A Note on Generalized Parabolic Marcinkiewicz Integrals with Grafakos–Stefanov Kernels

Abstract

This paper focuses on studying the generalized Marcinkiewicz integral operators with mixed homogeneity over symmetric spaces. By making an appropriate decomposition of the aforementioned operators and tracking certain estimates, the boundedness of these operators is established from the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{0,\tau }\left({\mathbb{R}}^d\right)$ to the $L^p\left({\mathbb{R}}^d\right)$ space for all $p,\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$ provided that the kernel functions belong to the Grafakos–Stefanov class. The main results generalize and improve some previously known results on Marcinkiewicz and generalized Marcinkiewicz operators.

1. Introduction

Let $d\ge 2$ and ${\mathbb{S}}^{d-1}$ be the unit sphere in the Euclidean space ${\mathbb{R}}^d$ equipped with the induced Lebesgue surface measure $d{\sigma }_d\left(w^{\prime }\right)$. For $j\in \{1,2,\cdots ,d\}$, let ${\kappa }_j\ge 1$ be fixed numbers. We define the real mapping $\Gamma$ on ${\mathbb{R}}_{+}\times {\mathbb{R}}^d$ as $\Gamma (\gamma ,w)={\displaystyle \sum _{j=1}^d}{w}_j^2{\gamma }^{-2{\kappa }_j}$, with $w=(w_1,w_2,\dots ,w_d)\in {\mathbb{R}}^d$. For each fixed $w\in {\mathbb{R}}^d$, we denote the unique solution to the equation $\Gamma (\gamma ,w)=1$ by $\gamma \equiv {\gamma }_d\left(w\right)$. The metric space $({\mathbb{R}}^d,\gamma )$ is known by the mixed homogeneity space associated to ${\left\{{\kappa }_j\right\}}_{j=1}^d$.

The change in variables concerning the space $({\mathbb{R}}^d,\gamma )$ is presented in the following transformation:

$$\begin{array}{c}{w}_1={\gamma }^{{\kappa }_1}cos{v}_1\cdots cos{v}_{d-2}cos{v}_{d-1},\hfill \\ w_2={\gamma }^{{\kappa }_2}cos{v}_1\cdots cos{v}_{d-2}sin{v}_{d-1},\hfill \\ ⋮\hfill \\ w_{d-1}={\gamma }^{{\kappa }_{d-1}}cos{v}_{1}sin{v}_2,\hfill \\ w_d={\kappa }^{{\kappa }_d}sin{v}_1.\hfill \end{array}$$

So, $dw={\gamma }^{\kappa -1}J\left(w^{\prime }\right)d\gamma d{\sigma }_d\left(w^{\prime }\right)$, where $\kappa ={\displaystyle \sum _{j=1}^d}{\kappa }_j,J\left(w^{\prime }\right)={\displaystyle \sum _{j=1}^d}{\kappa }_j{\left(w_j^{\prime }\right)}^2,w^{\prime }={\mathrm{M}}_{{\gamma }^{-1}}w\in {\mathbb{S}}^{d-1},$ ${\gamma }^{\kappa -1}J\left(w^{\prime }\right)$ is the Jacobian of the transformation, and ${\mathrm{M}}_{\gamma }$ is the $d\times d$ diagonal matrix given by

$${\mathrm{M}}_{\gamma }=\left[\begin{array}{ccc}{\gamma }^{{\kappa }_1}& & 0\\ & \ddots & \\ 0& & {\gamma }^{{\kappa }_d}\end{array}\right].$$

In [1], it was proved that $J\left(w^{\prime }\right)\in {\mathcal{C}}^{\infty }\left({\mathbb{S}}^{d-1}\right)$ and $1\le J\left(w^{\prime }\right)\le C$ for some $C>1$.

Let $\mho \in L^1\left({\mathbb{S}}^{d-1}\right)$ be a measurable mapping satisfying

$$\begin{array}{ccc}\hfill \mho \left({\mathrm{M}}_{\gamma }w\right)& =& \mho \left(w\right),\forall \gamma >0\hfill \end{array}$$

(1)

and

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

(2)

For $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$ and $\tau \in (1,\infty )$, we define the generalized parabolic Marcinkiewicz integral ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ by

$${\mathcal{G}}_{\mho }^{\left(\tau \right)}\left(f\right)\left(x\right)={\left({\int }_{{\mathbb{R}}_{+}}{\left|{\displaystyle \frac{1}{r}}{\int }_{\gamma \left(v\right)\le r}f(x-v){\displaystyle \frac{\mho \left(v\right)}{{\left(\gamma \left(v\right)\right)}^{\kappa -1}}}dv\right|}^{\tau }{\displaystyle \frac{dr}{r}}\right)}^{1/\tau }.$$

The study of singular integrals, as well as the corresponding Marcinkiewicz integrals, over symmetric spaces has attracted the attention of many authors in the past two decades. These operators fall within the broader category of Littlewood–Paley g-functions. Establishing $L^p$ bounds for them is valuable for analyzing the smoothness characteristics of functions and the behavior of integral transformations, including Poisson integrals, singular integrals, and more broadly, singular Radon transforms.

Marcinkiewicz integrals have been extensively investigated by numerous authors, tracing back to the foundational work by Zygmund on the circle and that by Stein on ${\mathbb{R}}^d$. For more information and further applications, readers may refer to [2,3,4].

We point out that when ${\kappa }_1={\kappa }_2=\cdots ={\kappa }_d=1$, we have $\kappa =d$, $\gamma \left(v\right)=\left|v\right|$ and $({\mathbb{R}}^d,\gamma )=({\mathbb{R}}^d,|·|)$. In this case, we denote ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ by ${\mathcal{G}}_{\mho }^{\left(\tau \right),c}$. In addition, when $\tau =2$, the operator ${\mathcal{G}}_{\mho }^{\left(\tau \right),c}$ reduces to the classical Marcinkiewicz integral operator, which is denoted by ${\mathcal{M}}_{\mho }$. The operator ${\mathcal{M}}_{\mho }$ was introduced by E. Stein in [2], in which he proved that the $L^p$ boundedness of ${\mathcal{M}}_{\mho }$ for $p\in (1,2)$ holds under the condition $\mho \in Li{p}_{\nu }\left({\mathbb{S}}^{d-1}\right)$, with $\nu \in (0,1]$. Subsequently, the operator ${\mathcal{M}}_{\mho }$ has been investigated by many authors. For example, Walsh [5] proved the $L^2$ boundedness of ${\mathcal{M}}_{\mho }$ provided that $\mho \in L{\left(logL\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)⊋Li{p}_{\nu }\left({\mathbb{S}}^{d-1}\right)$ and proved that the condition $\mho \in L{\left(logL\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$ is nearly optimal. Later, this result was extended and improved in [6], where the authors obtained that ${\mathcal{M}}_{\mho }$ is of type $(p,p)$ for all $p\in (1,\infty )$ if $\mho \in L{\left(logL\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$. The authors in [7] proved that if ℧ belongs to the block space $B_q^{(0,-1/2)}\left({\mathbb{S}}^{d-1}\right)$, then ${\mathcal{M}}_{\mho }$ is bounded on $L^p\left({\mathbb{R}}^d\right)$ for all $p\in (1,\infty )$, and this condition on ℧ is nearly optimal.

On the other hand, the authors in [8] showed that whenever ℧ belongs to the Grafakos–Stefanov class ${\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ with $\alpha >0$, the operator ${\mathcal{M}}_{\mho }$ is bounded on $L^p\left({\mathbb{R}}^d\right)$ for all $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. For more background information on ${\mathcal{M}}_{\mho }$, readers may consult [9,10,11,12]; for its extensions and developments [13,14,15,16,17,18]; and for recent advances [19,20,21,22,23,24,25,26,27,28,29,30].

The generalized Marcinkiewicz operator ${\mathcal{G}}_{\mho }^{\left(\tau \right),c}$ was introduced by the authors of [31]. They found that if $\mho \in L^q\left({\mathbb{S}}^{d-1}\right)$ with $q>1$, then the inequality

$${∥{\mathcal{G}}_{\mho }^{\left(\tau \right),c}\left(f\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{∥f∥}_{{\stackrel{.}{F}}_p^{0,\tau }\left({\mathbb{R}}^d\right)}$$

(3)

holds for all $p,\tau \in (1,\infty )$. Later, Le improved this result in [32] under the weaker condition $\mho \in L(logL)\left({\mathbb{S}}^{d-1}\right)$. These results were extended and improved by the authors of [33] who showed that if ℧ is in either $L{(logL)}^{1/\tau }\left({\mathbb{S}}^{d-1}\right)$ or $B_q^{(0,\frac{-1}{{\tau }^{\prime }})}\left({\mathbb{S}}^{d-1}\right)$, then ${\mathcal{G}}_{\mho }^{\left(\tau \right),c}$ is bounded on $L^p\left({\mathbb{R}}^d\right)$ for $p,\tau \in (1,\infty )$. The authors of [34] confirmed inequality (3) for all $p,\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$ provided that $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$. For relevant results, we refer the reader to [35,36,37,38].

We point out that the Grafakos–Stefanov class ${\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ (for $\alpha >0$), which is the set of all integrable functions ℧ over ${\mathbb{S}}^{d-1}$ such that

$$\underset{\zeta \in {\mathbb{S}}^{d-1}}{sup}{\int }_{{\mathbb{S}}^{d-1}}{log}^{\alpha +1}\left({\left|\zeta ·u\right|}^{-1}\right)\left|\mho \left(u\right)\right|d{\sigma }_d\left(u\right)<\infty ,$$

(4)

was introduced in [5] and then developed in [39].

Now, let us recall the definition of homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{ϵ,\tau }\left({\mathbb{R}}^d\right)$ [40]. For $ϵ\in \mathbb{R}$ and $\tau ,p\in (1,\infty )$, the space ${\stackrel{.}{F}}_p^{ϵ,\tau }\left({\mathbb{R}}^d\right)$ is the collection of all tempered distribution functions f defined on ${\mathbb{R}}^d$ such that

$${∥f∥}_{{\stackrel{.}{F}}_p^{ϵ,\tau }\left({\mathbb{R}}^d\right)}={∥{\left(\sum _{j\in \mathbb{Z}}{2}^{jϵ\tau }{\left|{\Psi }_j\ast f\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}<\infty ,$$

where $\widehat{{\Psi }_j}\left(\zeta \right)=\Gamma \left(2^{-j}\zeta \right)$ and $\Gamma \in C_0^{\infty }\left({\mathbb{R}}^{\eta }\right)$ is a radial mapping satisfying the following:

(1) $0\le \Gamma \left(\zeta \right)\le 1$;

(2) $supp(\Gamma )\subset \left\{\zeta :\left|\zeta \right|\in [{\displaystyle \frac{1}{2}},2]\right\}$;

(3) For $\left|\zeta \right|\in [{\displaystyle \frac{3}{5}},{\displaystyle \frac{5}{3}}]$, there is a positive constant C such that $\Gamma \left(\zeta \right)\ge C$;

(4) For $\zeta \ne 0$, we have ${\displaystyle \sum _{j\in \mathbb{Z}}}\Gamma \left(2^{-j}\zeta \right)=1$.

The authors of [35] proved the following:

(a) The Schwartz space $\mathcal{S}\left({\mathbb{R}}^d\right)$ is dense in ${\stackrel{.}{F}}_p^{ϵ,\tau }\left({\mathbb{R}}^d\right)$;

(b) For $p\in (1,\infty )$, ${\stackrel{.}{F}}_p^{0,2}\left({\mathbb{R}}^d\right)=L^p\left({\mathbb{R}}^d\right)$;

(c) ${\stackrel{.}{F}}_p^{ϵ,{\tau }_1}\left({\mathbb{R}}^d\right)\subseteq {\stackrel{.}{F}}_p^{ϵ,{\tau }_2}\left({\mathbb{R}}^d\right)$ whenever ${\tau }_1\le {\tau }_2$.

Motivated by the work performed in [34] regarding the $L^p$ boundedness of the generalized Marcinkiewicz integral ${\mathcal{G}}_{\mho }^{\left(\tau \right),c}$ in the classical form whenever $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ and by the work performed in [17] regarding the $L^p$ boundedness of the parabolic Marcinkiewicz integral ${\mathcal{G}}_{\mho }^{\left(2\right)}$ whenever $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$, we shall study the $L^p$ boundedness of the generalized parabolic Marcinkiewicz integral ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ provided that $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$. In fact, we shall prove the following.

Theorem 1.

Let $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$ and satisfy $\left(1\right)$–$\left(2\right)$. Then, there exists a positive constant $C_p>0$ such that

$${∥{\mathcal{G}}_{\mho }^{\left(\tau \right)}\left(f\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le C_p{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\tau }\left({\mathbb{R}}^d\right)}$$

for all $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$ and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$.

Remark 1.

(1) 

For the cases ${\kappa }_j=1$ ($1\le j\le d$) and $\tau =2$, the $L^p$ boundedness of ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ was proved in [2] for $p\in (1,2)$ provided that $\mho \in Li{p}_{\nu }\left({\mathbb{S}}^{d-1}\right)⊊{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$. Hence, our result generalizes and improves the result in [2].

(2) 

Since ${\displaystyle \bigcup _{q>1}}{L}^q\left({\mathbb{S}}^{d-1}\right)⊊{\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$, Theorem 1 generalizes and improves the result in [15] for the case $\tau =2$.

(3) 

If we take $\tau =2$ in Theorem 1, the main result in [17] is obtained.

(4) 

The authors of [18,20] proved the boundedness of ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ whenever ℧ belongs to the space $L{(logL)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$ and the space $B_q^{(0,-\frac{1}{2})}\left({\mathbb{S}}^{d-1}\right)$, which are totally different from the space ${\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$.

(5) 

For the special case ${\kappa }_j=1$ ($1\le j\le d$), we obtain the main result in [34].

Henceforth, the letter C stands for a positive constant which is independent of the main parameters and not necessarily the same at each occurrence.

2. Auxiliary Lemmas

We begin this section by presenting several definitions and lemmas. We define the family of measures $\{{\lambda }_{\mho ,r,j}:={\lambda }_{r,j}:r\in {\mathbb{R}}_{+},j\in \mathbb{Z}\}$ and its corresponding maximal operator ${\lambda }^{\ast }$ on ${\mathbb{R}}^d$ as

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{R}}^d}fd{\lambda }_{r,j}& =& {\displaystyle \frac{1}{2^{j}r}}{\int }_{2^{j-1}r\le \gamma \left(u\right)\le 2^{j}r}\mho \left(u\right)f\left(u\right)du\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\lambda }^{\ast }\left(f\right)& =& \underset{j\in \mathbb{Z}}{sup}\underset{r\in {\mathbb{R}}_{+}}{sup}\left|\right|{\lambda }_{r,j}| \ast f|,\hfill \end{array}$$

where $|{\lambda }_{r,j}|$ is defined in the same way as ${\lambda }_{r,j}$ but ℧ is replaced by $|\mho |$.

We shall need the following lemma from [41].

Lemma 1.

Let $P\left(\gamma \right)=(v_1{\gamma }^{{\kappa }_1},v_2{\gamma }^{{\kappa }_2},\cdots ,v_d{\gamma }^{{\kappa }_d})$, where $v_j,{\kappa }_j\in \mathbb{R}$ and $j\in \{1,2,\cdots ,d\}$. Let us suppose that ${\mathcal{M}}_P$ is the maximal function defined on ${\mathbb{R}}^d$ by

$${\mathcal{M}}_{P}f\left(w\right)=\underset{\rho >0}{sup}{\displaystyle \frac{1}{\rho }}{\int }_0^{\rho }\left|f(w-P(\gamma \left)\right)\right|d\gamma .$$

Then, there exists a positive $C_p$ (independent of $v_j^{\prime }s$) such that the estimate

$${∥{\mathcal{M}}_P\left(f\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le C_p{∥f∥}_{L^p\left({\mathbb{R}}^d\right)}$$

holds for all $1<p\le \infty$.

We shall need the following lemma by Chen and Ding [16].

Lemma 2.

Let us assume that ν denotes the number of distinct elements in ${\left\{{\kappa }_j\right\}}_{j=1}^d$ and that $\delta \in [0,1]$. Then, there exists $C>0$ such that for $u\in {\mathbb{S}}^{d-1}$ and $\zeta \in {\mathbb{R}}^d$,

$$\left|{\int }_{1/2}^1{e}^{-i{\mathrm{M}}_{\gamma }u·\zeta }{\displaystyle \frac{d\gamma }{\gamma }}\right|\le C{\left|u·\zeta \right|}^{-{\displaystyle \frac{\delta }{\nu }}}.$$

By using Lemma 1, we directly deduce the following result.

Lemma 3.

Let $\mho \in L^1\left({\mathbb{S}}^{d-1}\right)$ and $f\in L^p\left({\mathbb{R}}^d\right)$. Then, for each $p\in (1,\infty )$, there exists a positive constant $C_p$ such that

$$\parallel {\lambda }^{\ast }{\left(f\right)\parallel }_{L^p\left({\mathbb{R}}^d\right)}\le C_p{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{\parallel f\parallel }_{L^p\left({\mathbb{R}}^d\right)}.$$

(5)

Lemma 4.

Let $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$ and satisfy $\left(1\right)$–$\left(2\right)$. Then, there is a positive constant C such that

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

(6)

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,j}\left(\zeta \right)\right|& \le & Cmin\left\{\left|{\mathrm{M}}_{2^{j}r}\zeta \right|,{\left(log|{\mathrm{M}}_{2^{j}r}\zeta |\right)}^{-(\alpha +1)}\right\},\hfill \end{array}$$

(7)

where $∥{\lambda }_r∥$ is the total variation of ${\lambda }_r$.

Proof. 

By the definition of ${\lambda }_{r,j}$, it is easy to verify that $\left(6\right)$ holds. By Fubini’s theorem and Hölder’s inequality, we obtain

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,j}\left(\zeta \right)\right|& \le & C{\int }_{1/2}^1\left|{\int }_{{\mathbb{S}}^{d-1}}\mho \left(u\right)J\left(u\right)e^{-i{\mathrm{M}}_{2^{j}r\gamma }u·\zeta }d{\sigma }_d\left(u\right)\right|{\displaystyle \frac{d\gamma }{\gamma }}\hfill \\ & \le & C{\int }_{{\mathbb{S}}^{d-1}}\left|\mho \left(u\right)\right|{\int }_{1/2}^{1}I(\zeta ,u,r){\displaystyle \frac{d\gamma }{\gamma }}d{\sigma }_d\left(u\right),\hfill \end{array}$$

(8)

where

$$\begin{array}{c}\hfill I(\zeta ,u,r)={\int }_{1/2}^1{e}^{-i{\mathrm{M}}_{2^{j}r\gamma }u·\zeta }{\displaystyle \frac{d\gamma }{\gamma }}.\end{array}$$

Thanks to Lemma 2, we deduce that

$$\begin{array}{c}\hfill I(\zeta ,u,r)\le {\left|{\mathrm{M}}_{r{2}^j}u·\zeta \right|}^{-{\displaystyle \frac{\delta }{\nu }}}\end{array}$$

When the above is combined with the trivial estimate $\left|I(\zeta ,u,r)\right|\le 1$ and by using the fact $(t/{log}^{\alpha }t)$ is increasing over the interval $(2^{\alpha },\infty )$, we have that

$$\begin{array}{c}\hfill \left|I(\zeta ,u,r)\right|\le C{\displaystyle \frac{{log}^{\alpha +1}\left({\left|\eta ·u\right|}^{-1}\right)}{{log}^{\alpha +1}\left(\left|{\mathrm{M}}_{2^{j}r}\zeta \right|\right)}}if\left|{\mathrm{M}}_{2^{j}r}\zeta \right|>1,\end{array}$$

(9)

where $\eta ={\displaystyle \frac{{\mathrm{M}}_{2^{j}r}\zeta }{|{\mathrm{M}}_{2^{j}r}\zeta |}}$. Thus, by (8)–(9), we obtain

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,j}\left(\zeta \right)\right|& \le & C{\left(log\left|{\mathrm{M}}_{2^{j}r}\zeta \right|\right)}^{-\alpha -1}{\int }_{{\mathbb{S}}^{d-1}}\left|\mho \left(u\right)\right|{log}^{\alpha }\left({\left|\eta ·u\right|}^{-1}\right)d{\sigma }_d\left(u\right)\hfill \\ & \le & C{\left(log\left|{\mathrm{M}}_{2^{j}r}\zeta \right|\right)}^{-\alpha -1}if\left|{\mathrm{M}}_{2^{j}r}\zeta \right|>1.\hfill \end{array}$$

(10)

Now, by condition (2), we obtain

$$\begin{array}{ccc}\hfill \left|{\widehat{\lambda }}_{r,j}\left(\zeta \right)\right|& \le & C{\int }_{{\mathbb{S}}^{d-1}}\left|\mho \left(u\right)\right|{\int }_{1/2}^1\left|I(\zeta ,u,r)-1\right|{\displaystyle \frac{d\gamma }{\gamma }}d{\sigma }_d\left(u\right)\hfill \\ & \le & C|{\mathrm{M}}_{2^{j}r}\zeta |.\hfill \end{array}$$

(11)

Consequently, by (10)–(11), we obtain $\left(7\right)$. The proof of the lemma is complete. □

Now, we need to prove the following result.

Lemma 5.

Let $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$ satisfy $\left(1\right)$–$\left(2\right)$. Then, for any class of functions $\{{\mathcal{H}}_j(·),j\in \mathbb{Z}\}$ defined on ${\mathbb{R}}^d$, we have

$$\begin{array}{c}\hfill {∥{\left(\sum _{j\in \mathbb{Z}}\underset{1}{\overset{2}{\int }}{\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\right|}^{\tau }{\displaystyle \frac{dr}{r}}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

(12)

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

Proof. 

As $p>1$, by duality, a non-negative mapping $\phi \in L^{p^{\prime }}\left({\mathbb{R}}^d\right)$ exists such that ${∥\phi ∥}_{L^{p^{\prime }}\left({\mathbb{R}}^d\right)}\le 1$ and

$$\begin{array}{ccc}\hfill {∥\left(\sum _{j\in \mathbb{Z}}\underset{1}{\overset{2}{\int }}\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\right|{\displaystyle \frac{dr}{r}}\right)∥}_{L^p\left({\mathbb{R}}^d\right)}& =& {\int }_{{\mathbb{R}}^d}\sum _{j\in \mathbb{Z}}{\int }_1^2\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\left(w\right)\right|{\displaystyle \frac{dr}{r}}\phi \left(w\right)dw\hfill \\ & \le & {\int }_{{\mathbb{R}}^d}\sum _{j\in \mathbb{Z}}\left|{\mathcal{H}}_j\left(w\right)\right|{\lambda }^{\ast }(\widetilde{\phi )}(-w)dw,\hfill \end{array}$$

(13)

where $\tilde{\phi }\left(w\right)=\phi (-w)$. Thus, by Lemma 3, Hölder’s inequality, and (13), we obtain that

$$\begin{array}{c}\hfill {∥\left(\sum _{j\in \mathbb{Z}}\underset{1}{\overset{2}{\int }}\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\right|{\displaystyle \frac{dr}{r}}\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥\sum _{j\in \mathbb{Z}}\left|{\mathcal{H}}_j\right|∥}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

(14)

for $1<p<\infty$. On the other hand, by inequality (5), we obtain

$$\begin{array}{ccc}\hfill {∥\underset{j\in \mathbb{Z}}{sup}\underset{r\in [1,2]}{sup}\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\right|∥}_{L^p\left({\mathbb{R}}^d\right)}& \le & {∥{\lambda }^{\ast }\left(\underset{j\in \mathbb{Z}}{sup}\left|{\mathcal{H}}_j\right|\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\hfill \\ & \le & C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥\underset{j\in \mathbb{Z}}{sup}\left|{\mathcal{H}}_j\right|∥}_{L^p\left({\mathbb{R}}^d\right)}\hfill \end{array}$$

(15)

for all $p\in (1,\infty )$. Therefore, by interpolating between (13) and (15), we have

$$\begin{array}{c}\hfill {∥{\left(\sum _{j\in \mathbb{Z}}\underset{1}{\overset{2}{\int }}{\left|{\lambda }_{r,j}\ast {\mathcal{H}}_j\right|}^{\tau }{\displaystyle \frac{dr}{r}}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

for all $p,\tau \in (1,\infty )$. □

3. Proof of Theorem 1

Let us assume that $\mho \in {\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$. By Minkowski’s inequality, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{G}}_{\mho }^{\left(\tau \right)}\left(f\right)\left(x\right)& =& {\left({\int }_{{\mathbb{R}}_{+}}{\left|\sum _{j=0}^{\infty }{\displaystyle \frac{1}{r}}{\int }_{2^{-j-1}r<\gamma \left(v\right)\le 2^{-j}r}f(x-v){\displaystyle \frac{\mho \left(v\right)}{{\left(\gamma \left(v\right)\right)}^{\kappa -1}}}dv\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }\hfill \\ & \le & \sum _{j=0}^{\infty }{\left({\int }_{{\mathbb{R}}_{+}}{\left|{\displaystyle \frac{1}{r}}{\int }_{2^{-j-1}r<\gamma \left(v\right)\le 2^{-j}r}f(x-v){\displaystyle \frac{\mho \left(v\right)}{{\left(\gamma \left(v\right)\right)}^{\kappa -1}}}dv\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }\hfill \\ & \le & C{\left(\sum _{j=0}^{\infty }{\int }_1^2{\left|{\lambda }_{r,j}\ast f\left(x\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }.\hfill \end{array}$$

(16)

Let us choose a set of smooth mappings ${\left\{{\Lambda }_j\right\}}_{j\in \mathbb{Z}}$ on ${\mathbb{R}}_{+}$ satisfying the following:

$$\begin{array}{ccc}\hfill {\Lambda }_j\left(w\right)={\Lambda }_j\left(\gamma \left(w\right)\right),{\Lambda }_j& \subset & [0,1],\sum _{j\in \mathbb{Z}}{\Lambda }_j{\left(v\right)}^2=1,\hfill \\ \hfill \mathrm{supp} \left({\Lambda }_j\right)& \subseteq & [2^{-1-j},2^{1-j}],and\left|{\displaystyle \frac{d^m{\Lambda }_j\left(v\right)}{d{v}^m}}\right|\le {\displaystyle \frac{C_m}{v^m}}.\hfill \end{array}$$

For $\zeta \in {\mathbb{R}}^d$, we define the operators $\left(\widehat{{\Psi }_j}\left(\zeta \right)\right)={\Lambda }_j\left(\gamma \left(\zeta \right)\right)$ and $S_j\left(\zeta \right)=({\Psi }_j\ast f)\left(\zeta \right)$. Hence, for any $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$,

$${\left(\sum _{j=0}^{\infty }{\int }_1^2{\left|{\lambda }_{r,j}\ast f\left(x\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }\le {\left(\sum _{j\in \mathbb{Z}}^{\infty }{\int }_1^2{\left|\sum _{n\in \mathbb{Z}}{S}_{j+n}\left({\lambda }_{r,j}\ast \left(S_{j+n}f\right)\right)\left(x\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }:=D\left(f\right)\left(x\right).$$

Thus, to prove our main result, it suffices to prove that

$${∥D\left(f\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le C_p{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\tau }\left({\mathbb{R}}^d\right)}$$

(17)

for all $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$ and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. We define the mapping T as

$$T:{\left\{{\mathcal{H}}_{j,n}\left(x\right)\right\}}_{n,j\in \mathbb{Z}}⟶{\left\{\sum _{n=1}^{\infty }{S}_{j+n}\left({\mathcal{H}}_{j,n}\right)\left(x\right)\right\}}_{j\in \mathbb{Z}}.$$

Then, it is easy to show the following:

(i)

For $1<p<\tau$ and $1<s<p$,

$$\begin{array}{c}{∥{\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{\left|\sum _{n\in \mathbb{Z}}{S}_{j+n}\left({\mathcal{H}}_{j,n})\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\hfill \end{array}$$

$$\begin{array}{c}\le C{\left(\sum _{n\in \mathbb{Z}}{∥{\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{\left|{\mathcal{H}}_{j,n}\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}^s\right)}^{1/s}.\hfill \end{array}$$

(18)

(ii)

For $\tau <p<\infty$ and $1<s<p^{\prime }$,

$${∥{\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{\left|\sum _{n\in \mathbb{Z}}{S}_{j+n}\left({\mathcal{H}}_{j,n})\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}$$

$$\le C{\left(\sum _{n\in \mathbb{Z}}{\left({\int }_1^2{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_{j,n}\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{s/\tau }\right)}^{1/s}.$$

(19)

To prove (17), we need to consider three cases.

Case 1. $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},\tau )$, and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. By (18), we deduce that

$$\begin{array}{ccc}\hfill {∥D\left(f\right)∥}_{L^p\left({\mathbb{R}}^d\right)}\le & C& {\left(\sum _{n\in \mathbb{Z}}{∥{\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{\left|{\lambda }_{r,j}\ast \left(S_{j+n}f\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}^s\right)}^{1/s}\hfill \\ & \le & C{\left(\sum _{n\in \mathbb{Z}}{∥{\mathrm{A}}_{n,\tau }f∥}_{L^p\left({\mathbb{R}}^d\right)}^s\right)}^{1/s}\hfill \end{array}$$

(20)

for all $1<s<p$, where

$${\mathrm{A}}_{n,\tau }f={\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{\left|{\lambda }_{r,j}\ast \left(S_{j+n}f\right)\right|}^{\tau }{\displaystyle \frac{d\gamma }{\gamma }}\right)}^{1/\tau }.$$

(21)

Let us estimate the $L^p$ norm of ${\mathrm{A}}_{n,\tau }f$ whenever $p=\tau =2$. For this case, we have that ${\stackrel{.}{F}}_2^{0,\overrightarrow{2}}\left({\mathbb{R}}^d\right)=L^2\left({\mathbb{R}}^d\right)$. Thus, by Plancherel’s theorem along with Fubini’s theorem and Lemma 4, we obtain

$$\begin{array}{ccc}\hfill {∥{\mathrm{A}}_{n,\tau }f∥}_{L^2\left({\mathbb{R}}^d\right)}^2& \le & \sum _{j\in \mathbb{Z}}{\int }_{{\Omega }_{n+j}}\left({\int }_1^2{\left|\widehat{{\lambda }_{r,j}}\left(\zeta \right)\right|}^2{\displaystyle \frac{d\gamma }{\gamma }}\right){\left|\widehat{f}\left(\zeta \right)\right|}^{2}d\zeta \hfill \\ & \le & C\sum _{j\in \mathbb{Z}}{\int }_{{\Omega }_{n+j}}{\left|\widehat{f}\left(\zeta \right)\right|}^{2}min\left\{{\left|{\mathrm{M}}_{2^{j}r}\zeta \right|}^2,{\left(log|{\mathrm{M}}_{2^{j}r}\zeta |\right)}^{-2(\alpha +1)}\right\}d\zeta \hfill \\ & \le & C{(1+\left|n\right|)}^{-2(1+\alpha )}{∥f∥}_{L^2\left({\mathbb{R}}^d\right)}^2\hfill \\ & =& C{(1+\left|n\right|)}^{-2(1+\alpha )}{∥f∥}_{{\stackrel{.}{F}}_2^{\overrightarrow{0},2}\left({\mathbb{R}}^d\right)},\hfill \end{array}$$

(22)

where ${\Omega }_n=\left\{\zeta \in {\mathbb{R}}^d:\left|\zeta \right|\in [2^{-1-n},2^{1-n}]\right\}$.

It is clear that by Lemma 5,

$$\begin{array}{c}\hfill {∥{\mathrm{A}}_{n,\tau }f∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\Psi }_j\ast f\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\sim C{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\tau }\left({\mathbb{R}}^d\right)}\end{array}$$

(23)

for all $1<p,\tau <\infty$. Therefore, by interpolation of (22) with (23), there exists a constant $\theta \in (1/(1+\alpha ),1)$ such that for all $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},\tau )$,

$$\begin{array}{c}\hfill {∥{\mathrm{A}}_{n,\tau }f∥}_{L^p\left({\mathbb{R}}^d\right)}\le C{(1+\left|n\right|)}^{\theta }{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\tau }\left({\mathbb{R}}^d\right)}.\end{array}$$

(24)

For a fixed $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},\tau )$, let us choose $1<s<p$ so that $s\theta >1/(\alpha +1)$ and employ (20) along with (24); we confirm (17), which, by (16), proves our main result for the case $p\in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},\tau )$ and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$.

Case 2. $p\in (\tau ,2\alpha +2)$, and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. We construct the proof of Theorem 1 by following a similar argument as above, except we need to invoke (19) instead of (18). The details are omitted.

Case 3. $p=\tau$, and $\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. By the definition of $D\left(f\right)$ and Lemma 1, we have

$$\begin{array}{ccc}\hfill {∥Df∥}_{L^p\left({\mathbb{R}}^d\right)}& =& {\left({\int }_{{\mathbb{R}}^d}\sum _{j\in \mathbb{Z}}^{\infty }{\int }_1^2{\left|\sum _{n\in \mathbb{Z}}{S}_{j+n}\left({\lambda }_{r,j}\ast \left(S_{j+n}f\right)\right)\left(x\right)\right|}^{\tau }{\displaystyle \frac{dr}{r}}dx\right)}^{1/\tau }\hfill \\ & \le & C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{\left(\sum _{j\in \mathbb{Z}}{\int }_1^2{∥{\mathcal{M}}_P\left(\left|S_{j+n}f\right|\right)∥}_{L^p\left({\mathbb{R}}^d\right)}^{\tau }{\displaystyle \frac{dr}{r}}\right)}^{1/\tau }\hfill \\ & \le & C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\Psi }_j\ast f\right|}^{\tau }\right)}^{1/\tau }∥}_{L^p\left({\mathbb{R}}^d\right)}\hfill \\ & \le & C{∥\mho ∥}_{L^1\left({\mathbb{S}}^{d-1}\right)}{∥f∥}_{{\stackrel{.}{F}}_p^{\overrightarrow{0},\tau }\left({\mathbb{R}}^d\right)}.\hfill \end{array}$$

Consequently, the proof of Theorem 1 is complete.

4. Conclusions

In this work, we introduced the generalized parabolic Marcinkiewicz operator ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ under a very weak condition on the rough kernel ℧. Whenever ℧ belongs to the Grafakos–Stefanov class ${\mathcal{F}}_{\alpha }\left({\mathbb{S}}^{d-1}\right)$ for some $\alpha >0$, we proved that the operator ${\mathcal{G}}_{\mho }^{\left(\tau \right)}$ is bounded from homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_p^{0,\tau }\left({\mathbb{R}}^d\right)$ to the $L^p\left({\mathbb{R}}^d\right)$ space for all $p,\tau \in ({\displaystyle \frac{2+2\alpha }{1+2\alpha }},2\alpha +2)$. The results obtained in this paper improve and generalize a number of previously known results (see [2,8,15,17,31,34]).