Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions

Abstract

In this paper we investigate the weighted $L^p$ boundedness of generalized Marcinkiewicz integrals ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ over multiple symmetric domains. Under the conditions $\mathbf{K}\in L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, $q>1$, we establish suitable weighted $L^p$ bounds for the integrals ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$. These bounds are combined with Yano’s extrapolation argument to obtain the weighted $L^p$ boundedness of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ from the Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{0,\epsilon }({\omega }_1,{\omega }_2)$ to the space $L^p({\omega }_1,{\omega }_2)$ under the weaker conditions $\mathbf{K}\in B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\cup L{\left(\log L\right)}^{2/\epsilon }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. Our findings are essential improvements and extension of several known findings in the literature.

1. Introduction

Let $\mathbf{K}\in L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ be a measurable mapping on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ satisfying

$$\begin{array}{ccc}\hfill \mathbf{K}(ta,lb)& =& \mathbf{K}(a,b),\forall t,l>0,\hfill \end{array}$$

(1)

and

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{S}}^{m-1}}\mathbf{K}(a,b)d{\sigma }_m\left(a\right)& =& {\int }_{{\mathbb{S}}^{n-1}}\mathbf{K}(a,b)d{\sigma }_n\left(b\right)=0,\hfill \end{array}$$

(2)

where ${\mathbb{S}}^{\kappa -1}$ ($\kappa =m$ or n) is the unit sphere in the Euclidean space ${\mathbb{R}}^{\kappa }$ equipped with the normalized spherical measure $d{\sigma }_{\kappa }(·)$, and $\kappa \ge 2$.

For $\epsilon >1$ and $\Phi \in \mathcal{S}({\mathbb{R}}^m\times {\mathbb{R}}^n)$, consider the generalized Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ defined by

$${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}(\Phi )(u,v)={\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|H_{l,t}(\Phi )(u,v)\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon },$$

(3)

where

$$H_{l,t}(\Phi )(u,v)={\displaystyle \frac{1}{lt}}{\int }_{\left|b\right|\le l}{\int }_{\left|a\right|\le t}{\displaystyle \frac{\mathbf{K}(a,b)}{{\left|a\right|}^{m-1}{\left|b\right|}^{n-1}}}\Phi (u-a,v-b)dadb.$$

When $\epsilon =2$, we denote the operator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ by ${\mathcal{M}}_{\mathbf{K}}$, which is basically the classical Marcinkiewicz integral on product spaces that was introduced by Ding in [1]. Precisely, Ding proved that ${\mathcal{M}}_{\mathbf{K}}$ is bounded on $L^2({\mathbb{R}}^m\times {\mathbb{R}}^n)$ under the assumption $\mathbf{K}\in L{\left(\log L\right)}^2({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, where $L{\left(\log L\right)}^{\theta }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ is the space of all $\mathbf{K}\in L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ such that

$${\int \int }_{{\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}}\left|\mathbf{K}(a,b)\right|\log {\left(2+\left|\mathbf{K}(a,b)\right|\right)}^{\theta }d{\sigma }_m\left(a\right)d{\sigma }_n\left(b\right)<\infty ,\mathrm{for}\theta >0.$$

Subsequently, many authors have investigated the $L^p$ boundedness of the operator ${\mathcal{M}}_{\mathbf{K}}$ over symmetric regions under various conditions on the kernel functions $\mathbf{K}$. For instance, in [2] the authors established the boundedness of ${\mathcal{M}}_{\mathbf{K}}$ on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n)$$(1<p<\infty )$ whenever $\mathbf{K}\in L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. Further, they pointed out that, by adopting a similar argument as in [3], the condition $\mathbf{K}\in L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ is almost optimal in the sense that ${\mathcal{M}}_{\mathbf{K}}$ will lose the $L^2$ boundedness if $\mathbf{K}$ is not in the space $L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ but in the space $L{\left(\log L\right)}^{\theta }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ for any $\theta \in (0,1)$. The author of [4] proved that ${\mathcal{M}}_{\mathbf{K}}$ is of type $(p,p)$ for all $1<p<\infty$ if $\mathbf{K}$ belongs to the block space $B_q^{(0,0)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, $q>1$. Also, he showed that the condition $\mathbf{K}\in B_q^{(0,0)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ is nearly optimal. For relevant results, one may consult [5,6,7,8,9,10,11], among others.

Here, $B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ indicates to the block space that was introduced by Jiang and Lu in [12] and defined as the following: a q-block on ${\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}$ is an $L^q$ ($1<q\le \infty$) function $g\left(x,y\right)$ that satisfies the following: $\left(i\right)$ supp $\left(g\right)\subseteq I$; $\left(ii\right)$ ${∥g∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\le {\left|I\right|}^{-1/q^{\prime }}$, where $\left|I\right|={\sigma }_{\kappa }\left(I\right)$ and

$$I=B((x_0,y_0),(\alpha ,\beta ))\left\{x\in {\mathbb{S}}^{m-1}:\left|x-x_0\right|<\alpha \right\}\times \left\{y\in {\mathbb{S}}^{n-1}:\left|y-y_0\right|<\beta \right\}$$

for some $\alpha$, $\beta >0$ and $(x_0,y_0)\in {\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}$. The block spaces $B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ (for $q>1,v>-1$) is defined by the following:

$$B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})=\{\mathbf{K}\in L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}):\mathbf{K}=\sum _{\gamma =1}^{\infty }{c}_{\gamma }{g}_{\gamma }with{M}_q^{(0,v)}\left(\left\{c_{\gamma }\right\}\right)<\infty \},$$

where each $c_{\gamma }$ is a complex number, each $g_{\gamma }$ is a q-block function supported in an interval $I_{\gamma }$, and

$$M_q^{(0,v)}\left(\left\{c_{\gamma }\right\}\right)=\sum _{\gamma =1}^{\infty }\left|c_{\gamma }\right|\left\{1+{\log }^{v+1}\left({\left|I_{\gamma }\right|}^{-1}\right)\right\}<\infty .$$

Let ${∥\mathbf{K}∥}_{B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}=inf\{M_q^{(0,v)}\left(\left\{c_{\gamma }\right\}\right):\mathbf{K}={\displaystyle \sum _{\gamma =1}^{\infty }}{c}_{\gamma }{g}_{\gamma }\}$ and each $g_{\gamma }$ is a q-block function supported in an interval $I_{\gamma }$. Then ${∥·∥}_{B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}$ is a norm on the block space $B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ and $(B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}),{∥·∥}_{B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})})$ is a Banach space.

Let us now recall the definition of the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\mathbb{R}}^m\times {\mathbb{R}}^n)$. For $1<\epsilon ,p<\infty$, and $\overrightarrow{\tau }=(\zeta ,\nu )\in \mathbb{R}\times \mathbb{R}$, the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\mathbb{R}}^m\times {\mathbb{R}}^n)$ is the collection of all tempered distributions $\Phi$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ satisfying

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

where $\widehat{{\phi }_k^{\left(1\right)}}\left(\xi \right)=2^{-km}{g}_1\left(2^{-k}\xi \right)$, $\widehat{{\phi }_j^{\left(2\right)}}\left(\eta \right)=2^{-jn}{g}_2\left(2^{-j}\eta \right)$, and $g_1\in C_0^{\infty }\left({\mathbb{R}}^m\right)$, $g_2\in C_0^{\infty }\left({\mathbb{R}}^n\right)$ are radial functions satisfying the following:

(i)

$supp\left(g_1\right)\subset \left\{\xi :\left|\xi \right|\in [{\displaystyle \frac{1}{2}},2]\right\}$, $supp\left(g_2\right)\subset \left\{\eta :\left|\eta \right|\in [{\displaystyle \frac{1}{2}},2]\right\}$.

(ii)

$g_1\left(\xi \right),g_2\left(\eta \right)\in [0,1]$.

(iii)

$g_1\left(\xi \right),g_2\left(\eta \right)\ge C$ for all $\left|\xi \right|,\left|\eta \right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}]$, for some bounded constant $C>0$.

(iv)

For $\xi \ne 0$, $\sum _{k\in \mathbb{Z}}{g}_1\left(2^{-k}\xi \right)=1$; and for $\eta \ne 0$, ${\displaystyle \sum _{j\in \mathbb{Z}}}{g}_2\left(2^{-j}\eta \right)=1$.

The following properties were proved in [13].

(a)

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

(b)

If ${\epsilon }_1\le {\epsilon }_2$, then ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },{\epsilon }_2}({\mathbb{R}}^m\times {\mathbb{R}}^n)\supseteq {\stackrel{.}{F}}_p^{\overrightarrow{\tau },{\epsilon }_1}({\mathbb{R}}^m\times {\mathbb{R}}^n)$.

(c)

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

The study of the generalized Marcinkiewicz integral operator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ was initiated in [13] in which the authors obtained the boundedness of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n)$ for $1<p<\infty$ whenever $\mathbf{K}\in L{\left(\log L\right)}^{2/\epsilon }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. Recently, the authors of [14] employed an extrapolation argument of Yano in [15] to extend and improve all the above mentioned results. In fact, they established that

$${∥{\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}(\Phi )∥}_{L^p({\mathbb{R}}^m\times {\mathbb{R}}^n)}\le C_p{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\mathbb{R}}^m\times {\mathbb{R}}^n)}$$

(4)

for all $p,\epsilon \in (1,\infty )$ provided that $\mathbf{K}$ belongs to the space $L{\left(\log L\right)}^{2/\epsilon }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ or to the space $B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. For recent advances concerning the operator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$, readers are referred to [16,17,18,19,20] and the references therein.

Let us recall the definition, as well as some pertinent properties, of certain classes of weights which will be relevant to our current study.

Definition 1. 

A non-negative locally integrable mapping ω is said to be in the space $A_p\left({\mathbb{R}}^{\kappa }\right)$ with $p\in (1,\infty )$ if there is a bounded positive number C such that for any cube $\mathbf{Q}\subseteq {\mathbb{R}}^{\kappa }$ with its sides are parallel to the coordinate axes, we have

$$\left({\left|\mathbf{Q}\right|}^{-1}{\int }_{\mathbf{Q}}\omega \left(z\right)dz\right){\left({\left|\mathbf{Q}\right|}^{-1}{\int }_{\mathbf{Q}}\omega {\left(z\right)}^{-1/(p-1)}dz\right)}^{p-1}\le C<\infty ,$$

and ω is said to be in $A_1\left({\mathbb{R}}^{\kappa }\right)$ if

$$M^{\ast }\omega \left(z\right)\le C\omega \left(z\right)a.ez\in {\mathbb{R}}^{\kappa },$$

where $M^{\ast }\omega$ is the Hardy–Littlewood maximal function of ω.

Definition 2. 

For $p\in [1,\infty )$, we say that $\omega \in {\tilde{A}}_p\left({\mathbb{R}}^{\kappa }\right)$ if

$$\omega \left(z\right)=B_1{\left(\right|z\left|\right)}^{1-p}{B}_2\left(\right|z\left|\right),$$

where $B_1$, $B_2$ are functions defined on ${\mathbb{R}}_{+}$ and either $B_j^2\in A_1\left({\mathbb{R}}_{+}\right)$ or $B_j\in A_1\left({\mathbb{R}}_{+}\right)$ is decreasing, $j\in \{1,2\}$.

Definition 3. 

For $p\in (1,\infty )$, let ${\dot{A}}_p\left({\mathbb{R}}^{\kappa }\right)$ be the set of nonnegative locally integrable functions ω such that $\omega \left(z\right)=\omega \left(\left|z\right|\right)$ and ${\omega }^2\in A_1\left({\mathbb{R}}_{+}\right)$.

Let $A_p^I\left({\mathbb{R}}^{\kappa }\right)$ be the weight class defined by exchanging the cubes in the definition of $A_p\left({\mathbb{R}}^{\kappa }\right)$ for all $\kappa$-dimensional intervals with sides parallel to coordinate axes [21]. The authors of [22] showed that ${\dot{A}}_p\left({\mathbb{R}}_{+}\right)\subseteq {\tilde{A}}_p\left({\mathbb{R}}_{+}\right)$. Further, it is well known that whenever $\omega \left(l\right)\in {\dot{A}}_p\left({\mathbb{R}}_{+}\right)$, we have $\omega \left(\left|z\right|\right)\in A_p\left({\mathbb{R}}^{\kappa }\right)$, which is the Muckenhoupt weighted space defined in [23]. Let us present the weight class ${\tilde{A}}_p^I$ given by ${\tilde{A}}_p^I=A_p^I\cap {\tilde{A}}_p$ ($1\le p<\infty$). This class of weights has the following properties:

(1)

For $1\le p_1\le p_2<\infty$, we have ${\tilde{A}}_{p_1}^I\subseteq {\tilde{A}}_{p_2}^I$.

(2)

If $\omega \in {\tilde{A}}_p^I$, then a number $\theta >0$ exists such that ${\omega }^{1+\theta }\in {\tilde{A}}_p^I$.

(3)

If $\omega \in {\tilde{A}}_p^I$$(1<p<\infty )$, then there exists a positive number $\theta$ satisfying $p-\theta >1$ and $\omega \in {\tilde{A}}_{p-\theta }^I$.

(4)

For $1<p<\infty$, $\omega \in {\tilde{A}}_p^I$ if and only if ${\omega }^{1-p^{\prime }}\in {\tilde{A}}_{p^{\prime }}^I$, where $p^{\prime }$ denotes to the exponent conjugate of p.

The weighted $L^p$ space related to the weights ${\omega }_1,{\omega }_2$ is denoted by $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n,{\omega }_{1}du,{\omega }_{2}dv)\equiv L^p({\omega }_1,{\omega }_2)$ and is given by

$$L^p({\omega }_1,{\omega }_2)=\left\{\Phi :{∥\Phi ∥}_{L^p({\omega }_1,{\omega }_2)}={\left({\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}{\left|\Phi (u,v)\right|}^p\omega \left(u\right)\omega \left(v\right)dudv\right)}^{1/p}<\infty \right\}.$$

In the same manner, the weighted homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\omega }_1,{\omega }_2)$ related to the weights ${\omega }_1,{\omega }_2$ is defined by the space of all tempered distributions $\Phi \in {\mathcal{S}}^{\prime }$ satisfying

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

where $\tau$, $\epsilon$, p, ${\phi }_k^{\left(1\right)}$, and ${\phi }_j^{\left(2\right)}$ are defined as above.

Recently, the weighted $L^p$ boundedness of ${\mathcal{M}}_{\mathbf{K}}$ was obtained in [24] for all $1<p<\infty$ under the assumption $\mathbf{K}\in L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$.

In view of the $L^p$ boundedness results of the generalized Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ obtained in [14] and the weighted $L^p$ boundedness results of the classical Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}$ in [24], it is natural to ask the following: Does the weighted $L^p$ boundedness of generalized Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ hold under the same conditions as in [14]?

Our main purpose in this paper is to answer the above question in the affirmative. Our main results are formulated as follows:

Theorem 1. 

Let $\mathbf{K}\in L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ with $q\in (1,2]$ and satisfy the conditions (1)-(2). Suppose that ${\omega }_1\in {\tilde{A}}_p^I\left({\mathbb{R}}^m\right)$ and ${\omega }_2\in {\tilde{A}}_p^I\left({\mathbb{R}}^n\right)$. Then, for any $\epsilon >1$, the inequality

$${∥{\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}(\Phi )∥}_{L^p({\omega }_1,{\omega }_2)}\le C_p{\left(q-1\right)}^{-2/\epsilon }{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}$$

(5)

holds for all $p\in (1,\infty )$.

The proof of Theorem 1 will be given in Section 3.

The estimate in Theorem 1 along with Yano’s extrapolation argument (see [15,25]) leads to the following result.

Theorem 2. 

Let ${\omega }_1$ and ${\omega }_2$ be given as in Theorem 1. Assume that the kernel function $\mathbf{K}$ satisfies the conditions (1)-(2) and belongs to either the space $L{\left(\log L\right)}^{2/\epsilon }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ or to the space $B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ with $q>1$. Then, for any $\epsilon >1$, the operator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ is bounded on $L^p({\omega }_1,{\omega }_2)$ for all $p\in (1,\infty )$.

Remark 1. 

(1) 

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

$$C^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset Li{p}_{ϵ}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}),$$

$$L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset L{\left(\log L\right)}^{\alpha }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}),$$

$$\bigcup _{r>1}{L}^r({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset B_q^{(0,v)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})for anyv>-1,$$

$$L{\left(\log L\right)}^{{\alpha }_1}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset L{\left(\log L\right)}^{{\alpha }_2}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})for0<{\alpha }_2<{\alpha }_1,$$

$$B_q^{(0,v_1)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\subset B_q^{(0,v_2)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})for-1<v_2<v_1.$$

(2) 

In [1], it was only proved the $L^2$ boundedness of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ whenever $\mathbf{K}\in L{\left(\log L\right)}^2({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, $\epsilon =2$, and ${\omega }_1\equiv 1\equiv {\omega }_2$. Hence, our results generalize and improve the results in [1].

(3) 

For the special cases $\epsilon =2$ and ${\omega }_1\equiv 1\equiv {\omega }_2$, the authors of [7] proved that ${\mathcal{M}}_K$ is bounded on $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n)$ for all $p\in (1,\infty )$ if $\mathbf{K}$ belongs to the space $L^q\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$, $q>1$. Therefore, since $L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\cup B_q^{(0,0)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\supset L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, then our results are natural extensions to the results in [7].

(4) 

For the special cases ${\omega }_1\equiv 1\equiv {\omega }_2$ and $\epsilon =2$, the conditions $\mathbf{K}\in L\left(\log L\right)({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ and $\mathbf{K}\in B_q^{(0,0)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$ are the weakest known conditions in their particular classes, (see [2,4]).

(5) 

The results in Theorem 2 with $\epsilon =2$ give the $L^p({\mathbb{R}}^m\times {\mathbb{R}}^n)$ of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ for $p\in (1,\infty )$ if $\mathbf{K}\in L\left(\log L\right)({\mathbb{S}}^{n-1}\times {\mathbb{S}}^{m-1})$. The same result was obtained in [24].

(6) 

For the case ${\omega }_1\equiv 1\equiv {\omega }_2$, we observe that Theorem 2 generalizes Theorem 2.7 in [14].

Throughout this paper, the letter C stands for a positive constant that may vary at each occurrence, but it is independent of the essential variables.

2. Auxiliary Estimates

This section is devoted to giving some preliminary estimates that will play a significant role in proving the main results of this paper. Let $\mu \ge 2$, and consider the set of measures $\{{\mathrm{Y}}_{\mathbf{K},l,t}:={\mathrm{Y}}_{l,t}:l,t\in {\mathbb{R}}_{+}\}$ and its corresponding maximal operators ${\mathrm{M}}_{\mu }$ and ${\mathrm{Y}}^{\ast }$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ given by

$$\begin{array}{ccc}\hfill {\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}\Phi d{\mathrm{Y}}_{l,t}& =& {\displaystyle \frac{1}{lt}}{\int \int }_{{\Lambda }_{l,t}(a,b)}{\displaystyle \frac{\mathbf{K}(a,b)}{{\left|a\right|}^{m-1}{\left|b\right|}^{n-1}}}\Phi (u-a,v-b)dadb,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathrm{Y}}^{\ast }(\Phi )& =& \underset{l,t\in {\mathbb{R}}_{+}}{sup}|\Phi \ast |{\mathrm{Y}}_{l,t}\left|\right|,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathrm{M}}_{\mu }(\Phi )& =& \underset{j,k\in \mathbb{Z}}{sup}{\int }_{{\mu }^j}^{{\mu }^{j+1}}{\int }_{{\mu }^k}^{{\mu }^{k+1}}|\Phi \ast |{\mathrm{Y}}_{l,t}\left|\right|{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}},\hfill \end{array}$$

where ${\Lambda }_{l,t}(a,b)=\{(a,b)\in {\mathbb{R}}^m\times {\mathbb{R}}^n:l/2\le \left|b\right|\le l,t/2\le \left|a\right|\le t\}$, and $|{\mathrm{Y}}_{l,t}|$ is defined similarly as ${\mathrm{Y}}_{l,t}$ but $\mathbf{K}$ is replaced by $\left|\mathbf{K}\right|$.

The following lemma is a special case of Lemma 2.2 in [24].

Lemma 1. 

Assume that $\mathbf{K}$ satisfies (1) and (2) and belongs to $L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. Then for $p\in (1,\infty )$, ${\omega }_1\in {\mathrm{A}}_p\left({\mathbb{R}}^m\right)$, and ${\omega }_2\in {\mathrm{A}}_p\left({\mathbb{R}}^n\right)$, there exists a positive constant $C_p$ such that

$$\parallel {\mathrm{Y}}^{\ast }{(\Phi )\parallel }_{L^p({\omega }_1,{\omega }_2)}\le C_p{∥\mathbf{K}∥}_{L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}{\parallel \Phi \parallel }_{L^p({\omega }_1,{\omega }_2)}.$$

(6)

We notice that as a direct consequence of Lemma 1, we get that

$$\parallel {\mathrm{M}}_{\mu }{(\Phi )\parallel }_{L^p({\omega }_1,{\omega }_2)}\le C_p{ln}^2\left(\mu \right){\parallel \Phi \parallel }_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^1({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})},$$

(7)

for $1<p<\infty$.

The next lemma is found in [26].

Lemma 2. 

Let $q>1$ and $\mathbf{K}\in L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. Then, we have

$$\begin{array}{ccc}\hfill ∥{\mathrm{Y}}_{l,t}∥& \le & C{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\int }_{{\mu }^j}^{{\mu }^{j+1}}{\int }_{{\mu }^k}^{{\mu }^{k+1}}{\left|{\widehat{\mathrm{Y}}}_{l,t}(\xi ,\eta )\right|}^2{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}& \le & C{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}^2{(ln\mu )}^2\hfill \\ & \times & min\left\{{\left|\xi {\mu }^k\right|}^{-{\displaystyle \frac{\gamma }{ln\mu }}},{\left|\xi {\mu }^k\right|}^{{\displaystyle \frac{\gamma }{ln\mu }}}\right\}min\left\{{\left|\eta {\mu }^j\right|}^{{\displaystyle \frac{\gamma }{ln\mu }}},{\left|\eta {\mu }^j\right|}^{-{\displaystyle \frac{\gamma }{ln\mu }}}\right\},\hfill \end{array}$$

where C is a positive constant, $\gamma \in (0,1/2q^{\prime })$, and $∥{\mathrm{Y}}_{l,t}∥$ is the total variation of ${\mathrm{Y}}_{l,t}$.

Lemma 3. 

Let $q\in (1,2]$, $\mathbf{K}\in L^q\left({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1}\right)$, and $\mu =2^{q^{\prime }}$. Then for $p\in (1,\infty )$, ${\omega }_1\in {\tilde{A}}_p^I\left({\mathbb{R}}^m\right)$, ${\omega }_2\in {\tilde{A}}_p^I\left({\mathbb{R}}^n\right)$, and arbitrary set of functions $\{{\mathcal{J}}_{j,k};j,k\in \mathbb{Z}\}$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$, there is $C>0$ such that

$${∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{{\mu }^j}{\overset{{\mu }^{j+1}}{\int }}\underset{{\mu }^k}{\overset{{\mu }^{k+1}}{\int }}{\left|{\mathrm{Y}}_{l,t}\ast {\mathcal{J}}_{j,k}\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}\le C{(q-1)}^{-2/\epsilon }$$

$$\begin{array}{ccc}\hfill & & \hfill \times {∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{J}}_{j,k}\right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}.\end{array}$$

Proof. 

By invoking (6), we obtain

$$\begin{array}{ccc}& & {∥\underset{j,k\in \mathbb{Z}}{sup}\underset{(l,t)\in [1,\mu ]\times [1,\mu ]}{sup}\left|{\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}\right|∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \\ & \le & {∥{\mathrm{Y}}^{\ast }\left(\underset{j,k\in \mathbb{Z}}{sup}\left|{\mathcal{J}}_{j,k}\right|\right)∥}_{L^p({\omega }_1,{\omega }_2)}\le C{∥\underset{j,k\in \mathbb{Z}}{sup}\left|{\mathcal{J}}_{j,k}\right|∥}_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

(8)

for all $p\in (1,\infty )$, which means that

$$\begin{array}{c}\hfill {∥{∥\parallel {\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}{\parallel }_{L^{\infty }([1,\mu ]\times [1,\mu ],{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}})}∥}_{l^{\infty }(\mathbb{Z}\times \mathbb{Z})}∥}_{L^p({\omega }_1,{\omega }_2)}\end{array}$$

(9)

$$\begin{array}{c}\hfill \le C{∥{∥{\mathcal{J}}_{j,k}∥}_{l^{\infty }(\mathbb{Z}\times \mathbb{Z})}∥}_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}.\end{array}$$

(10)

Thanks to duality, there is a function $f\in L^{p^{\prime }}({\omega }_1^{1-p^{\prime }},{\omega }_2^{1-p^{\prime }})$ satisfying ${∥f∥}_{L^{p^{\prime }}({\omega }_1^{1-p^{\prime }},{\omega }_2^{1-p^{\prime }})}\le 1$ and

$$\begin{array}{ccc}\hfill & & {∥\sum _{j,k\in \mathbb{Z}}{\int }_1^{\mu }{\int }_1^{\mu }{\left|{\mathrm{Y}}_{{\mu }^{k}r,{\mu }^{j}s}\ast {\mathcal{J}}_{j,k}\right|}^{{\gamma }^{\prime }}{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}\sum _{j,k\in \mathbb{Z}}{\int }_1^{\mu }{\int }_1^{\mu }\left|{\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}\right|{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}f(a,b)dadb\hfill \\ & \le & {\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}\sum _{j,k\in \mathbb{Z}}{\int }_1^{\mu }{\int }_1^{\mu }\left|{\mathcal{J}}_{j,k}(a,b)\right|\left|({\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast \overline{f})(-a,-b)\right|{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}dadb\hfill \\ & \le & C{\int \int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}\left(\sum _{j,k\in \mathbb{Z}}\left|{\mathcal{J}}_{j,k}(a,b)\right|\right){\mathrm{Y}}^{\ast }\left(\overline{f}\right)(-a,-b)dadb\hfill \\ & \le & C{(q-1)}^{-2}{∥\sum _{j,k\in \mathbb{Z}}\left|{\mathcal{J}}_{j,k}\right|∥}_{L^p({\omega }_1,{\omega }_2)}{∥{\mathrm{Y}}^{\ast }\left(\overline{f}\right)∥}_{L^{p^{\prime }}({\omega }_1^{1-p^{\prime }},{\omega }_2^{1-p^{\prime }})},\hfill \end{array}$$

(11)

where $\overline{f}(a,b)=f(-a,-b)$. Let ${\mathbf{T}}_{j,k}$ be the linear operator defined on ${\mathcal{J}}_{j,k}$ by ${\mathbf{T}}_{j,k}\left({\mathcal{J}}_{j,k}\right)={\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}$. Thus, by a simple change of variables along with interpolating between (10) and (11), we conclude

$$\begin{array}{c}\hfill {∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{{\mu }^j}{\overset{{\mu }^{j+1}}{\int }}\underset{{\mu }^k}{\overset{{\mu }^{k+1}}{\int }}{\left|{\mathrm{Y}}_{l,t}\ast {\mathcal{J}}_{j,k}\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}={∥\sum _{j,k\in \mathbb{Z}}\underset{{\mu }^j}{\overset{{\mu }^{j+1}}{\int }}\underset{{\mu }^k}{\overset{{\mu }^{k+1}}{\int }}{\left|{\mathrm{Y}}_{l,t}\ast {\mathcal{J}}_{j,k}\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}∥}_{L^{p/\epsilon }({\omega }_1,{\omega }_2)}^{1/\epsilon }\end{array}$$

$$\begin{array}{ccc}& \le & C{∥{\left(\sum _{j,k\in \mathbb{Z}}\underset{1}{\overset{\mu }{\int }}\underset{1}{\overset{\mu }{\int }}{\left|{\mathrm{Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \\ & \le & C{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{J}}_{j,k}\right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}{\left(\ln \mu \right)}^{2/\epsilon }\hfill \\ & \le & C{(q-1)}^{-2/\epsilon }{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|{\mathcal{J}}_{j,k}\right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

for all $1<p<\infty$. □

3. Proof of Theorem 1

Let $\epsilon >1$ and $\mathbf{K}\in L^q({\mathbb{S}}^{n-1}\times {\mathbb{S}}^{m-1})$ with $q\in (1,2]$. Suppose that ${\omega }_1\in {\tilde{A}}_p^I\left({\mathbb{R}}^m\right)$, ${\omega }_2\in {\tilde{A}}_p^I\left({\mathbb{R}}^n\right)$ with $p\in (1,\infty )$. Then, Minkowski’s inequality gives

$$\begin{array}{ccc}\hfill {\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}(\Phi )(u,v)& =& \left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}\left|\sum _{j,k=0}^{\infty }{\displaystyle \frac{1}{lt}}{\int }_{2^{-j-1}l<\left|b\right|\le 2^{-j}l}{\int }_{2^{-k-1}t<\left|a\right|\le 2^{-k}t}\Phi (u-a,v-b)\right.\right.\hfill \\ & \times & {\left.{\left.{\displaystyle \frac{\mathbf{K}(a,b)}{{\left|a\right|}^{m-1}{\left|b\right|}^{n-1}}}dadb\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }\hfill \\ & \le & \sum _{j,k=0}^{\infty }\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}\left|{\displaystyle \frac{1}{lt}}{\int }_{2^{-j-1}l<\left|b\right|\le 2^{-j}l}{\int }_{2^{-k-1}t<\left|a\right|\le 2^{-k}t}\right.\right.\hfill \\ & \times & {\left.{\left.{\displaystyle \frac{\mathbf{K}(a,b)}{{\left|a\right|}^{m-1}{\left|b\right|}^{n-1}}}\Phi (u-a,v-b)dadb\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }\hfill \\ & \le & C{\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|{\mathrm{Y}}_{l,t}\ast \Phi (u,v)\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }.\hfill \end{array}$$

(12)

Choose a collection of mappings ${\left\{h_i\right\}}_{i\in \mathbb{Z}}$ such that

$$\begin{array}{c}\hfill h_i\in C^{\infty }(0,\infty ),0\le h_i\le 1,\sum _{i\in \mathbb{Z}}{h}_i\left(l\right)=1,\\ \hfill \mathrm{supp}\left(h_i\right)\subseteq [2^{-q^{\prime }(i+1)},2^{-q^{\prime }(i-1)}],and\left|{\displaystyle \frac{d^j{h}_i\left(l\right)}{d{l}^j}}\right|\le {\displaystyle \frac{C_j}{l^j}},\end{array}$$

where $C_j$ does not depend on q. Define two measures $\left\{{\Lambda }_{i,m}:i\in \mathbb{Z}\right\}$ on ${\mathbb{R}}^m$ and $\left\{{\Lambda }_{i,n}:i\in \mathbb{Z}\right\}$ on ${\mathbb{R}}^n$ by

$$\left(\widehat{{\Lambda }_{i,m}}\left(\xi \right)\right)=h_i\left(\left|\xi \right|\right)and\left(\widehat{{\Lambda }_{i,n}}\left(\eta \right)\right)=h_i\left(\left|\eta \right|\right),$$

where $\xi \in {\mathbb{R}}^m$ and $\eta \in {\mathbb{R}}^n$. Therefore, we deduce that for $\Phi \in \mathcal{S}({\mathbb{R}}^m\times {\mathbb{R}}^n)$,

$${\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|{\mathrm{Y}}_{l,t}\ast \Phi (u,v)\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }\le C\sum _{r,s\in \mathbb{Z}}{\mathcal{Q}}_{r,s}(\Phi )(u,v),$$

(13)

where

$${\mathcal{Q}}_{r,s}(\Phi )(u,v)={\left({\int \int }_{{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}{\left|A_{r,s}^{l,t}(\Phi )(u,v)\right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }$$

and

$$A_{r,s}^{l,t}(\Phi )(u,v)=\sum _{j,k\in \mathbb{Z}}{\mathrm{Y}}_{l,t}\ast \left({\Lambda }_{k+s,m}\otimes {\Lambda }_{j+r,n}\right)\ast \Phi (u,v){\chi }_{[2^{k{q}^{\prime }},2^{(k+1)q^{\prime }})\times [2^{j{q}^{\prime }},2^{(j+1)q^{\prime }})}(l,t).$$

Hence, to complete the proof of Theorem 1, it is sufficient to find a positive number $\vartheta$ such that

$$\begin{array}{c}\hfill {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1,{\omega }_2)}\le C_p{2}^{-{\displaystyle \frac{\vartheta }{2}}\left(\right|r|+\left|s\right|)}{(q-1)}^{-2/\epsilon }{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\end{array}$$

(14)

for all $1<p<\infty$.

If $p=\epsilon =2$, then by employing Lemma 2 with $\mu =2^{q^{\prime }}$ and invoking Plancherel’s theorem, we get

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^2({\omega }_1,{\omega }_2)}^2\hfill \\ & \le & \sum _{j,k\in \mathbb{Z}}{\int \int }_{{\mathcal{A}}_{j+r,k+s}}\left({\int }_{2^{j{q}^{\prime }}}^{2^{(j+1)q^{\prime }}}{\int }_{2^{k{q}^{\prime }}}^{2^{(k+1)q^{\prime }}}{\left|{\widehat{\mathrm{Y}}}_{l,t}(\xi ,\eta )\right|}^2{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right){\left|\widehat{\Phi }(\xi ,\eta )\right|}^2{\omega }_1\left(\xi \right){\omega }_2\left(\eta \right)d\xi d\eta \hfill \\ & \le & C_p{(q-1)}^{-2}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}^2\sum _{j,k\in \mathbb{Z}}{\int \int }_{{\mathcal{A}}_{j+r,k+s}}{\left|2^k\xi \right|}^{\pm {\displaystyle \frac{\gamma }{ln2}}}{\left|2^j\eta \right|}^{\pm {\displaystyle \frac{\gamma }{ln2}}}\hfill \\ & \times & {\left|\widehat{\Phi }(\xi ,\eta )\right|}^2{\omega }_1\left(\xi \right){\omega }_2\left(\eta \right)d\xi d\eta \hfill \\ & \le & C_p{(q-1)}^{-2}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}^2{2}^{-\gamma \left(\right|r|+|s\left|\right)}\sum _{j,k\in \mathbb{Z}}{\int \int }_{{\mathcal{A}}_{j+r,k+s}}{\left|\widehat{\Phi }(\xi ,\eta )\right|}^2{\omega }_1\left(\xi \right){\omega }_2\left(\eta \right)d\xi d\eta \hfill \\ & \le & C_p{(q-1)}^{-2}{2}^{-\gamma \left(\right|r|+|s\left|\right)}{∥\Phi ∥}_{L^2({\omega }_1,{\omega }_2)}^2{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}^2,\hfill \end{array}$$

where ${\mathcal{A}}_{j,k}=\left\{(\xi ,\eta )\in {\mathbb{R}}^m\times {\mathbb{R}}^n:(\left|\xi \right|,\left|\eta \right|)\in [2^{-q^{\prime }(k+1)},2^{-q^{\prime }(k-1)}]\times [2^{-q^{\prime }(j+1)},2^{-q^{\prime }(j-1)}]\right\}$ and ${\alpha }^{\pm \beta }=min\{{\alpha }^{+\beta },{\alpha }^{-\beta }\}$. Therefore,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^2({\omega }_1,{\omega }_2)}\le C_p{(q-1)}^{-1}{2}^{-{\displaystyle \frac{\gamma }{2}}\left(\right|r|+\left|s\right|)}{∥\Phi ∥}_{L^2({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}.\hfill \end{array}$$

(15)

For the other cases, we invoke Lemma 3 to obtain

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & C{∥{\left(\sum _{j,k\in \mathbb{Z}}{\int }_{2^{j{q}^{\prime }}}^{2^{(j+1)q^{\prime }}}{\int }_{2^{k{q}^{\prime }}}^{2^{(k+1)q^{\prime }}}{\left|{\mathrm{Y}}_{l,t}\ast \left({\Lambda }_{k+s}\otimes {\Lambda }_{j+r}\right)\ast \Phi \right|}^{\epsilon }{\displaystyle \frac{dt}{t}}{\displaystyle \frac{dl}{l}}\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \\ & \le & C{(q-1)}^{-2/\epsilon }{∥{\left(\sum _{j,k\in \mathbb{Z}}{\left|\left({\Lambda }_{k+s}\otimes {\Lambda }_{j+r}\right)\ast \Phi \right|}^{\epsilon }\right)}^{1/\epsilon }∥}_{L^p({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \\ & \le & C_p{(q-1)}^{-2/\epsilon }{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

(16)

for all $1<p<\infty$. Now, we consider three cases:

Case 1. $p>2$. Choose $\tilde{p}>p$ and $\delta >0$ such that ${\omega }_1^{1+\delta }\in {\tilde{A}}_p^I\left({\mathbb{R}}^m\right)\subset {\tilde{A}}_{\tilde{p}}^I\left({\mathbb{R}}^m\right)$ and ${\omega }_2^{1+\delta }\in {\tilde{A}}_p^I\left({\mathbb{R}}^n\right)\subset {\tilde{A}}_{\tilde{p}}^I\left({\mathbb{R}}^n\right)$. Hence, the estimate (16) leads to

$$\begin{array}{ccc}\hfill {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1^{1+\delta },{\omega }_2^{1+\delta })}& \le & C_p{(q-1)}^{-2/\epsilon }{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1^{1+\delta },{\omega }_2^{1+\delta })}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

which when combined with (15) gives that

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1,{\omega }_2)}\le C_p{(q-1)}^{-2/\epsilon }{2}^{-{\displaystyle \frac{{\theta }_1\gamma }{2}}\left(\right|r|+\left|s\right|)}{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

(17)

for $p>2$ and ${\theta }_1\in (0,1)$.

Case 2. $1<p<2$. Choose $1<\tilde{p}<p$ and $\delta >0$ such that ${\omega }_1\in {\tilde{A}}_p^I\left({\mathbb{R}}^m\right)$, ${\omega }_2\in {\tilde{A}}_p^I\left({\mathbb{R}}^n\right)$ and ${\omega }_1^{1+\delta }\in {\tilde{A}}_{\tilde{p}}^I\left({\mathbb{R}}^m\right)$, ${\omega }_2^{1+\delta }\in {\tilde{A}}_{\tilde{p}}^I\left({\mathbb{R}}^n\right)$. Therefore,

$$\begin{array}{ccc}\hfill {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1^{1+\delta },{\omega }_2^{1+\delta })}& \le & C_p{(q-1)}^{-2/\epsilon }{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1^{1+\delta },{\omega }_2^{1+\delta })}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

which when combined with (15) gives that

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^p({\omega }_1,{\omega }_2)}\le C_p{(q-1)}^{-2/\epsilon }{2}^{-{\displaystyle \frac{{\theta }_2\gamma }{2}}\left(\right|r|+\left|s\right|)}{∥\Phi ∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

(18)

for $1<p<2$ and ${\theta }_2\in (0,1)$.

Case 3. $p=2$. Choose $\delta >0$ such that ${\omega }_1^{1+\delta }\in {\tilde{A}}_2^I\left({\mathbb{R}}^m\right)$, ${\omega }_2^{1+\delta }\in {\tilde{A}}_2^I\left({\mathbb{R}}^n\right)$. Thus,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{Q}}_{r,s}(\Phi )∥}_{L^2({\omega }_1,{\omega }_2)}\le C{(q-1)}^{-2/\epsilon }{2}^{-{\displaystyle \frac{{\theta }_3\gamma }{2}}\left(\right|r|+\left|s\right|)}{∥\Phi ∥}_{{\stackrel{.}{F}}_2^{\overrightarrow{0},\epsilon }({\omega }_1,{\omega }_2)}{∥\mathbf{K}∥}_{L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})}\hfill \end{array}$$

(19)

where ${\theta }_3\in (0,1)$. Consequently, by interpolating between (17)–(19) we obtain (14), which leads along with (12) and (13) to (5).

4. Conclusions

In this work, we proved certain weighted $L^p$ estimates for a class of generalized Marcinkiewicz integrals ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ whenever the kernel function $\mathbf{K}$ belongs to the space $L^q({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$, $q>1$. By using these estimates and using Yano’s extrapolation argument, we obtained the boundedness of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ on $L^p({\omega }_1,{\omega }_2)$ spaces under the weak conditions on the kernel functions $\mathbf{K}\in B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})\cup L{\left(\log L\right)}^{2/\epsilon }({\mathbb{S}}^{m-1}\times {\mathbb{S}}^{n-1})$. The results in this paper extend and improve several known results on generalized Marcinkiewicz operators as those in [1,2,4,7,9,10,14,24]. In future work, we aim to prove the boundedness of ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ from the weighted homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\omega }_1,{\omega }_2)$ to the weighted homogeneous Triebel-Lizorkin space ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\omega }_1,{\omega }_2)$.