On Certain Rough Marcinkiewicz Integral Operators with Grafakos-Stefanov Kernels on Product Spaces

1. Introduction

Let ${\mathbb{R}}^q$ ($q=m$ or n) 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 $d{\sigma }_q(·)\equiv d\sigma$. Also, let $z^{\prime }=z/\left|z\right|$ for $z\in {\mathbb{R}}^q\setminus \left\{0\right\}$.

Let ℧ be a measurable function defined on ${\mathbb{R}}^m\times {\mathbb{R}}^n$, integrable over ${\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}$ and satisfy the following:

$$\begin{array}{ccc}\hfill \mho (rx,sy)& =& \mho (x,y),\forall r,s>0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{S}}^{m-1}}\mho (x,y)d\sigma \left(x\right)& =& {\int }_{{\mathbb{S}}^{n-1}}\mho (x,y)d\sigma \left(y\right)=0.\hfill \end{array}$$

(2)

For an appropriate mapping $\mathsf{\Phi }:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$, we consider the parametric Marcinkiewicz integral operator ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ along the surface of revolution ${\mathsf{\Lambda }}_{\mathsf{\Phi }}(u,v)=\left(u,v,\mathsf{\Phi }(\left|u\right|,\left|v\right|)\right)$ defined, initially for $g\in C_0^{\infty }({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$, by

$${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}\left(g\right)(u,v,w)={\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|F_{r,s}\left(g\right)(u,v,w)\right|}^2\frac{drds}{rs}\right)}^{1/2},$$

(3)

where

$$F_{r,s}\left(g\right)(u,v,w)=\frac{1}{rs}{\int }_{\left|y\right|\le s}{\int }_{\left|x\right|\le r}f(u-x,v-y,w-\mathsf{\Phi }(\left|x\right|,\left|y\right|))\frac{\mho (x,y)}{{\left|x\right|}^{m-1}{\left|y\right|}^{n-1}}dxdy.$$

We point out her that the Marcinkiewicz integral ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is a natural generalization of the Marcinkiewicz inegral ${\mathfrak{M}}_{\mho }^{\varphi }$ along surface of revolution ${\mathsf{\Lambda }}_{\mathsf{\Phi }}\left(u\right)=\left(u,\mathsf{\Phi }\left(\left|u\right|\right)\right)$ in the one parameter setting which is given by

$${\mathfrak{M}}_{\mho }^{\varphi }\left(g\right)(u,u_{m+1})={\left({\int }_{{\mathbb{R}}_{+}}{\left|\frac{1}{r}{\int }_{\left|x\right|\le r}g(u-x,u_{m+1}-\varphi \left(\left|x\right|\right))\frac{\mho \left(x\right)}{{\left|x\right|}^{m-1}}dx\right|}^2\frac{dr}{r}\right)}^{1/2}.$$

(4)

The investigation of the $L^p$ boundedeness of ${\mathfrak{M}}_{\mho }^{\varphi }$ under certain assumptions on ℧ and $\varphi$ has received a large amount of attention by many mathematicians. For a sample of some known results relevant to our study, we refer the readers to see [1,2,3,4,5,6,7].

Let $G_{\alpha }\left({\mathbb{S}}^{m-1}\right)$ (for $\alpha >0$) be the class of all functions $\mho \in L^1\left({\mathbb{S}}^{m-1}\right)$ that satisfy the condition

$$\underset{\xi \in {\mathbb{S}}^{m-1}}{sup}{\int }_{{\mathbb{S}}^{m-1}}\left|\mho \left(x\right)\right|{log}^{\alpha }\left({\left|\xi ·x\right|}^{-1}\right)d\sigma \left(x\right)<\infty .$$

(5)

It is worth mentioning that the class $G_{\alpha }\left({\mathbb{S}}^{m-1}\right)$ was defined by Walsh in [8] and developed by Grafakos and Stefanov in [9].

An interesting result related to our work in the one parameter setting was obtained in [10] as described in the following theorem:

Theorem A.

Let $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\right)$ for some $\alpha >1/2$. Then ${\mathfrak{M}}_{\mho }^{\varphi }$ is bounded on $L^p({\mathbb{R}}^m\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$ if one of the following conditions is satisfied:

(a) 

$\varphi \in C^1\left({\mathbb{R}}_{+}\right)$, ${\varphi }^{\prime }$ is increasing and convex function with either $m=2$ or $m\ge 3$ and ${\varphi }^{\prime }\left(0\right)=0$.

(b) 

$\varphi$ is a polynomial with either $m=2$ or $m\ge 3$ and ${\varphi }^{\prime }\left(0\right)=0$.

Our focus will be on the operator ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$. If $\mathsf{\Phi }\equiv 0$, we denote ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ by ${\mathfrak{M}}_{\mho }$ which is the classical Marcinkiewicz integral on product domains. The study of the boundedness of ${\mathfrak{M}}_{\mho }$ started by Ding in [11] and then continued by many authors. For a sample of past studies and for more information about the applications and development of the operator ${\mathfrak{M}}_{\mho }$, we refer the readers to consult [14–21] and the references therein.

For $\alpha >0$, let $G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ be the set of all functions $\mho \in L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ which satisfy Grafakos and Stefanov condition:

$$\underset{(\xi ,\zeta )\in {\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}{sup}{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mho \left(x,y\right)\right|{log}^{\alpha }\left({\left|\xi ·x\right|}^{-1}\right){log}^{\alpha }\left({\left|\zeta ·y\right|}^{-1}\right)d\sigma \left(x\right)d\sigma \left(y\right)<\infty .$$

(6)

By following the same arguments as those in [9], we get

$$\begin{array}{ccc}\hfill \bigcup _{\kappa >1}{L}^{\kappa }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)& ⊈& G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)forany\alpha >0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \bigcap _{\alpha >0}{G}_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)& ⊈& L(logL)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})⊈\bigcup _{\alpha >0}{G}_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right).\hfill \end{array}$$

(8)

Very recently, in [23] the authors studied the related singular integral operators along surfaces of revolution ${\mathcal{T}}_{\mho ,\mathsf{\Phi }}$ which are given by

$${\mathcal{T}}_{\mho ,\mathsf{\Phi }}\left(g\right)(u,v,w)={\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}f(u-x,v-y,w-\mathsf{\Phi }(\left|x\right|,\left|y\right|))\frac{\mho (x,y)}{{\left|x\right|}^m{\left|y\right|}^n}dxdy,$$

(9)

where $\mathsf{\Phi }:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$ is an appropriate mapping. When $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$, the authors of [23] showed that ${\mathcal{T}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$ under various assumptions on $\mathsf{\Phi }$.

In light of the assumptions imposed on $\mathsf{\Phi }$ in [23] and of the results in [10] concerning the boundedness of Marcinkiewicz operator ${\mathfrak{M}}_{\mho }^{\varphi }$ in the one parameter setting whenever $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\right)$ as well as of the results in [21] concerning the boundedness of Marcinkiewicz operator ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ whenever $\mho \in L(logL)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\cup B_{\kappa }^{(0,0)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, a question arises naturally is the following:

Question: Under the same conditions on $\mathsf{\Phi }$ in [23], does the operator ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ satisfy the $L^p$ boundedness provided that $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$?

Our main purpose in this paper is on giving a positive answer to this question. Indeed, we obtain the following:

Theorem 1.

Assume that ℧ belongs to the space $G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$ and satisfies the conditions $\left(1\right)$-$\left(2\right)$. Assume that $\mathsf{\Phi }\in C^1({\mathbb{R}}_{+}\times {\mathbb{R}}_{+})$. If one of the following conditions holds, then ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$.

(i) $m=2=n$, $D_t\mathsf{\Phi }(t,\ell )$ and $D_{\ell }\mathsf{\Phi }(t,\ell )$ are convex increasing functions,

(ii) $m=2$, $n\ge 3$, $D_t\mathsf{\Phi }(t,\ell )$ and $D_{\ell }\mathsf{\Phi }(t,\ell )$ are convex increasing functions with $D_{\ell }(t,0)=0$,

(iii) $n=2$, $m\ge 3$, $D_t\mathsf{\Phi }(t,\ell )$ and $D_{\ell }\mathsf{\Phi }(t,\ell )$ are convex increasing functions with $D_t\mathsf{\Phi }(0,\ell )=0$,

(iv) $m\ge 3$, $n\ge 3$, ${\mathsf{\Phi }}_t(t,\ell )$ and $D_{\ell }\mathsf{\Phi }(t,\ell )$ are convex increasing functions with $D_t\mathsf{\Phi }(0,\ell )=0=D_{\ell }\mathsf{\Phi }(t,0)$.

Theorem 2.

Assume that $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$ and satisfies the conditions $\left(1\right)$-$\left(2\right)$. Assume that $\mathsf{\Phi }(t,\ell )={\displaystyle \sum _{i=0}^{d_1}}{\displaystyle \sum _{j=0}^{d_2}}{a}_{j,i}{t}^{{\alpha }_i}{\ell }^{{\beta }_j}$ is a generalized polynomial on ${\mathbb{R}}^2$, where the exponents in each set $\{{\alpha }_i:0\le i\le d_1\}$ and $\{{\beta }_j:0\le j\le d_2\}$ are distinct positive numbers. Then ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$ whenever one of the following conditions holds:

(i) ${\alpha }_i\ne 1$ and ${\beta }_j\ne 1$ for all $0\le i\le d_1$ and $0\le j\le d_2$,

(ii) There is ${\alpha }_{i_0}=1$ for some $0\le i_0\le d_1$ and ${\beta }_j\ne 1$ for all $0\le j\le d_2$; and if $m\ge 3$, we need $D_t\mathsf{\Phi }(0,\ell )=0$,

(iii) There is ${\beta }_{j_0}=1$ for some $0\le j_0\le d_2$ and ${\alpha }_i\ne 1$ for all $0\le i\le d_1$; and if $n\ge 3$, we need $D_{\ell }\mathsf{\Phi }(t,0)=0$,

(iv) There exists ${\alpha }_{i_0}=1$ for some $0\le i_0\le d_1$ and there exists ${\beta }_{j_0}=1$ for some $0\le j_0\le d_2$. Moreover, whenever $m\ge 3$ we need $D_t\mathsf{\Phi }(0,\ell )=0$ and whenever $n\ge 3$ we need $D_{\ell }\mathsf{\Phi }(t,0)=0$.

Theorem 3.

Assume that ℧ is given as in Theorem 1. Assume that $\mathsf{\Phi }(t,\ell )=\varphi \left(t\right)P(\ell )$, where $\varphi \left(t\right)$ is in $C^1\left({\mathbb{R}}_{+}\right)$, ${\varphi }^{\prime }$ is increasing and convex function; and P is a generalized polynomials given by $P(\ell )={\displaystyle \sum _{j=0}^{d_2}}{a}_j{\ell }^{{\beta }_j}$. Then ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$ if one of the following conditions is satisfied:

(i) ${\beta }_j\ne 1$ for all $0\le j\le d_2$, and if $m\ge 3$ we need ${\varphi }^{\prime }\left(0\right)=0$,

(ii) There is ${\beta }_{j_0}=1$ for some $0\le j_0\le d_2$, and if $m,n\ge 3$ we need ${\varphi }^{\prime }\left(0\right)=0=P\left(0\right)$.

Theorem 4.

Assume $\mathsf{\Phi }(t,\ell )={\varphi }_1\left(t\right)+{\varphi }_2(\ell )$, where ${\varphi }_j(·)$ ($j=1,2$) is either a generalized polynomial or is in $C^1\left({\mathbb{R}}_{+}\right)$, ${\varphi }_j^{\prime }$ is increasing and convex function. If $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$ and one of the following conditions is held, then ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$.

Case 1: ${\varphi }_j(·)$ ($j=1,2$) is in $C^1\left({\mathbb{R}}_{+}\right)$, ${\varphi }_j^{\prime }$ is increasing and convex function.

(i) $m=2$, $n=2$,

(ii) $m=2$, $n\ge 3$ and ${\varphi }_1^{\prime }\left(0\right)$,

(iii) $n=2$, $m\ge 3$ and ${\varphi }_2^{\prime }\left(0\right)$,

(iv) $m\ge 3$, $n\ge 3$ and ${\varphi }_1^{\prime }\left(0\right)=0={\varphi }_2^{\prime }\left(0\right)$.

Case 2: ${\varphi }_1(·)$ is in $C^1\left({\mathbb{R}}_{+}\right)$, ${\varphi }_2(·)$ is increasing and convex function; ${\varphi }_2(·)$ is a generalized polynomial on $\mathbb{R}$ given by ${\varphi }_2(\ell )={\displaystyle \sum _{j=0}^{d_2}}{a}_j{\ell }^{{\beta }_j}$, where the exponents in the set $\{{\beta }_j:0\le j\le d_2\}$ are distinct positive numbers.

(i) ${\beta }_j\ne 1$ for all $0\le j\le d_2$, and if $n\ge 3$ we need ${\varphi }_1^{\prime }\left(0\right)=0$,

(ii) There is ${\beta }_{j_0}=1$ for some $0\le j_0\le d_2$, and if $m,n\ge 3$ we need ${\varphi }_1^{\prime }\left(0\right)=0={\varphi }_2^{\prime }\left(0\right)$.

Theorem 5.

Assume that $\mathsf{\Phi }(t,\ell )=\varphi (t\ell )$ for all $t,\ell >0$, where $\varphi \in C^1\left({\mathbb{R}}_{+}\right)$. If $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$ and one of the following conditions is satisfied, then ${\mathfrak{M}}_{\mho ,\mathsf{\Phi }}$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$ for all for all $p\in (\frac{1+2\alpha }{2\alpha },1+2\alpha )$.

(i) $m=2=n$ and ${\varphi }^{\prime }(·)$ is an increasing and convex function,

(ii) If m or $n\ge 3$ and ${\varphi }^{\prime }(·)$ is an increasing and convex function with ${\varphi }^{\prime }\left(0\right)=0$.

We point out that the surfaces of revolution ${\mathsf{\Lambda }}_{\mathsf{\Phi }}(u,v)=\left(u,v,\mathsf{\Phi }(\left|u\right|,\left|v\right|)\right)$ considered in Theorems 1–5 cover several important natural classical surfaces. For example, our theorems allow surfaces of the type ${\mathsf{\Lambda }}_{\mathsf{\Phi }}$, where $\mathsf{\Phi }(t,\ell )=t^2{\ell }^2(e^{-1/t}+e^{-1/\ell }),t,\ell >0)$; $\mathsf{\Phi }(t,\ell )=t^{\alpha }{\ell }^{\beta }$ with $\alpha ,\beta >0$; $\mathsf{\Phi }(t,\ell )=P(t,\ell )$ is a polynomial; $\mathsf{\Phi }(t,\ell )={\varphi }_1\left(t\right){\varphi }_2(\ell )$, where for $i\in \{1,2$}, each ${\varphi }_i$$\in C^2\left({\mathbb{R}}_{+}\right)$ is a convex increasing function with ${\varphi }_i\left(0\right)=0$.

Henceforward, the constant C denotes a positive real constant which is not necessary the same at each occurrence but it is independent of the essential variables.

2. Some Lemmas

For a suitable mapping $\mathsf{\Phi }(r,s)$ on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$, we define the family of measures $\{{\lambda }_{\mho ,\mathsf{\Phi },r,s}^{k,j}:={\lambda }_{r,s}^{k,j}:k,j\in \mathbb{Z},r,s\in {\mathbb{R}}_{+}\}$ and its related maximal operator ${\lambda }^{*}$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}$ by

$$\begin{array}{ccc}\hfill {\int \int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}}gd{\lambda }_{r,s}^{k,j}& =& \frac{1}{2^{k+j}rs}{\int }_{2^{j}s\le \left|u\right|\le 2^{j+1}s}{\int }_{2^{k}r\le \left|v\right|\le 2^{k+1}r}g(v,u,\mathsf{\Phi }(\left|v\right|,\left|u\right|))K_{\mho ,h}(v,u)dvdu\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\lambda }^{*}\left(g\right)(x,y,w)& =& \underset{k,j\in \mathbb{Z}}{sup}\underset{r,s\in {\mathbb{R}}_{+}}{sup}\left|\right|{\lambda }_{r,s}^{k,j}|*g(x,y,w)|,\hfill \end{array}$$

where $|{\lambda }_{r,s}^{k,j}|$ is defined in the same way as ${\lambda }_{r,s}^{k,j}$ but with replacing ℧ by $|\mho |$ .

The following lemma can be proved directly by the results in [24,25], see also [23].

Lemma 1.

Let $P:{\mathbb{R}}^{+}\to {\mathbb{R}}^q$ be a function given by $P\left(t\right)=(c_1{t}^{{\gamma }_1},\cdots ,c_n{t}^{{\gamma }_q})$, where $c_j^{\prime }$s are real numbers and ${\gamma }_j^{\prime }$s are distinct positive exponents (not necessarily integer). Define the maximal function related to P by

$${\mathcal{M}}_{P}g\left(z\right)=\underset{h>0}{sup}\frac{1}{h}{\int }_0^h\left|g(z-P(t\left)\right)\right|dt.$$

Then, for all $1<p\le \infty$, there exists a constant $C_p>0$ (independent of g and $c_j^{\prime }s$) such that

$${∥{\mathcal{M}}_P\left(g\right)∥}_{L^p\left({\mathbb{R}}^q\right)}\le C_p{\left|g\right.∥}_{L^p\left({\mathbb{R}}^q\right)}.$$

We shall need the following estimate from [26].

Lemma 2.

Let $\phi :{\mathbb{R}}_{+}\to \mathbb{R}$ be a $C^1$ function such that ${\phi }^{\prime }$ is convex and increasing with ${\phi }^{\prime }\left(0\right)=0$. Then, there exists a positive constant C such that

$$\left|{\int }_1^b{e}^{-i(2^{k}tr+\eta \phi \left(2^{k}t\right))}dt\right|\le C{\left|2^{k}r\right|}^{-1/2}$$

holds for all $b\ge 1$, $r,\eta \in \mathbb{R}$ and $k\in \mathbb{Z}$.

Lemma 3.

[27] Let $\mathsf{\Gamma }\left(x\right)={\displaystyle \sum _{\left|\alpha \right|\le d}}{b}_{\alpha }{x}^{\alpha }$, where $b_{\alpha }\in \mathbb{R}$. Then

$$\left|{\int }_{{[0,1]}^q}{e}^{-i\mathsf{\Gamma }\left(x\right)}dx\right|\le C_{d,q}{\left(\sum _{0<\left|\alpha \right|\le d}\left|b_{\alpha }\right|\right)}^{-1/d}.$$

Now we need to prove the following lemma which will play a key role on proving our main results.

Lemma 4.

Let $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for $\alpha >1/2$ and satisfy the conditions $\left(1\right)$-$\left(2\right)$. Suppose that Φ is given as in any of Theorems 1-5. Then there is a real number $C>0$ such that the inequalities

$$\begin{array}{ccc}\hfill ∥{\lambda }_{r,s}^{k,j}∥& \le & C,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & Cmin\left\{\left|2^k\xi \right|,{\left(log|2^k\xi |\right)}^{-\alpha }\right\},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & Cmin\left\{\left|2^j\zeta \right|,{\left(log|2^j\zeta |\right)}^{-\alpha }\right\}\hfill \end{array}$$

(12)

hold for all $k,j\in \mathbb{Z}$ and $r,s>0$, where $∥{\lambda }_{r,s}^{k,j}∥$ stands for the total variation of ${\lambda }_{r,s}^{k,j}$.

Proof. 

By the definition of ${\lambda }_{r,s}^{k,j}$, it is easy to prove the inequality $\left(10\right)$. Next, by Hölder’s inequality, we have

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\int }_1^2{\int }_1^2\left|{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}{e}^{-i\{2^{k}rtx·\xi +2^{j}s\ell y·\zeta +\mathsf{\Phi }(2^{k}rt,2^{j}s\ell )\eta \}}\right.\hfill \\ & \times & \left.\mho \left(x,y\right)d\sigma \left(x\right)d\sigma \left(y\right)\right|\frac{dtd\ell }{t\ell }\hfill \\ & \le & C{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mho \left(x,y\right)\right|{\int }_1^2{I}_k(\xi ,\eta ,x,t)\frac{d\ell }{\ell }d\sigma \left(x\right)d\sigma \left(y\right),\hfill \end{array}$$

(13)

where

$$\begin{array}{c}\hfill I_k(\xi ,\eta ,x,t)={\int }_1^2{e}^{-i\{2^{k}rtx·\xi +\mathsf{\Phi }(2^{k}rt,2^{j}s\ell )\eta \}}\frac{dt}{t}.\end{array}$$

Hence, we have

$$\begin{array}{c}\hfill I_k(\xi ,\eta ,x,t)=\left|{\int }_1^2{e}^{-i\{2^{k}rtx·\xi +\eta 2^{k}rt{D}_t\mathsf{\Phi }(0,2^{j}s\ell )+[\mathsf{\Phi }(2^{k}rt,2^{j}s\ell )-2^{k}rt{D}_t\mathsf{\Phi }(0,2^{j}s\ell )]\eta \}}\frac{dt}{t}\right|.\end{array}$$

Now we need to consider several cases.

Case 1. If $\mathsf{\Phi }$ is given as in Theorem 1, then by Lemma 2, we deduce that

$$\begin{array}{c}\hfill I_k(\xi ,\eta ,x,t)\le {\left|2^{k}r\left|\xi \right|\left|x·{\xi }^{\prime }+\right|\eta {\left|\xi \right|}^{-1}{D}_t\mathsf{\Phi }(0,2^{j}s\ell )\right|}^{-1/2}.\end{array}$$

(14)

Let $\delta =min\left\{2,\eta {\left|\xi \right|}^{-1}{D}_t\mathsf{\Phi }(0,2^{j}s\ell )\right\}sgn\left(\eta D_t\mathsf{\Phi }(0,2^{j}s\ell )\right)$. Combine the inequality (14) with the trivial estimate $\left|I_k(\xi ,\eta ,x,t)\right|\le 1$ and use the fact that $(t/{log}^{\alpha }t)$ is increasing over the interval $(2^{\alpha },\infty )$, we get that

$$\begin{array}{c}\hfill \left|I_k(\xi ,\eta ,x,t)\right|\le C\frac{{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x+\delta \right|}^{-1}\right)}{{log}^{\alpha }\left|2^{k}r\xi \right|}if\left|2^{k}r\xi \right|>2^{\alpha }.\end{array}$$

(15)

Thus, if $n\ge 3$, then by the additional assumption $D_t\mathsf{\Phi }(0,2^{j}s\ell )=0$, we get $\delta =0$. Hence,

$$\begin{array}{c}\hfill \left|I_k(\xi ,\eta ,x,t)\right|\le C\frac{{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x\right|}^{-1}\right)}{{log}^{\alpha }\left|2^{k}r\xi \right|}if\left|2^{k}r\xi \right|>2^{\alpha }.\end{array}$$

(16)

Therefore, by (13) we acquire that

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\left(log\left|2^{k}r\xi \right|\right)}^{-\alpha }{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mho \left(x,y\right)\right|{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x\right|}^{-1}\right)d\sigma \left(x\right)d\sigma \left(y\right)\hfill \\ & \le & C{\left(log\left|2^{k}r\xi \right|\right)}^{-\alpha }if\left|2^{k}r\xi \right|>2^{\alpha }.\hfill \end{array}$$

(17)

Now, if $n=2$, then follow the argument similar to that in [28]; we may assume that $\delta >0$. Set ${\delta }^{\prime }=min\{1,\delta \}$ and $\theta =arcsin\left({\delta }^{\prime }\right)$, and let $e_{+}$ denote the vector obtained by rotating ${\xi }^{\prime }$ by angles $\theta$ and $e_{-}$ denote the vector obtained by rotating ${\xi }^{\prime }$ by angles $-\theta$. Then we conclude that a constant $C_0\in (0,1)$ exists such that

$${\xi }^{\prime }·x+{\delta }^{\prime }\ge C_{0}min\left\{\right|x·e_{-}{|}^2,|x·e_{+}{|}^2\}$$

for $x\in {\mathbb{S}}^1$. So, the estimate in (16) holds and therefore (17) holds for the case $n=2$.

We notice her that the conclusions of this lemma when $n=2,m=2$ are proved without the additional assumptions $D_t\mathsf{\Phi }(0,\ell )=0$ and $D_{\ell }\mathsf{\Phi }(t,o)=0$.

In the same manner, we can prove that

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\left(log\left|2^{j}s\zeta \right|\right)}^{-\alpha }if\left|2^{j}s\zeta \right|>2^{\alpha }.\hfill \end{array}$$

(18)

Now by using the cancellation conditions on ℧, we get

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mho \left(x,y\right)\right|{\int }_1^2\left|J_k(\xi ,\eta ,x,t)\right|\frac{d\ell }{\ell }d\sigma \left(x\right)d\sigma \left(y\right)\hfill \\ & \le & C\left|2^{k}r\xi \right|,\hfill \end{array}$$

(19)

where

$$\begin{array}{c}\hfill J_k(\xi ,\eta ,x,t)={\int }_1^2{e}^{-i\{2^{k}rtx·\xi +\mathsf{\Phi }(2^{k}rt,2^{j}s\ell )\eta \}}-e^{\mathsf{\Phi }(2^{k}rt,2^{j}s\ell )\eta }\frac{dt}{t}.\end{array}$$

Similarly, we obtain that

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C\left|2^{k}r\xi \right|.\hfill \end{array}$$

(20)

Consequently, by (17)–(20), we finish the proof of this lemma for the first case.

We notice here that the proofs of (19)–(20) do not depend on $\mathsf{\Phi }$.

Case 2. If $\mathsf{\Phi }$ is given as in Theorem 2. Notice that

$$\mathsf{\Phi }(t,\ell )=\sum _{j=1}^{d_2}\sum _{i=1}^{d_1}{a}_{j,i}{t}^{{\alpha }_i}{\ell }^{{\beta }_j}.$$

Hence, $\mathsf{\Phi }(t,\ell )$ can be written as

$$\mathsf{\Phi }(t,\ell )=P_{\ell }\left(t\right)=\sum _{i=0}^{d_1}{b}_i(\ell )t^{{\alpha }_i}$$

and

$$\mathsf{\Phi }(t,\ell )=Q_t(\ell )=\sum _{j=0}^{d_2}{c}_j\left(t\right){\ell }^{{\beta }_j},$$

where

$$b_i(\ell )=\sum _{j=0}^{d_2}{a}_{j,i}{\ell }^{{\beta }_j}and{c}_j\left(t\right)=\sum _{i=0}^{d_1}{a}_{j,i}{t}^{{\alpha }_i}.$$

If ${\alpha }_i\ne 1$ for all $1\le i\le d_1$, then by Lemma 3, we get that

$$I_k(\xi ,\eta ,x,t)\le {\left|2^k\left|\xi \right|({\xi }^{\prime }·x)\right|}^{-1/d_1}.$$

(21)

So, by following the same argument as above, we get

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & {\left(log\left|2^{k}r\xi \right|\right)}^{-\alpha }if\left|2^{k}r\xi \right|>2^{\alpha }.\hfill \end{array}$$

(22)

If ${\alpha }_{i_0}=1$ for some $1\le i_0\le d_1$. Since

$$\begin{array}{c}\hfill I_k(\xi ,\eta ,x,t)=\left|{\int }_1^2{e}^{-i{2}^{k}rt\left|\xi \right|{\{x·{\xi }^{\prime }+\eta {\left|\xi \right|}^{-1}{b}_{i_0}(2^{j}s\ell )+\eta \sum _{i=0,i\ne i_0}^{d_1}{b}_i(2^{j}s\ell )\left(2^{k}rt\right))}^{{\alpha }_i}\}}\frac{dt}{t}\right|,\end{array}$$

then by invoking Lemma 3 we obtain

$$\left|I_k(\xi ,\eta ,x,t)\right|\le C{\left|2^{k}r\left|\xi \right|\left(x·{\xi }^{\prime }+\eta {\left|\xi \right|}^{-1}{b}_{i_0}(2^{j}s\ell )\right)\right|}^{-1/D_t}.$$

(23)

Set $Y=min\left\{2,\eta {\left|\xi \right|}^{-1}{b}_{i_0}(2^{j}s\ell )\right\}sgn\left(b_{i_0}(2^{j}s\ell )\eta \right)$. By combining (23) with the trivial estimate $\left|I_k(\xi ,\eta ,x,t)\right|\le 1$ and using the fact that $(t/{log}^{\alpha }t)$ is increasing over the interval $(2^{\alpha },\infty )$, we get

$$\begin{array}{c}\hfill \left|I_k(\xi ,\eta ,x,t)\right|\le C\frac{{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x+Y\right|}^{-1}\right)}{{log}^{\alpha }\left|2^{k}r\xi \right|}if\left|2^{k}r\xi \right|>2^{\alpha }.\end{array}$$

(24)

When $n\ge 3$, by the additional condition $D_t\mathsf{\Phi }(0,2^{j}s\ell )=0$, we deduce that $Y=0$; which yields that

$$\begin{array}{c}\hfill \left|I_k(\xi ,\eta ,x,t)\right|\le C\frac{{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x\right|}^{-1}\right)}{{log}^{\alpha }\left|2^{k}r\xi \right|}if\left|2^{k}r\xi \right|>2^{\alpha }.\end{array}$$

(25)

By the last estimate and (13) we get

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\left(log\left|2^{k}r\xi \right|\right)}^{-\alpha }{\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mho \left(x,y\right)\right|{log}^{\alpha }\left(2{\left|{\xi }^{\prime }·x\right|}^{-1}\right)d\sigma \left(x\right)d\sigma \left(y\right)\hfill \\ & \le & C{\left(log\left|2^{k}r\xi \right|\right)}^{-\alpha }if\left|2^{k}r\xi \right|>2^{\alpha }.\hfill \end{array}$$

(26)

When $n=2$, we follow the same arguments similar to those in Case 1, we obtain (26).

Similarly, we have that

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,s}^{k,j}(\xi ,\zeta ,\eta )\right|& \le & C{\left(log\left|2^{j}s\zeta \right|\right)}^{-\alpha }if\left|2^{j}s\zeta \right|>2^{\alpha }.\hfill \end{array}$$

(27)

Case 3. This lemma can be proved for the other cases of $\mathsf{\Phi }$ by following the same argument as in the proof adopted for the cases 1 and 2. We omit the details. □

Lemma 5.

Suppose that $\mho \in L^1\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ and satisfies the conditions $\left(1\right)$-$\left(2\right)$. Let Φ be given as in any one of Theorems 1-5. Then there exists $C_p>0$ such that

$$\parallel {\lambda }^{*}{\left(g\right)\parallel }_{L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})}\le C_p{\parallel g\parallel }_{L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}).}$$

(28)

We can easily prove this lemma by using iterated integration and using at least one of results in Corollary 5.3 in [26] or Lemma 1.

3. Proof of Main Theorems

Assume that $\mho \in G_{\alpha }\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$ for some $\alpha >1/2$ and satisfies the conditions $\left(1\right)$-$\left(2\right)$, and let $\mathsf{\Phi }$ be given as in any of Theorems 1-5. Then by Minkowski’s inequality, we have

$$\begin{array}{ccc}\hfill {\mathfrak{M}}_{\mho ,\mathsf{\Phi }}\left(g\right)(u,v,w)& =& {\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|\sum _{j=-\infty }^{-1}\sum _{k=-\infty }^{-1}{2}^{j+k}\left({\lambda }_{r,s}^{k,j}*g\right)(u,v,w)\right|}^2\frac{drds}{rs}\right)}^{1/2}\hfill \\ & \le & \sum _{j=-\infty }^{-1}\sum _{k=-\infty }^{-1}{2}^{j+k}{\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|\left({\lambda }_{r,s}^{0,0}*g\right)(u,v,w)\right|}^2\frac{drds}{rs}\right)}^{1/2}\hfill \\ & =& {\left({\int }_1^2{\int }_1^2{\left|\sum _{j\in \mathbb{Z}}\sum _{k\in \mathbb{Z}}\left({\lambda }_{r,s}^{k,j}*g\right)(u,v,w)\right|}^2\frac{drds}{rs}\right)}^{1/2}\hfill \\ & \le & {\left({\int }_1^2{\int }_1^2{\left|\sum _{j\in \mathbb{Z}}\sum _{k\in \mathbb{Z}}\left({\lambda }_{r,s}^{k,j}*g\right)(u,v,w)\right|}^{2}drds\right)}^{1/2}.\hfill \end{array}$$

(29)

Choose two radial Schwartz functions ${\mathsf{\Psi }}_1\in \mathcal{S}\left({\mathbb{R}}^m\right)$ and ${\mathsf{\Psi }}_2\in \mathcal{S}\left({\mathbb{R}}^n\right)$ which satisfy the following properties:

$\left(i\right)0\le {\mathsf{\Psi }}_1\le 1,0\le {\mathsf{\Psi }}_2\le 1,$

$\left(ii\right)\mathrm{supp}\left({\mathsf{\Psi }}_1\right)\subseteq \{u\in {\mathbb{R}}^m:1/4\le \left|u\right|\le 4\},\mathrm{supp}\left({\mathsf{\Psi }}_2\right)\subseteq \{v\in {\mathbb{R}}^n:1/4\le \left|v\right|\le 4\}.$

$\left(iii\right){\displaystyle \sum _{a_1\in \mathbb{Z}}}{\left({\mathsf{\Psi }}_1\left(2^{a_1}r\right)\right)}^2=1,{\displaystyle \sum _{a_2\in \mathbb{Z}}}{\left({\mathsf{\Psi }}_2\left(2^{a_2}s\right)\right)}^2=1$.

Define the multiplier operators $\left\{T_{a_1,a_2}\right\}$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}$ by

$$\widehat{T_{a_1,a_2}\left(g\right)}(\xi ,\zeta ,\eta )={\mathsf{\Psi }}_1\left(2^{a_1}\left|\xi \right|\right){\mathsf{\Psi }}_2\left(2^{a_2}\left|\zeta \right|\right)\widehat{g}(\xi ,\zeta ,\eta ).$$

Thus for any $g\in C_0^{\infty }({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})$, we have

$$g(u,v,w)=\sum _{a_2\in \mathbb{Z}}\sum _{a_1\in \mathbb{Z}}{T}_{a_1,a_2}^2\left(g\right)(u,v,w),$$

which in turn by (29) leads to

$$\begin{array}{ccc}\hfill {\mathfrak{M}}_{\mho ,\mathsf{\Phi }}\left(g\right)(u,v,w)& \le & \mathcal{I}\left(g\right)(u,v,w),\hfill \end{array}$$

(30)

where

$$\begin{array}{c}\hfill \mathcal{I}\left(g\right)(u,v,w)={\left(\sum _{j\in \mathbb{Z}}\sum _{k\in \mathbb{Z}}{\int }_1^2{\int }_1^2{\left|\sum _{a_1\in \mathbb{Z}}\sum _{a_2\in \mathbb{Z}}{T}_{j-a_1,k-a_2}\left({\lambda }_{r,s}^{k,j}*\left(T_{j-a_1,k-a_2}\left(g\right)\right)\right)(u,v,w)\right|}^{2}drds\right)}^{1/2}.\end{array}$$

Therefore, by Lemmas 4–5 and employing the same argument as in [22], p. 301], we obtain

$$\begin{array}{c}\hfill {∥\mathcal{I}\left(g\right)∥}_{L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})}\le C{∥f∥}_{L^p({\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R})}\end{array}$$

(31)

for all $p\in (1+1/(2\alpha ),1+2\alpha )$. Now the proof is complete by the last inequality and using (30).