On the Functions of Generalized Marcinkiewicz Integral Operators along Subvarieties via Extrapolation

Abstract

Under weak conditions assumed on the kernels, appropriate $L_p$ estimates for certain class of generalized Marcinkiewicz integrals over subvarieties are established. By the virtue of the obtained estimates along with Yano’s extrapolation argument, the $L_p$ boundedness of the aforementioned integral operators under weaker conditions on the kernels is proved. The results in this paper represent essential extensions and improvements of many known results on the generalized Marcinkiewicz over symmetric spaces.

1. Introduction

Let ${\mathbf{R}}^m$, $m\ge 2$, be the Euclidean space of dimension m and ${\mathbb{U}}^{m-1}$ denote the unit sphere in ${\mathbf{R}}^m$ equipped with the normalized Lebesgue surface measure $d\mu$. Let h be a measurable function on ${\mathbf{R}}_{+}$ and ℧ be a function of homogeneous degree zero on ${\mathbf{R}}^m$ enjoying the properties that

$$\mho \in L_1\left({\mathbb{U}}^{m-1}\right)$$

(1)

and

$${\int }_{{\mathbb{U}}^{m-1}}\mho \left(w\right)d\mu \left(w\right)=0.$$

(2)

For a convenient mapping $\psi :{\mathbf{R}}_{+}\to \mathbf{R}$ and $u\in \mathcal{S}\left({\mathbf{R}}^m\right)$, we define the generalized parametric Marcinkiewicz operator ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ on the symmetric space ${\mathbf{R}}^m$ by

$${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)\left(x\right)={\left({\int }_0^{\infty }{\left|\frac{1}{s^{\alpha }}{\int }_{\left|w\right|\le s}u(x-\psi (\left|w\right|)w^{\prime })\frac{\mho \left(w\right)h\left(\left|w\right|\right)}{{\left|w\right|}^{m-\alpha }}dw\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa },$$

where $\alpha ={\alpha }_1+i{\alpha }_2$ (${\alpha }_1,{\alpha }_2\in \mathbf{R}$ with ${\alpha }_1>0$) and $\kappa >1$.

If $\psi \left(s\right)=s$, $h\equiv 1$, $\alpha =1$, and $\kappa =2$, we denote ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ by ${\mathfrak{M}}_{\mho }$. It is known that ${\mathfrak{M}}_{\mho }$ is the classical Marcinkiewicz integral operator which was introduced by Stein in [1] in which he proved the $L_p$ boundedness of ${\mathfrak{M}}_{\mho }$ for all $p\in (1,2]$ whenever $\mho \in Li{p}_{\sigma }\left({\mathbb{U}}^{m-1}\right)$ with $\sigma \in (0,1]$. Subsequently, the boundedness of ${\mathfrak{M}}_{\mho }$ was investigated by many researchers. For instance, it was shown in [2] that if ℧ lies in the space $C^1\left({\mathbb{U}}^{m-1}\right)$, then ${\mathfrak{M}}_{\mho }$ is bounded on $L_p\left({\mathbb{R}}^m\right)$ for all $p\in (1,\infty )$. This result was improved by Walsh in [3]. In fact, he confirmed the $L_2\left({\mathbf{R}}^m\right)$ boundedness of ${\mathfrak{M}}_{\mho }$ provided that $\mho \in L{\left(logL\right)}^{1/2}\left({\mathbb{U}}^{m-1}\right)$. Moreover, he established that the condition $\mho \in L{(logL)}^{1/2}\left({\mathbb{U}}^{m-1}\right)$ is optimal in the sense that there is an $\mho \in L{\left(logL\right)}^{\delta }\left({\mathbb{U}}^{m-1}\right)$ for all $0<\delta <1/2$ such that ${\mathfrak{M}}_{\mho }$ is not bounded on $L_2\left({\mathbf{R}}^m\right)$. Later on, the authors of [4] established the $L_p$ boundedness of ${\mathfrak{M}}_{\mho }$ for all $1<p<\infty$ under the assumption ℧ belongs to the block space $B_q^{(0,-1/2)}\left({\mathbb{U}}^{m-1}\right)$ with $q>1$. Furthermore, they found that the condition ℧ in $B_q^{(0,-1/2)}\left({\mathbb{U}}^{m-1}\right)$ is optimal in the sense that ${\mathfrak{M}}_{\mho }$ will not be bounded on $L_2\left({\mathbb{R}}^m\right)$ if $\mho \in B_q^{(0,\epsilon )}\left({\mathbb{U}}^{m-1}\right)$ for any $\epsilon \in (-1,-1/2)$.

On the other side, the parametric Marcinkiewicz operator was firstly studied by Hörmander in [5]. Actually, he emphasized the $L_p$ boundedness of ${\mathfrak{M}}_{\mho ,\psi ,1,\alpha }^{\left(2\right)}$ for any $p\in (1,\infty )$ if $\alpha >1$, $\psi \left(s\right)=s$ and $\mho \in Li{p}_{\sigma }\left({\mathbb{U}}^{m-1}\right)$ with $\sigma >0$. Thereafter, the discussion of the $L_p$ boundedness of ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(2\right)}$ under diverse conditions on ℧, $\psi$ and h has received much attention from many mathematicians. For a sampling of the past studies of these operators, see [6,7,8,9,10,11,12,13,14] and the references therein.

Recently, the study of generalized parametric Marcinkiewicz integrals ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ has begun. Historically, the operator ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ was introduced in [15] in which the authors proved that if $h\equiv 1$, $\mho \in L_q\left({\mathbb{U}}^{m-1}\right)$ for some $q>1$, $\psi \left(s\right)=s$, and $1<\kappa <\infty$, then for all $p\in (1,\infty )$,

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,1}^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}.$$

(3)

However, this result was improved by Le in [16]. In fact, he confirmed that Equation (3) holds under the same above assumptions but with replacing $h\equiv 1$ by a weaker assumption $h\in {\mathrm{\Theta }}_{max\{{\kappa }^{\prime },2\}}\left({\mathbf{R}}_{+}\right)$, where ${\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ ($\beta \ge 1$) is the class of all functions $h:[0,\infty )\to \mathbf{C}$ which are measurable and satisfying

$${\parallel h\parallel }_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}=\underset{j\in \mathbf{Z}}{sup}{\left({\int }_{2^j}^{2^{j+1}}{\left|h\left(s\right)\right|}^{{}^{\beta }}\frac{ds}{s}\right)}^{1/\beta }<\infty .$$

Very recently, Al-Qassem improved and extended the above results in [17]. As a matter of fact, he showed that if $\psi \left(s\right)=s$, $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ with $\beta >2$, $\mho \in L{(logL)}^{1/\kappa }\left({\mathbb{U}}^{m-1}\right)\cup B_q^{(0,\frac{1}{\kappa }-1)}\left({\mathbb{U}}^{m-1}\right)$ with $q>1$, then ${\mathfrak{M}}_{\mho ,s,h,\alpha }^{\left(\kappa \right)}$ is bounded on $L_p\left({\mathbf{R}}^m\right)$ for all $1<p<\kappa$ with $\kappa \le {\beta }^{\prime }$ and also for all ${\beta }^{\prime }<p<\infty$ with $\kappa >{\beta }^{\prime }$. Moreover, under the same assumptions on ℧ and $\psi$, but replacing the condition $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ by a very weaker condition $h\in {\mathcal{M}}^{1/\kappa }\left({\mathbf{R}}_{+}\right)$ with $\kappa >1$, he proved the $L_p$ boundedness of ${\mathfrak{M}}_{\mho ,s,h,\alpha }^{\left(\kappa \right)}$ for all $\kappa \le p<\infty$. For the significance and recent advances on the study of such operators, readers may consult [17,18,19,20,21] among others.

Let us recall some classes we shall use in this work. For $\beta >0$, we let ${\mathcal{M}}^{\beta }\left({\mathbf{R}}_{+}\right)$ denote the collection of all functions $h:[0,\infty )\to \mathbf{C}$ which are measurable and satisfy the condition

$$M^{\beta }\left(h\right)=\sum _{j=1}^{\infty }{2}^j{j}^{\beta }{d}_j\left(h\right)<\infty ,$$

where $d_j\left(h\right)=\underset{l\in \mathbf{Z}}{sup}{2}^{-l}\left|W(l,j)\right|$ with $W(l,j)=\left\{s\in (2^l,2^{l+1}]:2^{j-1}<\left|h\left(s\right)\right|\le 2^j\right\}$ for $j\ge 2$ and $W(l,1)=\left\{s\in (2^l,2^{l+1}]:\left|h\left(s\right)\right|\le 2\right\}$.

It is known that ${\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)\subset {\mathcal{M}}^r\left({\mathbf{R}}_{+}\right)$ for any $\beta \ge 1$ and $r>0$.

Let us recall the definition of the Triebel–Lizorkin spaces. For $\gamma \in \mathbf{R}$, $\kappa \in (1,\infty ]$ and $p\in (1,\infty )$, the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_{\gamma }^{\kappa ,p}\left({\mathbf{R}}^m\right)$ is defined by

$${\stackrel{.}{F}}_{\gamma }^{\kappa ,p}\left({\mathbf{R}}^m\right)=\left\{u\in {\mathcal{S}}^{\prime }\left({\mathbf{R}}^m\right):{∥u∥}_{{\stackrel{.}{F}}_{\gamma }^{\kappa ,p}\left({\mathbf{R}}^m\right)}={∥{\left(\sum _{l\in \mathbf{Z}}{2}^{l\gamma \kappa }{\left|{\mathrm{\Lambda }}_l\ast u\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}<\infty \right\}.$$

Here, ${\mathcal{S}}^{\prime }$ denotes the tempered distribution class on ${\mathbf{R}}^m$, $\widehat{{\mathrm{\Lambda }}_l}\left(\eta \right)=\phi \left(2^{-l}\eta \right)$ for $l\in \mathbf{Z}$ and $\phi$ is a radial function satisfying the following conditions:

(a)

$0\le \phi \le 1$;

(b)

$supp\phi \subset \left\{\eta :\left|\eta \right|\in [1,2]\right\}$;

(c)

$\phi \left(\eta \right)\ge C>0$ whenever $\left|\eta \right|\in [\frac{3}{5},\frac{5}{3}]$;

(d)

${\displaystyle \sum _{l\in \mathbf{Z}}}\phi \left(2^{-l}\eta \right)=1$$(\eta \ne 0)$.

The following properties are well known.

(i)

$\mathcal{S}\left({\mathbf{R}}^m\right)$ is dense in ${\stackrel{.}{F}}_{\gamma }^{\kappa ,p}\left({\mathbf{R}}^m\right)$;

(ii)

${\stackrel{.}{F}}_0^{2,p}\left({\mathbf{R}}^m\right)=L_p\left({\mathbf{R}}^m\right)$ for $1<p<\infty$ , and ${\stackrel{.}{F}}_0^{2,\infty }\left({\mathbf{R}}^m\right)=BMO$;

(iii)

${\stackrel{.}{F}}_{\gamma }^{{\kappa }_1,p}\left({\mathbf{R}}^m\right)\subseteq {\stackrel{.}{F}}_{\gamma }^{{\kappa }_2,p}\left({\mathbf{R}}^m\right)$ if ${\kappa }_1\le {\kappa }_2$;

(iv)

${\left({\stackrel{.}{F}}_{\gamma }^{\kappa ,p}\left({\mathbf{R}}^m\right)\right)}^{*}={\stackrel{.}{F}}_{-\gamma }^{{\kappa }^{\prime },p^{\prime }}\left({\mathbf{R}}^m\right)$.

In this work, we let I be the class of all nonnegative $C^1$ functions $\psi :{\mathbf{R}}_{+}\to \mathbf{R}$ that satisfy the following conditions:

(1)

$\psi$ is strictly increasing on ${\mathbf{R}}_{+}$ and ${\psi }^{\prime }$ is monotone;

(2)

$\psi \left(2s\right)\ge C_1\psi \left(s\right)$ for some fixed $C_1>1$ and $\psi \left(2s\right)\le C_2\psi \left(s\right)$ for some constant $C_2\ge C_1$;

(3)

$s{\psi }^{\prime }\left(s\right)\ge C_3\psi \left(s\right)$ on ${\mathbf{R}}_{+}$ for some fixed $0<C_3<log\left(C_2\right)$.

Additionally, we let D be the set of all nonnegative $C^1$ functions $\psi :{\mathbf{R}}_{+}\to \mathbf{R}$ that satisfy the following conditions:

(1)

$\psi$ is strictly decreasing on ${\mathbf{R}}_{+}$ and ${\psi }^{\prime }$ is monotone;

(2)

$\psi \left(s\right)\ge C_1\psi \left(2s\right)$ for some fixed $C_1>1$ and $\psi \left(s\right)\le C_2\psi \left(2s\right)$ for some constant $C_2\ge C_1$;

(3)

$\left|s{\psi }^{\prime }\left(s\right)\right|\ge C_3\psi \left(s\right)$ on ${\mathbf{R}}_{+}$ for some fixed $0<C_3<log\left(C_2\right)$.

We point out that the classes $\mathbf{I}$ and $\mathbf{D}$ were introduced in [22]. Some model examples for the class $\mathbf{D}$ are $\psi \left(s\right)=s^{-\tau }{e}^{-\lambda s}$ for $\tau \ge 0$ and $\lambda \ge 0$, and for the class $\mathbf{I}$ are $\psi \left(s\right)=s^{\tau }{e}^{\lambda s}$ for $\tau \ge 0$ and $\lambda \ge 0$.

It is worth mentioning that whenever $\psi$ belongs to the classes $\mathbf{I}$ or $\mathbf{D}$, $\mho \in B_q^{(0,-1/2)}\left({\mathbb{U}}^{m-1}\right)$ with $q>1$ and $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ with $\beta >1$, then the $L^p$ boundedness of ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(2\right)}$ for $\left|1/p-1/2\right|<min\{1/{\beta }^{\prime },1/2\}$ was proved in [14] only for the case $\kappa =2$. In view of the result in [14] and of the result in [17], a natural question arises. Does the boundedness of ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ hold under the same conditions assumed in [14] but with replacing the condition $\kappa =2$ by a weaker condition $\kappa >1$?

The motivation of this work is to answer the above question in the affirmative. Precisely, we will improve and extend the results in [1,2,3,4,5,7,14] which are just special cases of our results. Furthermore, we will generalize what were established in [15,16,17]. To achieve this, we need to prove the following theorems.

Theorem 1.

Let ℧ satisfy the condition Equation (2) and belong to $L_q\left({\mathbb{U}}^{m-1}\right)$ for some $1<q\le 2$. Suppose that h belongs to ${\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ for some $1<\beta \le 2$ and ψ lies inIorD. Then for any $u\in {\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)$, there is a constant $C_p>0$ (independent of ℧, ψ, h, κ, β, and q) such that

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p{(q-1)}^{-1}{(\beta -1)}^{-1}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}if1<p<\kappa ,$$

(4)

and

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p{(q-1)}^{-1/\kappa }{(\beta -1)}^{-1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}if\kappa \le p<\infty .$$

(5)

Theorem 2.

Let ψ and ℧ be given as in the above Theorem, and $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ for some $\beta >2$. Then there exists a constant $C_p>0$ satisfies

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p{(q-1)}^{-1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

(6)

for $1<p<\kappa$ if $\beta \in (2,\infty )$ and $\beta \ge {\kappa }^{\prime }$, and

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p{(q-1)}^{-1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

(7)

for ${\beta }^{\prime }<p<\infty$ if $\beta \in (2,\infty ]$ and $\beta >{\kappa }^{\prime }$.

Here and in what follows, the letter C refers to a positive constant whose value may vary at each appearance, but independent of the fundamental variables.

This paper is organised as follows: In Section 2, some auxiliary lemmas are proved. In Section 3, the proofs of Theorems 1 and 2 are presented. Finally, some consequences of Theorems 1 and 2 are given is Section 4.

2. Preliminary Lemmas

In this section, some auxiliary lemmas will be given. Let us begin by introducing some necessary notations. Let $\nu \ge 2$. For appropriate functions $\psi :{\mathbf{R}}_{+}\to \mathbf{R}$, $\mho :{\mathbb{U}}^{m-1}\to \mathbf{R}$ and $h:{\mathbf{R}}_{+}\to \mathbf{C}$; we define the family of measures $\{{\mu }_{\mho ,\psi ,h,s}:s\in {\mathbf{R}}_{+}\}$ and its corresponding maximal integrals ${\mu }_{\mho ,\psi ,h}^{*}$ and $M_{\mho ,\psi ,h,\nu }$ on ${\mathbf{R}}^m$ by

$$\begin{array}{ccc}\hfill {\int }_{{\mathbf{R}}^m}ud{\mu }_{\mho ,\psi ,h,s}& =& s^{-\alpha }{\int }_{s/2\le \left|w\right|\le s}u\left(\psi \right(\left|w\right|)w^{\prime })\frac{\mho \left(w\right)h\left(\left|w\right|\right)}{{\left|w\right|}^{m-\alpha }}dw,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mu }_{\mho ,\psi ,h}^{*}\left(u\right)& =& \underset{s\in {\mathbf{R}}_{+}}{sup}\left|\right|{\mu }_{\mho ,\psi ,h,s}|\ast u|,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill M_{\mho ,\psi ,h,\nu }\left(u\right)& =& \underset{j\in \mathbf{Z}}{sup}{\int }_{{\nu }^j}^{{\nu }^{j+1}}\left|\right|{\mu }_{\mho ,\psi ,h,s}|\ast u|\frac{ds}{s}.\hfill \end{array}$$

Here, we define $|{\mu }_{\mho ,\psi ,h,s}|$ in the same way as ${\mu }_{\mho ,\psi ,h,s}$, but with replacing $h\mho$ by $|h\mho |$.

The following two lemmas will play a substantial role in the proof of our main results. They can be obtained by following similar procedures (with only minor changes) used in [14], Lemmas 2.7–2.8].

Lemma 1.

Let $\nu \ge 2$, $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ and $\mho \in L_q\left({\mathbb{U}}^{m-1}\right)$ for some $\beta ,q>1$. Assume that ψ belongs toD or I. Then there are constants C and ε with $0<2\epsilon <1/q^{\prime }$ such that for all $j\in \mathbf{Z}$,

$$\begin{array}{cc}∥{\mu }_{\mho ,\psi ,h,s}∥\le C;\hfill & \end{array}$$

(8)

$$\begin{array}{ccc}& & \underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\widehat{\mu }}_{\mho ,\psi ,h,s}\left(\eta \right)\right|}^2\frac{ds}{s}\le C(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^2{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{2}min\left\{{\left|\eta \psi \left({\nu }^j\right)\right|}^{+\frac{2\epsilon }{ln\nu }},{\left|\eta \psi \left({\nu }^j\right)\right|}^{-\frac{2\epsilon }{ln\nu }}\right\},\hfill \end{array}$$

(9)

where $∥{\mu }_{\mho ,\psi ,h,s}∥$ is the total variation of ${\mu }_{\mho ,\psi ,h,s}$.

Lemma 2.

Let ν, ψ and h be given as in Lemma 1. Assume that $\mho \in L_q\left({\mathbb{U}}^{m-1}\right)$ for some $1<q\le 2$. Then there exists a constant $C_p>0$ such that

$$\parallel M_{\mho ,\psi ,h,\nu }{\left(u\right)\parallel }_{L_p\left({\mathbf{R}}^m\right)}\le C_p(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{\parallel u\parallel }_{L_p\left({\mathbf{R}}^m\right)},$$

(10)

$$\parallel {\mu }_{\mho ,\psi ,h}^{*}{\left(u\right)\parallel }_{L_p\left({\mathbf{R}}^m\right)}\le C_p(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{\parallel u\parallel }_{L_p\left({\mathbf{R}}^m\right)}$$

(11)

for all $1<p\le \infty$ with $1<\beta \le 2$, and

$$\parallel {\mu }_{\mho ,\psi ,h}^{*}{\left(f\right)\parallel }_{L_p\left({\mathbf{R}}^m\right)}\le C_p{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{\parallel u\parallel }_{L_p\left({\mathbf{R}}^m\right)};$$

(12)

for all ${\beta }^{\prime }<p<\infty$ with $\beta \ge 2$.

By employing analogous arguments that utilized in [17], we obtain the following:

Lemma 3.

Let $\mho \in L_q\left({\mathbb{U}}^{m-1}\right)$ and $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ for some $q,\beta \in (1,2]$. Let $\nu \ge 2$ and ψ be given as in Lemma 1. Then a positive constant $C_p$ exists such that for arbitrary functions $\{g_j(·),j\in \mathbf{Z}\}$ on ${\mathbf{R}}^m$, we have

$$\begin{array}{ccc}\hfill {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{\frac{1}{\kappa }}∥}_{L_p\left({\mathbf{R}}^m\right)}& \le & C_p{(ln\nu )}^{1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(13)

for all $p\in [\kappa ,\infty )$, and

$$\begin{array}{ccc}\hfill {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{\frac{1}{r}}∥}_{L_p\left({\mathbf{R}}^m\right)}& \le & C_p(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(14)

for all $p\in (1,\kappa )$.

Proof. 

Let us start with the inequality Equation (13). Consider the case $p=\kappa$. By using Hölder’s inequality and then Equation (10), we obtain

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\nu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{\frac{1}{\kappa }}∥}_{L_p\left({\mathbf{R}}^m\right)}^{\kappa }\le C{∥h∥}_{{\mathrm{\Theta }}_1\left({\mathbf{R}}^{+}\right)}^{(\kappa /{\kappa }^{\prime })}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })}\hfill \\ & \times & \sum _{j\in \mathbf{Z}}\underset{{\mathbf{R}}^m}{\int }\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}\underset{\frac{1}{2}s}{\overset{s}{\int }}\underset{{\mathbb{U}}^{m-1}}{\int }{\left|b_j(x-\psi \left(t\right)w)\right|}^r\left|\mho \left(w\right)\right|\left|h\left(t\right)\right|d\mu \left(w\right)\frac{dt}{t}\frac{ds}{s}dx\hfill \\ & \le & C(ln\nu ){∥h∥}_{{\mathrm{\Theta }}_1\left({\mathbf{R}}_{+}\right)}^{(\kappa /{\kappa }^{\prime })+1}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })+1}\underset{{\mathbf{R}}^m}{\int }{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\left(x\right)\right|}^{\kappa }dx\right)}^{p/\kappa }.\hfill \end{array}$$

(15)

Thus, when we take the ${\kappa }^{th}$ root to the both sides, Equation (13) is satisfied for the case $p=\kappa$. Now consider the case $p>\kappa$. By duality, there is a non-negative function $g\in L_{{(p/\kappa )}^{\prime }}\left({\mathbf{R}}^m\right)$ with ${∥g∥}_{L_{{(p/\kappa )}^{\prime }}\left({\mathbf{R}}^m\right)}\le 1$ such that

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}^{\kappa }=\underset{{\mathbf{R}}^m}{\int }\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\left(x\right)\right|}^{\kappa }\frac{ds}{s}g\left(x\right)dx.\hfill \end{array}$$

(16)

It is clear that Hölder’s inequality leads to

$$\begin{array}{ccc}& & {\left|{\mu }_{\mho ,\psi ,h,s}*b_j\left(x\right)\right|}^{\kappa }\le C{∥h∥}_{{\mathrm{\Theta }}_1\left({\mathbf{R}}^{+}\right)}^{(\kappa /{\kappa }^{\prime })}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })}\underset{\frac{1}{2}s}{\overset{s}{\int }}\underset{{\mathbb{U}}^{m-1}}{\int }{\left|b_j(x-\psi \left(t\right)w)\right|}^{\kappa }\left|\mho \left(w\right)\right|\left|h\left(t\right)\right|d\mu \left(w\right)\frac{dt}{t}.\hfill \end{array}$$

Hence, by a simple change of variables and applying Hölder’s inequality on Equation (16), we deduce

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}^{\kappa }\hfill \\ & \le & C{∥h∥}_{{\mathrm{\Theta }}_1\left({\mathbf{R}}_{+}\right)}^{(\kappa /{\kappa }^{\prime })}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })}\underset{{\mathbf{R}}^m}{\int }\left(\sum _{j\in \mathbf{Z}}{\left|b_j\left(x\right)\right|}^{\kappa }\right)M_{\left|\mho \right|,\psi ,\left|h\right|,\nu }\tilde{g}(-x)dx\hfill \\ & \le & C{∥h∥}_{{\mathrm{\Theta }}_1\left({\mathbf{R}}_{+}\right)}^{(\kappa /{\kappa }^{\prime })}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })}{∥\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }∥}_{L_{(p/\kappa )}\left({\mathbf{R}}^m\right)}{∥M_{\left|\mho \right|,\psi ,\left|h\right|,\nu }\tilde{g}∥}_{L_{{(p/\kappa )}^{\prime }}\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p(ln\nu ){∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{(\kappa /{\kappa }^{\prime })+1}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^{(\kappa /{\kappa }^{\prime })+1}{∥\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }∥}_{L_{(p/\kappa )}\left({\mathbf{R}}^m\right)}{∥\tilde{g}∥}_{L_{{(p/\kappa )}^{\prime }}\left({\mathbf{R}}^m\right)},\hfill \end{array}$$

where $\tilde{g}(-x)=g\left(x\right)$. Consequently, Equation (13) holds for all $\kappa \le p<\infty$. Finally, we consider the case $1<p<\kappa$. This gives that ${\kappa }^{\prime }<p^{\prime }$ and, hence, by the duality, there are functions ${\xi }_j(x,s)$ defined on ${\mathbf{R}}^m\times {\mathbf{R}}_{+}$ such that ${∥{∥\parallel {\xi }_j{\parallel }_{L_{{\kappa }^{\prime }}([{\nu }^j,{\nu }^{j+1}],\frac{ds}{s})}∥}_{L_{{\kappa }^{\prime }}}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}\le 1$ and

$$\begin{array}{c}\hfill {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^j}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}=\underset{{\mathbf{R}}^m}{\int }\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}\left({\mu }_{\mho ,\psi ,h,s}\ast b_j\left(x\right)\right){\xi }_j(x,s)\frac{ds}{s}dx.\end{array}$$

(17)

For simplicity, let $\mathrm{\Gamma }\left({\xi }_j\right)$ be given by

$$\mathrm{\Gamma }\left({\xi }_j\right)\left(x\right)=\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^j}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast {\xi }_j(x,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}.$$

Again, by the duality we deduce that there exists a function $\varphi$ which belongs to the space $L_{{(p^{\prime }/{\kappa }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)$ with norm 1 and satisfies

$$\begin{array}{ccc}& & {∥{\left(\Gamma \left({\xi }_j\right)\right)}^{1/{\kappa }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}^{{\kappa }^{\prime }}=\sum _{j\in \mathbf{Z}}\underset{{\mathbf{R}}^m}{\int }\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast {\xi }_j(x,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}\varphi \left(x\right)dx\hfill \\ & \le & C{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}^{+}\right)}^{({\kappa }^{\prime }/\kappa )}{∥{{\mu }^{*}}_{\left|\mho \right|,\psi ,\left|h\right|}\left(\varphi \right)∥}_{L_{{(p^{\prime }/{\kappa }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}{∥\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\xi }_j(·,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}\right)∥}_{L_{(p^{\prime }/{\kappa }^{\prime })}\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )+1}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}^{+}\right)}^{({\kappa }^{\prime }/\kappa )+1}{∥\varphi ∥}_{L_{{(p^{\prime }/{\kappa }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}.\hfill \end{array}$$

Hence, Hölder’s inequality together with Equation (17) give

$$\begin{array}{ccc}\hfill {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^j}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}& \le & C_p{(ln\nu )}^{1/\kappa }{∥{\left(\Gamma \left({\xi }_j\right)\right)}^{1/{\kappa }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p(ln\nu ){∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}^{+}\right)}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(18)

hold for all $1<p<\kappa$. Consequently, the proof of this lemma is complete. □

In the same manner, we obtain the following:

Lemma 4.

Let ℧, ψ, ν, and κ be given as in Lemma 3. Assume that $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ for some $2\le \beta <\infty$. Then there exists a constant $C_p$ such that

(i)

If $\kappa \le {\beta }^{\prime }$, we have

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{\frac{1}{\kappa }}∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p{(ln\nu )}^{1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}for1<p<\kappa ;\hfill \end{array}$$

(19)

(ii)

If $\kappa >{\beta }^{\prime }$, we have

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{\frac{1}{\kappa }}∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p{(ln\nu )}^{1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}for{\beta }^{\prime }<p<\infty ,\hfill \end{array}$$

(20)

where $\{b_j(·),j\in \mathbf{Z}\}$ is a set of functions on ${\mathbf{R}}^m$.

Proof. 

Firstly, we consider the case $1<p<\kappa$ with $\kappa \le {\beta }^{\prime }$. By following the same above arguments, we obtain by the duality that there are functions ${\zeta }_j(x,s)$ defined on ${\mathbf{R}}^m\times {\mathbf{R}}_{+}$ with ${∥{∥\parallel {\zeta }_j{\parallel }_{L_{{\kappa }^{\prime }}([{\nu }^j,{\nu }^{j+1}],\frac{ds}{s})}∥}_{L_{{\kappa }^{\prime }}}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}\le 1$ and satisfy

$$\begin{array}{ccc}\hfill {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^j}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}*b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}=& & \underset{{\mathbf{R}}^m}{\int }\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}\left({\mu }_{\mho ,\psi ,h,s}\ast b_j\left(x\right)\right){\zeta }_j(x,s)\frac{ds}{s}dx\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & C_{p}ln{\left(\nu \right)}^{1/\kappa }{∥{\left({\rm Y}\left({\zeta }_j\right)\right)}^{1/{\kappa }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)},\hfill \end{array}$$

(21)

where

$${\rm Y}\left({\zeta }_j\right)\left(x\right)=\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^j}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast {\zeta }_j(x,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}.$$

Since $\kappa \le {\beta }^{\prime }\le \beta$, then Hölder’s inequality gives

$$\begin{array}{ccc}\hfill {\left|{\mu }_{\mho ,\psi ,h,s}\ast {\zeta }_j(x,s)\right|}^{{\kappa }^{\prime }}& \le & C{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\kappa }^{\prime }}{\int }_{{\nu }^j}^{{\nu }^{j+1}}{\int }_{{\mathbb{U}}^{m-1}}\left|\mho \left(w\right)\right|\hfill \\ & \times & {\left|{\zeta }_j(x-\psi \left(t\right)w,s)\right|}^{{\kappa }^{\prime }}d\mu \left(w\right)\frac{dt}{t}\hfill \\ & \le & C{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\kappa }^{\prime }}{\int }_{{\nu }^j}^{{\nu }^{j+1}}{\int }_{{\mathbb{U}}^{m-1}}\left|\mho \left(u\right)\right|\hfill \\ & \times & {\left|{\zeta }_j(x-\psi \left(t\right)w,s)\right|}^{{\kappa }^{\prime }}d\mu \left(u\right)\frac{dt}{t}.\hfill \end{array}$$

(22)

Again, since $p^{\prime }>{\kappa }^{\prime }$, we obtain that there exists a function $\rho$ belonging to $L_{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)$ such that

$$\begin{array}{ccc}& & {∥{\left({\rm Y}\left({\zeta }_j\right)\right)}^{1/{\kappa }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}^{{\kappa }^{\prime }}=\sum _{j\in \mathbf{Z}}\underset{{\mathbf{R}}^m}{\int }\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast {\zeta }_j(x,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}\rho \left(x\right)dx.\hfill \end{array}$$

Thus, by Hölder’s inequality, a simple change of variables plus the inequalities Equations (11) and (22), we conclude

$$\begin{array}{ccc}\hfill {∥{\left({\rm Y}\left({\zeta }_j\right)\right)}^{1/{\kappa }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbf{R}}^m\right)}^{{\kappa }^{\prime }}& \le & C{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\kappa }^{\prime }}{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )}{∥{{\mu }^{*}}_{\left|\mho \right|,\psi ,1}\left(\rho \right)∥}_{L_{{(p^{\prime }/{\kappa }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}\hfill \\ & \times & {∥\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\zeta }_j(·,s)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}\right)∥}_{L_{(p^{\prime }/{\kappa }^{\prime })}\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\kappa }^{\prime }/\kappa )+1}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\kappa }^{\prime }}{∥\rho ∥}_{L_{{(p^{\prime }/{\kappa }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}.\hfill \end{array}$$

(23)

Therefore, by Equations (21) and (23), we attain Equation (19) for any $1<p<\kappa$ with $\kappa \le {\beta }^{\prime }$.

Now, let us consider the case ${\beta }^{\prime }<p<\infty$ with $\kappa >{\beta }^{\prime }$. By Equation (12), we have

$$\begin{array}{ccc}\hfill {∥\underset{j\in \mathbf{Z}}{sup}\underset{s\in [1,\nu ]}{sup}\left|{\mu }_{\mho ,\psi ,h,{\nu }^{j}s}*b_j\right|∥}_{L_p\left({\mathbf{R}}^m\right)}& \le & {∥{\mu }_{\mho ,\psi ,h}^{*}\left(\underset{j\in \mathbf{Z}}{sup}\left|b_j\right|\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C_p{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥\underset{j\in \mathbf{Z}}{sup}\left|b_j\right|∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(24)

for all ${\beta }^{\prime }<p<\infty$ with $\beta \ge 2$. This gives that

$$\begin{array}{ccc}\hfill {∥{∥\parallel {\mu }_{\mho ,\psi ,h,{\nu }^{j}s}\ast b_j{\parallel }_{L_{\infty }([1,\nu ],\frac{ds}{s})}∥}_{l_{\infty }\left(\mathbf{Z}\right)}∥}_{L_p\left({\mathbf{R}}^m\right)}& \le & C_p{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{∥b_j∥}_{l_{\infty }\left(\mathbf{Z}\right)}∥}_{L_p\left({\mathbf{R}}^m\right)}.\hfill \end{array}$$

(25)

Again, by duality, there exists $\varrho \in L_{{(p/{\beta }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)$ such that ${∥\varrho ∥}_{L_{{(p/{\beta }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}\le 1$ and

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{1}{\overset{\nu }{\int }}{\left|{\mu }_{\mho ,\psi ,h,{\nu }^{j}s}\ast b_j\right|}^{{\beta }^{\prime }}\frac{ds}{s}\right)}^{\frac{1}{{\beta }^{\prime }}}∥}_{L_p\left({\mathbf{R}}^m\right)}^{{\beta }^{\prime }}=\underset{{\mathbf{R}}^m}{\int }\sum _{j\in \mathbf{Z}}\underset{1}{\overset{\nu }{\int }}{\left|{\mu }_{\mho ,\psi ,h,{\nu }^{j}s}\ast b_j\left(x\right)\right|}^{{\kappa }^{\prime }}\frac{ds}{s}\varrho \left(x\right)dx\hfill \\ & & \le C{∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\beta }^{\prime }/\beta )}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\beta }^{\prime }}\underset{{\mathbf{R}}^m}{\int }\sum _{j\in \mathbf{Z}}{\left|b_j\left(x\right)\right|}^{{\beta }^{\prime }}{\mu }_{\mho ,\psi ,1}^{*}\overline{\varrho }(-x)dx\hfill \\ & & \le Cln\left(\nu \right){∥\mho ∥}_{L_1\left({\mathbb{U}}^{m-1}\right)}^{({\beta }^{\prime }/\beta )}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{s^{\prime }}{∥\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{{\beta }^{\prime }}∥}_{{}_{L_{(p//bet{a}^{\prime })}\left({\mathbf{R}}^m\right)}}{∥{\mu }_{\mho ,\psi ,1}^{*}\left(\overline{\varrho }\right)∥}_{{}_{L_{{(p/{\beta }^{\prime })}^{\prime }}\left({\mathbf{R}}^m\right)}}\hfill \\ & & \le Cln\left(\nu \right){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^{({\beta }^{\prime }/\beta )+1}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^{{\beta }^{\prime }}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{{\beta }^{\prime }}\right)}^{\frac{1}{{\beta }^{\prime }}}∥}_{{}_{L_p\left({\mathbf{R}}^m\right)}}^{{\beta }^{\prime }},\hfill \end{array}$$

(26)

where $\overline{\varrho }\left(x\right)=\varrho (-x)$. By using the following linear operator $\mathcal{G}$ which is defined on any function $b_j\left(x\right)$ by $\mathcal{G}\left(b_j\left(x\right)\right)={\mu }_{\mho ,\psi ,h,{\nu }^{j}s}\ast b_j\left(x\right)$, we can interpolate Equations (25) and (26) to obtain

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbf{Z}}\underset{{\nu }^j}{\overset{{\nu }^{j+1}}{\int }}{\left|{\mu }_{\mho ,\psi ,h,s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\le {∥{\left(\sum _{j\in \mathbf{Z}}\underset{1}{\overset{\nu }{\int }}{\left|{\mu }_{\mho ,\psi ,h,{\nu }^{j}s}\ast b_j\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & & \le C_p{(ln\nu )}^{1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥{\left(\sum _{j\in \mathbf{Z}}{\left|b_j\right|}^{\kappa }\right)}^{1/\kappa }∥}_{{}_{L_p\left({\mathbf{R}}^m\right)}}\hfill \end{array}$$

for all ${\beta }^{\prime }<p<\infty$ with $\beta \ge 2$. The proof of Lemma 4 is complete. □

3. Proof of the Main Results

Proof of Theorem 1. The idea of the proof of this theorem depends on the arguments taken from [14,17]. Assume that $\mho \in L_q\left({\mathbb{U}}^{m-1}\right)$ with $q\in (1,2]$, $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ with $\beta \in (1,2]$ and $\psi$ belongs to I or D. By using Minkowski’s inequality, we directly obtain

$$\begin{array}{ccc}\hfill {\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(f\right)\left(x\right)& \le & \sum _{j=0}^{\infty }{\left({\int }_0^{\infty }{\left|s^{-\alpha }{\int }_{2^{-j-1}s<\left|w\right|\le 2^{-j}s}u(x-\psi (\left|w\right|)w^{\prime })\frac{\mho \left(w\right)h\left(\left|w\right|\right)}{{\left|w\right|}^{m-\alpha }}dw\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }\hfill \\ & =& \frac{2^{{\alpha }_1}}{2^{{\alpha }_1}-1}{\left({\int }_{{\mathbf{R}}_{+}}{\left|{\mu }_{\mho ,\psi ,h,s}*u\left(x\right)\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa }.\hfill \end{array}$$

(27)

Let $\nu =2^{q^{\prime }{s}^{\prime }}$. For $j\in \mathbf{Z}$, let ${\left\{{\mathcal{K}}_j\right\}}_{-\infty }^{\infty }$ be a smooth partition of unity in $(0,\infty )$ adapted to the interval ${\mathcal{I}}_{j,\nu }=[{\left(\psi \left(\nu \right)\right)}^{-j-1},{\left(\psi \left(\nu \right)\right)}^{-j+1}]$. In fact, we require the folowing:

$$\begin{array}{ccc}\hfill & & 0\le {\mathcal{K}}_j\le 1,\sum _j{\mathcal{K}}_j\left(s\right)=1,\hfill \\ \hfill \mathrm{supp}{\mathcal{K}}_j& \subseteq & {\mathcal{I}}_{j,\nu },and\left|\frac{d^k{\mathcal{K}}_j\left(s\right)}{d{s}^k}\right|\le \frac{C_k}{s^k}.\hfill \end{array}$$

Let $\widehat{{\Phi }_j}\left(\eta \right)={\mathcal{K}}_j\left(\left|\eta \right|\right)$. This gives that for $u\in \mathcal{S}\left({\mathbf{R}}^m\right)$,

$${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)\le \frac{2^{{\alpha }_1}}{2^{{\alpha }_1}-1}\sum _{j\in \mathbf{Z}}{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}\left(u\right),$$

(28)

where

$${\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}u\left(x\right)={\left(\underset{0}{\overset{\infty }{\int }}{\left|{\mathcal{F}}_{\mho ,\psi ,h,j,\nu }(x,s)\right|}^{\kappa }\frac{ds}{s}\right)}^{1/\kappa },$$

$${\mathcal{F}}_{\mho ,\psi ,h,j,\nu }(x,s)=\sum _{l\in \mathbf{Z}}({\Phi }_{l+j}*{\mu }_{\mho ,\psi ,h,s}*u)\left(x\right){\chi }_{{}_{[{\nu }^l,{\nu }^{l+1})}}\left(s\right).$$

By Equation (28), it is clear that the proof of Theorem 1 is finished once we prove that

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C{2}^{-ϵ\left|j\right|}{\left(q-1\right)}^{-1/\kappa }{\left(\beta -1\right)}^{-1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(29)

for all $\kappa \le p<\infty$; and

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\hfill \\ & \le & C{2}^{-ϵ\left|j\right|}{\left(q-1\right)}^{-1}{\left(\beta -1\right)}^{-1}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(30)

for all $1<p<\kappa$ and for some $ϵ\in (0,1)$.

Let us start with inequality Equation (29). Consider the case $p=\kappa =2$. So we have ${∥u∥}_{{\stackrel{.}{F}}_0^{2,2}\left({\mathbf{R}}^m\right)}={∥u∥}_{L_2\left({\mathbf{R}}^m\right)}$. Thanks to Plancherel’s theorem and Lemma 1, we have

$$\begin{array}{ccc}\hfill {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(2\right)}\left(u\right)∥}_{L_2\left({\mathbf{R}}^m\right)}^2& \le & \sum _{l\in \mathbf{Z}}\underset{{\mathcal{A}}_{l+j,\nu }}{\int }\left(\underset{{\nu }^l}{\overset{{\nu }^{l+1}}{\int }}{\left|{\widehat{\mu }}_{\mho ,\psi ,h,s}\left(\eta \right)\right|}^2\frac{ds}{s}\right){\left|\widehat{u}\left(\eta \right)\right|}^{2}d\eta \hfill \\ & \le & C(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^2{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^2\sum _{l\in \mathbf{Z}}\underset{{\mathcal{A}}_{l+j,\nu }}{\int }\left({\left|\left(\psi \left({\nu }^j\right)\eta \right)\right|}^{\pm \frac{2\epsilon }{q^{\prime }{\beta }^{\prime }}}\right){\left|\widehat{u}\left(\eta \right)\right|}^{2}d\eta \hfill \\ & \le & C(ln\nu ){∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^2{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^2{2}^{-ϵ\left|j\right|}\sum _{l\in \mathbf{Z}}{\int }_{{\mathcal{A}}_{l+j,\nu }}{\left|\widehat{u}\left(\eta \right)\right|}^{2}d\eta \hfill \\ & \le & C{\left(\beta -1\right)}^{-1}{\left(q-1\right)}^{-1}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}^2{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}^2{2}^{-ϵ\left|j\right|}{∥u∥}_{L_2\left({\mathbf{R}}^m\right)}^2,\hfill \end{array}$$

where ${\left|\left(\psi \left({\nu }^j\right)\eta \right)\right|}^{\pm \frac{2\epsilon }{q^{\prime }{\beta }^{\prime }}}=min\{{\left|\left(\psi \left({\nu }^j\right)\eta \right)\right|}^{+\frac{2\epsilon }{q^{\prime }{\beta }^{\prime }}},{\left|\left(\psi \left({\nu }^j\right)\eta \right)\right|}^{-\frac{2\epsilon }{q^{\prime }{\beta }^{\prime }}}\}$, ${\mathcal{A}}_{l,\nu }=\left\{\eta \in {\mathbf{R}}^m:\left|\eta \right|\in {\mathcal{I}}_{l,\nu }\right\}$ and $0<ϵ<1$. This gives that

$$\begin{array}{c}\hfill {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(2\right)}\left(u\right)∥}_{L_2\left({\mathbf{R}}^m\right)}\le C{2}^{-\frac{ϵ}{2}\left|j\right|}{\left(\beta -1\right)}^{-1/2}{\left(q-1\right)}^{-1/2}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{2,2}\left({\mathbf{R}}^m\right)}.\end{array}$$

(31)

However, by Lemma 3, we obtain that

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C{\left(q-1\right)}^{-1/\kappa }{\left(\beta -1\right)}^{-1/\kappa }{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}\hfill \end{array}$$

(32)

for $\kappa \le p<\infty$, and

$$\begin{array}{c}\hfill {∥{\mathcal{N}}_{\mho ,\psi ,h,j}^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C{\left(q-1\right)}^{-1}{\left(\beta -1\right)}^{-1}{∥\mho ∥}_{L_q\left({\mathbb{U}}^{m-1}\right)}{∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_p^{0,\kappa }\left({\mathbf{R}}^m\right)}\end{array}$$

(33)

for $1<p<\kappa$. Therefore, when we Interpolate Equation (31) with Equations (32) and (33), we immediately obtain Equations (29) and (30).

Proof of Theorem 2. It is easy to prove this lemma by employing the same above arguments with $\nu =2^{q^{\prime }}$ instead of $\nu =2^{q^{\prime }{\beta }^{\prime }}$, and invoke Lemma 4 instead of Lemma 3.

4. Further Applications

In this section we give some consequent results that follow by applying our results in Section 1. Precisely, by using the conclusions of Theorems 1 and 2 together with Yano’s extrapolation arguments, we obtain the following:

Theorem 3.

Let ψ be in the classesDorI .

$\left(i\right)$ If $h\in {\mathcal{M}}^{1/\kappa }\left({\mathbf{R}}_{+}\right)$ and $\mho \in B_q^{(0,\frac{1}{\kappa }-1)}\left({\mathbb{U}}^{m-1}\right)$ for some $q,\kappa >1$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{B_q^{(0,\frac{1}{\kappa }-1)}\left({\mathbb{U}}^{m-1}\right)}\right)\left(1+M^{1/\kappa }\left(h\right)\right){∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for all $\kappa \le p<\infty$;

$\left(ii\right)$ If $h\in {\mathcal{M}}^1\left({\mathbf{R}}_{+}\right)$ and $\mho \in B_q^{(0,0)}\left({\mathbb{U}}^{m-1}\right)$ for some $q>1$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{B_q^{(0,0)}\left({\mathbb{U}}^{m-1}\right)}\right)\left(1+M^1\left(h\right)\right){∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for all $1<p<\kappa$;

$\left(iii\right)$ If $h\in {\mathcal{M}}^{1/\kappa }\left({\mathbf{R}}_{+}\right)$ and $\mho \in L{(logL)}^{1/\kappa }\left({\mathbb{U}}^{m-1}\right)$ for some $q,\kappa >1$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{L{(logL)}^{1/\kappa }\left({\mathbb{U}}^{m-1}\right)}\right)\left(1+M^{1/\kappa }\left(h\right)\right){∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for $\kappa \le p<\infty$;

$\left(iv\right)$ If $h\in {\mathcal{M}}^1\left({\mathbf{R}}_{+}\right)$ and $\mho \in L(logL)\left({\mathbb{U}}^{m-1}\right)$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{L(logL)\left({\mathbb{U}}^{m-1}\right)}\right)\left(1+M^1\left(h\right)\right){∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for $1<p<\kappa$, where $C_p$ is a bounded positive constant independent of h, ℧ and ψ.

Theorem 4.

Let ℧ satisfy the conditions Equations (1) and (2), $h\in {\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)$ for some $\beta >2$ and ψ be inDorI .

$\left(i\right)$ If $\mho \in B_q^{(0,\frac{1}{\kappa }-1)}\left({\mathbb{U}}^{m-1}\right)$ for some $q>1$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{B_q^{(0,\frac{1}{\kappa }-1)}\left({\mathbb{U}}^{m-1}\right)}\right){∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for all $1<p<\kappa$ with $\kappa \le {\beta }^{\prime }$ and $2<\beta <\infty$; and also for all ${\beta }^{\prime }<p<\infty$ with $\kappa >{\beta }^{\prime }$ and $2<\beta \le \infty$.

$\left(ii\right)$ If $\mho \in L{(logL)}^{1/\kappa }\left({\mathbb{U}}^{m-1}\right)$, then

$${∥{\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}\left(u\right)∥}_{L_p\left({\mathbf{R}}^m\right)}\le C_p\left(1+{∥\mho ∥}_{L{(logL)}^{1/\kappa }\left({\mathbb{U}}^{m-1}\right)}\right){∥h∥}_{{\mathrm{\Theta }}_{\beta }\left({\mathbf{R}}_{+}\right)}{∥u∥}_{{\stackrel{.}{F}}_0^{\kappa ,p}\left({\mathbf{R}}^m\right)}$$

for all $1<p<\kappa$ with $\kappa \le {\beta }^{\prime }$ and $2<\beta <\infty$ and also for all ${\beta }^{\prime }<p<\infty$ with $\kappa >{\beta }^{\prime }$ and $2<\beta \le \infty$.

It is clear that our obtained results generalize what was established in [17] and also improve and extend [14], Theorems 1.3–1.4]. In particular, the results in [17] can be reached when we take $\psi \left(s\right)=s$ in our results. However, when we take $\kappa =2$, we obtain the results in [14].

It is worth mentioning that, by Theorem 3, we establish the boundedness of ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ whenever the condition on $\mho \in B_q^{(0,-1/2)}\left({\mathbb{U}}^{m-1}\right)\cup L{(logL)}^{1/2}\left({\mathbb{U}}^{m-1}\right)$ is optimal (see [3,4]). In addition, by taking $\kappa =\infty$ in Theorem 4, we obtain that ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ is bounded on $L_p\left({\mathbb{R}}^m\right)$ for the full range $p\in (1,\infty )$.

5. Conclusions

In this article, we obtained sharp $L_p$ bounds for the integral operator ${\mathfrak{M}}_{\mho ,\psi ,h,\alpha }^{\left(\kappa \right)}$ when the rough kernels belong to the space $L^q\left({\mathbb{U}}^{m-1}\right)$ for some $q>1$. These bounds allowed us to use Yano’s extrapolation argument to confirm the $L_p$ boundedness of the aforementioned operator under much weaker conditions on the kernels. Our results extend and improve all the above cited results.