infty$ with ${\ell }^{\prime }\le \mu$ and for all $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ with ${\ell }^{\prime }\ge \mu$.

First, we estimate the norm of ${\mathcal{H}}_{n,m}\left(g\right)$ for the case $p=\mu =2$. By using Fubini’s theorem along with Plancherel’s theorem and the inequality (5) we get

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{H}}_{n,m}\left(g\right)∥}_{L^2({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^2\hfill \\ & \le & \sum _{j,k\in \mathbb{Z}}{\int \int }_{B_{n+j,m+k}}\left({\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\widehat{\sigma }}_{s,r}(\zeta ,\xi )\right|}^2\frac{dsdr}{sr}\right){\left|\widehat{g}(\zeta ,\xi )\right|}^{2}d\zeta d\xi \hfill \\ & \le & A_p{\left(\frac{1}{(q-1)(\ell -1)}\right)}^2{A}_{h,\mho }^2\sum _{j,k\in \mathbb{Z}}{\int \int }_{B_{n+j,m+k}}{\left|D_{{\gamma }^k}\zeta \right|}^{\pm \frac{2\delta }{n_{1}ln\left(\gamma \right)}}{\left|D_{{\gamma }^j}\xi \right|}^{\pm \frac{2\delta }{n_{2}ln\left(\gamma \right)}}{\left|\widehat{g}(\zeta ,\xi )\right|}^{2}d\zeta d\xi \hfill \\ & \le & A_p{\left(\frac{1}{(q-1)(\ell -1)}\right)}^2{2}^{-\epsilon \left(\right|n|+|m\left|\right)}{A}_{h,\mho }^2\sum _{j,k\in \mathbb{Z}}{\int \int }_{B_{n+j,m+k}}{\left|\widehat{g}(\zeta ,\xi )\right|}^{2}d\zeta d\xi \hfill \\ & \le & A_p{\left(\frac{1}{(q-1)(\ell -1)}\right)}^2{2}^{-\epsilon \left(\right|n|+|m\left|\right)}{A}_{h,\mho }^2{∥g∥}_{L^2({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}^2,\hfill \end{array}$$

(19)

where $B_{j,k}=\left\{(\zeta ,\xi )\in {\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta }:(\left|\zeta \right|,\left|\xi \right|)\in [{\gamma }^{-1-k},{\gamma }^{1-k}]\times [{\gamma }^{-1-j},{\gamma }^{1-j}]\right\}$ and $\epsilon \in (0,1)$.

Now, let us estimate the $L^p$-norm of ${\mathcal{H}}_{n,m}\left(g\right)$. By Littlewood–Paley theory, Lemma 3, inequality (15), and invoking Lemma 2.3 in [13], we get

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{H}}_{m,n}\left(g\right)∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \\ & \le & A{∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{{\gamma }^j}^{{\gamma }^{j+1}}{\int }_{{\gamma }^k}^{{\gamma }^{k+1}}{\left|{\sigma }_{s,r}\ast \left({\mathrm{\Psi }}_{m+k}\otimes {\mathrm{\Psi }}_{n+j}\right)\ast g\right|}^{\mu }\frac{dsdr}{sr}\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \\ & \le & A_{h,\mho }\frac{1}{{\left[(q-1)(\ell -1)\right]}^{2/\mu }}{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|\left({\mathrm{\Psi }}_{m+k}\otimes {\mathrm{\Psi }}_{n+j}\right)\ast g\right|}^{\mu }\right)}^{1/\mu }∥}_{L^p({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \\ & \le & A_p\frac{1}{{\left[(q-1)(\ell -1)\right]}^{2/\mu }}{A}_{h,\mho }{∥g∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\mu }({\mathbb{R}}^{\kappa }\times {\mathbb{R}}^{\eta })}\hfill \end{array}$$

(20)

for ${\ell }^{\prime }<p<\infty$ with $\mu \ge {\ell }^{\prime }$, and also for $p\in (\frac{\mu {\ell }^{\prime }}{\mu +{\ell }^{\prime }-1},\frac{{\mu }^{\prime }\ell }{{\mu }^{\prime }-\ell })$ with $\mu \le {\ell }^{\prime }$. Therefore, by interpolating (19) with (20), we obtain (18). The proof of Theorem 1 is complete.

Proof of Theorem 2. A proof can be constructed by following a similar approach as that used in the proof of Theorem 1 except that we employ Lemma 4 instead of Lemma 3. We omit the details.