Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions

1. Introduction

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

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

(1)

and

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

(2)

where ${\mathbb{B}}^{\kappa -1}$ ($\kappa =m$ 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

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

(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 which was introduced by Ding in [1]. The author proved that ${\mathcal{M}}_{\mathbf{K}}$ is bounded on $L^2({\mathbb{R}}^m\times {\mathbb{R}}^n)$ under the assumption $\mathbf{K}\in L{(logL)}^2({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$, where $L{(logL)}^{\theta }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ is the space of all $\mathbf{K}\in L^1({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ such that

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

Subsequently, many authors investigated the $L^p$ boundedness of the operator ${\mathcal{M}}_{\mathbf{K}}$ 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 $\mathbf{K}\in L(logL)({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$, and they pointed out that by adopting a similar argument as in [3], the condition $\mathbf{K}\in L(logL)({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ is almost optimal in the sense that ${\mathcal{M}}_{\mathbf{K}}$ will be unbounded on $L^2({\mathbb{R}}^m\times {\mathbb{R}}^n)$ if $\mathbf{K}\in L{(logL)}^{\theta }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ for any $\theta \in (0,1)$. A similar result was obtained in [4] when the kernel function belongs to a cerain class of block spaces $B_q^{(0,\theta )}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$. The author proved that ${\mathcal{M}}_{\mathbf{K}}$ is of type $(p,p)$ for all $1<p<\infty$ if $\mathbf{K}\in B_q^{(0,0)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$, $q>1$ and that the condition $\mathbf{K}\in B_q^{(0,0)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ is nearly optimal. For relevant results, one may consult [5,6,7,8,9,10], among others.

Let $1<\epsilon ,p<\infty$ and $\overrightarrow{\tau }=(\zeta ,\nu )\in \mathbb{R}\times \mathbb{R}$. Then, a tempered distribution function $\Phi$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ belongs to the space of homogeneous Triebel-Lizorkin functions, ${\stackrel{.}{F}}_p^{\overrightarrow{\tau },\epsilon }({\mathbb{R}}^m\times {\mathbb{R}}^n)$, if

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

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 mappings satisfy 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 $\xi \ne 0$, ${\displaystyle \sum _{k\in \mathbb{Z}}}{g}_1\left(2^{-k}\xi \right)=1$; 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 [11].

(a)

The 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 opoerator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ was initiated in [11] 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 $1<p<\infty$ if $\mathbf{K}\in L{(logL)}^{2/\epsilon }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$. Recently, the authors of [12] employed an extrapolation argument of Yano in [13] 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 $L{(logL)}^{2/\epsilon }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ or to the space $\mathbf{K}\in B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$. For recent advances concerning the operator ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$, readers are referred to [14,15,16,17,18] and the references therein.

At this point, let us recall the definition, as well as some pertinent properties, of certain classes of weighs 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 B such that for any cube $\mathbf{S}\subseteq {\mathbb{R}}^{\kappa }$ with its sides are parallel to the coordinate axes, we have

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

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

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

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

$$\omega \left(z\right)={\mathcal{B}}_1{\left(\right|z\left|\right)}^{1-p}{\mathcal{B}}_2\left(\right|z\left|\right),$$

where either ${\mathcal{B}}_j^2\in A_1\left({\mathbb{R}}_{+}\right)$ or ${\mathcal{B}}_j\in A_1\left({\mathbb{R}}_{+}\right)$ 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 ${\omega }^2\in A_1\left({\mathbb{R}}_{+}\right)$.

The weighted class $A_p^I\left({\mathbb{R}}^{\kappa }\right)$ is defined as the definition of $A_p\left({\mathbb{R}}^{\kappa }\right)$ with replacing the cubes $\mathbf{S}$ by all $\kappa$-dimensional intervals whose sides are parallel to coordinate axes [19]. The authors of [20] 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 Mukenhoupt weighted space defined in [21]. 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

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

Recently, the weighted $L^p$ boundedness of ${\mathcal{M}}_{\mathbf{K}}$ was obtained in [22] for all $1<p<\infty$ under the assumption $\mathbf{K}\in L(logL)({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$.

In view of the $L^p$ boundedness results of the generalized Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}^{\left(\epsilon \right)}$ obtained in [12] and the weighted $L^p$ boundedness results of the classical Marcinkiewicz integral ${\mathcal{M}}_{\mathbf{K}}$ in [22] 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 [12]?

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

Theorem 1. 

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

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

(5)

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

By the results in Theorem 1 and following an extrapolation argument employed in [13,23] we get the following result.

Theorem 2. 

Let ${\omega }_1$ and ${\omega }_2$ be given as in Theorem 1. Assume that the kernel function $\mathbf{K}$ is in the space $L{(logL)}^{2/\epsilon }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ or in the space $B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$ 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 $p\in (1,\infty )$.

$\mathbf{Remarks}$

(1)

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 $L^q\left({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1}\right)$, $q>1$. Therefore, since $L(logL)({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})\cup B_q^{(0,0)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})\supset L^q({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$, then our results are natural extensions to the results in [7].

(2)

The conditions on $\mathbf{K}$ are the weakest known conditions for the cases ${\omega }_1\equiv 1\equiv {\omega }_2$ and $\epsilon =2$, (see [2,4]).

(3)

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 $\mathbf{K}\in L(logL)({\mathbb{B}}^{n-1}\times {\mathbb{B}}^{m-1})$. The same result was obtained in [22].

(4)

For the case ${\omega }_1\equiv 1\equiv {\omega }_2$, we observe that Theorem 2 generalizes Theorem 2.7 in [12].

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 $\{{{\rm Y}}_{\mathbf{K},l,t}:={{\rm Y}}_{l,t}:l,t\in {\mathbb{R}}_{+}\}$ and its corresponding maximal operators ${\mathrm{M}}_{\mu }$ and ${{\rm Y}}^{\ast }$ on ${\mathbb{R}}^m\times {\mathbb{R}}^n$ given by

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

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

and

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

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 $|{{\rm Y}}_{l,t}|$ is defined similarly as ${{\rm Y}}_{l,t}$ but $\mathbf{K}$ is replaced by $\left|\mathbf{K}\right|$.

The following lemma is a special case of Lemma 2.2 in [22].

Lemma 1. 

Assume that $\mathbf{K}$ satisfies (1)-(2) and belongs to $L^1({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$. 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

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

(6)

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

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

(7)

for $1<p<\infty$.

The next lemma is found in [24].

Lemma 2. 

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

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

(8)

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

(9)

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

Lemma 3. 

Let $q\in (1,2]$, $\mathbf{K}\in L^q\left({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1}\right)$, 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

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

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

Proof. 

By invoking (6), we obtain

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

(10)

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

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

(11)

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

(12)

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

$$\begin{array}{c}{∥\sum _{j,k\in \mathbb{Z}}{\int }_1^{\mu }{\int }_1^{\mu }{\left|{{\rm Y}}_{{\mu }^{k}r,{\mu }^{j}s}\ast {\mathcal{J}}_{j,k}\right|}^{{\gamma }^{\prime }}{\displaystyle \frac{dtdl}{tl}}∥}_{L^p({\omega }_1,{\omega }_2)}\hfill \end{array}$$

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

(13)

where $\overline{f}(a,b)=f(-a,-b)$. Let $\mathbf{T}$ be the linear operator defined on ${\mathcal{J}}_{j,k}$ by $\mathbf{T}\left({\mathcal{J}}_{j,k}\right)={{\rm Y}}_{{\mu }^{k}l,{\mu }^{j}t}\ast {\mathcal{J}}_{j,k}$. Thus, when we interpolate (12) with (13), we conclude

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

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

for all $1<p<\infty$. □

3. Proof of Theorem 1

Let $\epsilon >1$ and $\mathbf{K}\in L^q({\mathbb{B}}^{n-1}\times {\mathbb{B}}^{m-1})$ 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

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

(14)

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

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

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 $\Phi \in \mathcal{S}({\mathbb{R}}^m\times {\mathbb{R}}^n)$,

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

(15)

where

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

and

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

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

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

(16)

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

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

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,

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

(17)

For the other cases, by employing Lemma 3 and following the same argument as that used in the proof of Lemma 2.1 in [22], we obtain

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

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

(18)

for all $1<p<\infty$. Now, by interpolating between (17) and (18) we obtain (16) which when combined with (14)-(15) gives (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 $L^q({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$, $q>1$. 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 $\mathbf{K}\in B_q^{(0,{\displaystyle \frac{2}{\epsilon }}-1)}({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})\cup L{(logL)}^{2/\epsilon }({\mathbb{B}}^{m-1}\times {\mathbb{B}}^{n-1})$. Our results in this paper extend and improve several known results on generalized Marcinkiewicz operators as those in [1,2,4,7,9,10,12,22].