Generalized Marcinkiewicz Integral Operators Along Certain Surfaces

1. Introduction

Throughout this paper, we assume that ${\mathbb{R}}^s$ is the $2\le s$-dimensional Euclidean space and ${\mathbb{B}}^{s-1}$ is the unit sphere in ${\mathbb{R}}^s$ equipped with the normalized Lebesgue surface measure $d{\sigma }_s(·)$. Also, we assume that $y=v/\left|v\right|$ for $v\in {\mathbb{R}}^s\setminus \left\{0\right\}$.

Let $\zeta$ be a measurable function on ${\mathbb{R}}_{+}$ and $\Phi \in L_1\left({\mathbb{B}}^{s-1}\right)$ be a homogeneous function of degree zero on ${\mathbb{R}}^s$ that satisfies the condition

$${\int }_{{\mathbb{B}}^{s-1}}\Phi \left(y\right)d{\sigma }_s\left(y\right)=0.$$

(1)

For suitable mappings $\mathcal{P}:{\mathbb{R}}^s\to {\mathbb{R}}^d$ and $\Psi :{\mathbb{R}}_{+}\to \mathbb{R}$, we define the generalized Marcinkiewicz integral ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$, initially for $\mathcal{V}\in C_0^{\infty }\left({\mathbb{R}}^{d+1}\right)$, by

$${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)\left(\dot{x}\right)={\left({\int }_{{\mathbb{R}}_{+}}{\left|{\displaystyle \frac{1}{t^a}}{\int }_{\left|v\right|\le t}\mathcal{V}(x-\mathcal{P}\left(v\right),x_{d+1}-\Psi \left(\right|v\left|\right)){\displaystyle \frac{\Phi \left(v\right)\zeta \left(\left|v\right|\right)}{{\left|v\right|}^{s-a}}}dv\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha },$$

where $\dot{x}=(x,x_{d+1})\in {\mathbb{R}}^{d+1}$, $\alpha >1$, and $a=a_1+i{a}_2$ with $a_1>0$ and $a_2\in \mathbb{R}$.

When $\alpha =2$, $s=d$, $\mathcal{P}\left(v\right)=I\left(v\right)=v$, $\Psi \equiv 0$, and $\zeta \equiv 1$, we denote ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ by ${\mathcal{M}}_{\Phi ,a}$; and when $a=1$, we denote ${\mathcal{M}}_{\Phi ,a}$ by ${\mathcal{M}}_{\Phi }$ which is the traditional Marcinkiewicz operator that Stein initiated in [1]. The boundedness of ${\mathcal{M}}_{\Phi }$ has been considered by many authors for a long time. Historically, Stein [1] proved that ${\mathcal{M}}_{\Phi }$ is bounded on $L_p\left({\mathbb{R}}^s\right)$ for $1<p<2$ provided that $\Phi \in Li{p}_{ϵ}\left({\mathbb{B}}^{s-1}\right)$ with $ϵ\in (0,1]$. Then, the authors of [2] improved this result; they confirmed the $L_p$ boundedness of ${\mathcal{M}}_{\Phi }$ for all $p\in (1,\infty )$ whenever $\Phi \in C^1\left({\mathbb{B}}^{s-1}\right)$. Later, Walsh [3] obtained only $L^2\left({\mathbb{R}}^s\right)$ boundedness of ${\mathcal{M}}_{\Phi }$ if $\Phi \in L{\left(logL\right)}^{1/2}\left({\mathbb{B}}^{s-1}\right)$, and also he found that the condition $\Phi \in L{(logL)}^{1/2}\left({\mathbb{B}}^{s-1}\right)$ is optimal in the sense that if $\Phi \in L{\left(logL\right)}^{\beta }\left({\mathbb{B}}^{s-1}\right)$ for any $\beta \in (0,1/2)$, then ${\mathcal{M}}_{\Phi }$ will forfeit the $L_2$ boundedness. Thereafter, the above results were extended and improved for all $p\in (1,\infty )$ whenever $\Phi \in L{(logL)}^{1/2}\left({\mathbb{B}}^{s-1}\right)$, see [4]; and also whenever $\Phi \in B_q^{(0,-1/2)}\left({\mathbb{B}}^{s-1}\right)$ for some $q>1$, see [5]. Here $B_q^{(0,\epsilon )}\left({\mathbb{B}}^{s-1}\right)$ indicates to the special class of block space that was introduced in [6].

It is worth mentioning that the investigation of $L_p$ boundedness of parametric Marcinkiewicz operator ${\mathcal{M}}_{\Phi ,a}$ was begun by Hörmander [7] and then continued by many researchers. Let us present some past results related to our findings. For instance, the operator ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^2$ was studied in [8] for the special case $\Psi \equiv 0$, $\Phi \in L{(logL)}^{1/2}\left({\mathbb{B}}^{s-1}\right)\cup B_q^{(0,-1/2)}\left({\mathbb{B}}^{s-1}\right)$, $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta >1$, and $\mathcal{P}\left(v\right)=\left(P_1\left(v\right),P_2\left(v\right),\dots ,P_d\left(v\right)\right)$ is a polynomial mapping, where each $P_j$ is a real valued polynomial on ${\mathbb{R}}^s$. In fact, it was proved the $L_p$ boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^2$ for all $\left|1/p-1/2\right|<min\{1/{\eta }^{\prime },1/2\}$. Here, ${\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ (with $\eta >1$) denotes the collection of all mappings $\zeta$ satisfying

$${∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}=\underset{m\in \mathbb{Z}}{sup}{\left({\int }_{2^m}^{2^{m+1}}{\left|\zeta \left(t\right)\right|}^{\eta }{\displaystyle \frac{dt}{t}}\right)}^{1/\eta }<\infty .$$

The authors of [9] got the same results in [8] with replacing the conditions $\mathcal{P}$ and $\Psi$ by $\mathcal{P}=I$ and $\Psi \in C^2\left({\mathbb{R}}_{+}\right)$ is convex and increasing function with $\Psi \left(0\right)=0$; respectively. Very recently, the authors of [10] generalized the results in [8,9] by obtaining $L_p$ boundedness of the operator ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^2$ for all $\left|1/p-1/2\right|<min\{1/{\eta }^{\prime },1/2\}$ provided that $\mathcal{P}\left(v\right)=\left(P_1\left(v\right),P_2\left(v\right),\dots ,P_d\left(v\right)\right)$ is a polynomial mapping, $\Phi \in L{(logL)}^{1/2}\left({\mathbb{B}}^{s-1}\right)\cup B_q^{(0,-1/2)}\left({\mathbb{B}}^{s-1}\right)$, $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta >1$, and $\Psi$ is a function satisfying

$$\Psi \left(t\right)={\psi }_1\left(t\right)+{\psi }_2\left(t\right),$$

where ${\psi }_1$ is a polynomial, ${\psi }_2^{\left(n\right)}\left(0\right)=0$ for all $1\le n\le K$, ${\psi }_2^{\left(n\right)}$ is positive non-decreasing on ${\mathbb{R}}_{+}$ for all $1\le n\le K+1$, and $K=max\{deg\left({\psi }_1\right),deg\left(\mathcal{P}\right)\}$. Afterwards, the $L_p$ boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^2$ was investigated by many authors under various conditions on $\Phi$, $\mathcal{P}$, $\Psi$, and $\zeta$. For a sample of past discussions relevant to our current study, we refer the readers to consult [11,12,13,14,15], and for developments and recent advances, see [16,17,18,19,20].

On the other side, the discussion of generalized Marcinkiewicz operator ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ was started in [21] where the authors established the $L_p$($1<p<\infty$) boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},0,1}^{\alpha }$ whenever $\Phi \in L^q\left({\mathbb{B}}^{s-1}\right)$, $\mathcal{P}\left(v\right)\equiv I\left(v\right)=v$ and $\alpha \in (1,\infty )$. Later on, Le [22] improved this result by proving the boundedness of ${\mathbf{G}}_{\Phi ,I,0,\zeta }^{\alpha }$ under weaker conditions that $\zeta \in {\Theta }_{max\{{\eta }^{\prime },2\}}\left({\mathbb{R}}_{+}\right)$ and $\Phi \in L(logL)\left({\mathbb{B}}^{s-1}\right)$. The authors of [23] improved and extended the above results. Precisely, they employed Yano’s extrapolation argument to obtain the boundedness of ${\mathbf{G}}_{\Phi ,I,0,\zeta }^{\alpha }$ for all $p\in (1,\alpha )$ with ${\alpha }^{\prime }\ge \eta$ and also for all $p\in ({\eta }^{\prime },\infty )$ with $\alpha >{\eta }^{\prime }$ if $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ with $\eta >2$ and $\Phi \in L{(logL)}^{1/\alpha }\left({\mathbb{B}}^{s-1}\right)\cup B_q^{(0,{\displaystyle \frac{1}{\alpha }}-1)}\left({\mathbb{B}}^{s-1}\right)$. Thereafter, the operator ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ was studied under specific conditions on $\alpha ,\Phi ,\mathcal{P},\Psi$, and $\zeta$ (see [24,25,26,27,28], and the references therein.)

For $u\in (-\infty ,\infty )$ and $\alpha ,p\in (1,\infty )$, the space of homogeneous Triebel-Lizorkin functions ${\stackrel{.}{F}}_p^{u,\alpha }\left({\mathbb{R}}^s\right)$ is the collection of all tempered distribution mappings $\mathcal{V}$ on ${\mathbb{R}}^s$ such that

$${∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{u,\alpha }\left({\mathbb{R}}^s\right)}={∥{\left(\sum _{j\in \mathbb{Z}}{2}^{ju\alpha }{\left|{\mathsf{ϝ}}_j\ast \mathcal{V}\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^s\right)}<\infty ,$$

where $\widehat{{\mathsf{ϝ}}_j}\left(v\right)=\lambda (v/2^j)$ and $\lambda \in C_0^{\infty }\left({\mathbb{R}}^s\right)$ is a radial mapping satisfying:

(1) $0\le \lambda \le 1$,

(b) There exists $C>0$ such that, for all v with ${\displaystyle \frac{3}{5}}\le \left|v\right|\le {\displaystyle \frac{5}{3}}]$, we have $\lambda \left(v\right)\ge C$,

(c) $\mathbf{S}\supset supp\left(\lambda \right)$, where $\mathbf{S}=\left\{v:\left|v\right|\in [1/2,2]\right\}$,

(d) $\sum _{j\in \mathbb{Z}}\lambda (v/2^j)=1$ for all non-zero $v\in {\mathbb{R}}^s$.

We point out that the authors of [25] proved the following:

(i) The Schwartz space $\mathcal{S}\left({\mathbb{R}}^s\right)$ is dense in ${\stackrel{.}{F}}_p^{u,\alpha }\left({\mathbb{R}}^s\right)$,

(ii) When $\alpha =2$ and $u=0$, we have ${\stackrel{.}{F}}_p^{0,2}\left({\mathbb{R}}^s\right)=L_p\left({\mathbb{R}}^s\right)$ for all $p\in (1,\infty )$,

(iii) If ${\alpha }_2\ge {\alpha }_1$, we have ${\stackrel{.}{F}}_p^{u,{\alpha }_2}\left({\mathbb{R}}^s\right)\supseteq {\stackrel{.}{F}}_p^{u,{\alpha }_1}\left({\mathbb{R}}^s\right)$.

Motivated by the results obtained in [10] concerning the boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^2$ on $L_p\left({\mathbb{R}}^{d+1}\right)$, it is natural discuss the $L_p$ boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ under the same conditions in [10] with replacing $\alpha =2$ by $\alpha >1$. The prime findings of this paper are described as follows:

Theorem 1.

Let $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ with $\eta \in (1,2]$ and $\Phi \in L_q\left({\mathbb{B}}^{s-1}\right)$ be a homogeneous function of degree zero satisfying (1) with $q\in (1,2]$. Suppose that $\mathcal{P}\left(v\right)=\left(P_1\left(v\right),P_2\left(v\right),\dots ,P_d\left(v\right)\right)$ is a polynomial mapping and Ψ is a function satisfying

$$\Psi \left(t\right)={\psi }_1\left(t\right)+{\psi }_2\left(t\right),$$

where ${\psi }_1$ is a polynomial, ${\psi }_2^{\left(n\right)}\left(0\right)=0$ for all $1\le n\le K$, ${\psi }_2^{\left(n\right)}$ is positive non-decreasing on ${\mathbb{R}}_{+}$ for all $1\le n\le K+1$, and $K=max\{deg\left({\psi }_1\right),deg\left(\mathcal{P}\right)\}$. Then, there exists $C_{p,\Phi ,\zeta }$ such that

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{-1/\alpha }{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\mathrm{for}p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}];$$

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }(\mathcal{V}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha -\eta }{\eta \alpha }}}{∥\mathcal{V})∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\mathrm{for}p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha );$$

and

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\mathrm{for}p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha ),$$

where $C_{p,\Phi ,\zeta }=C_p{C}_{\Phi ,\zeta }$, $C_{\Phi ,\zeta }=C{∥\Phi ∥}_{L_q\left({\mathbb{B}}^{s-1}\right)}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}$, and $C_p$ is a positive number independent of $\Phi ,\mathcal{P},\Psi ,\zeta$, and α.

Theorem 2.

Assume that $\Phi ,\mathcal{P}$, and Ψ are given as in the Theorem 1 and that $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ with $\eta \in (2,\infty )$.

(a) 

If $\alpha >{\eta }^{\prime }$, then for $p\in ({\eta }^{\prime },\infty )$, we have

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[q-1\right]}^{-1/{\eta }^{\prime }}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)};$$

(b) 

If $\alpha \le {\eta }^{\prime }$, then for $p\in (1,\alpha )$, we have

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[q-1\right]}^{-1}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}.$$

Theorems 1-2 along with Yano’s extrapolation technique ([29,30]) lead to the following results.

Theorem 3.

Let $\mathcal{P}$, Ψ, and ζ be given as in the Theorem 1.

(i)

If $\Phi \in L{(logL)}^{1/\alpha }\left({\mathbb{B}}^{s-1}\right)$, then for $p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{L{(logL)}^{1/\alpha }\left({\mathbb{B}}^{s-1}\right)}\right).$$

(ii)

If $\Phi \in L{(logL)}^{{\displaystyle \frac{\eta \alpha -\alpha +\eta }{\eta \alpha }}}\left({\mathbb{B}}^{s-1}\right)$, then for $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }(\mathcal{V}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{L{(logL)}^{{\displaystyle \frac{\eta \alpha -\alpha +\eta }{\eta \alpha }}}\left({\mathbb{B}}^{s-1}\right)}\right).$$

(iii)

If $\Phi \in L{(logL)}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}\left({\mathbb{B}}^{s-1}\right)$, then for $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{L{(logL)}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}\left({\mathbb{B}}^{s-1}\right)}\right).$$

(iv)

If $\Phi \in B_q^{(0,-1/{\alpha }^{\prime })}\left({\mathbb{B}}^{s-1}\right)$ with $q>1$, then for $p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{q^{(0,-1/{\alpha }^{\prime })}\left({\mathbb{B}}^{s-1}\right)}\right).$$

(v)

If $\Phi \in B_q^{(0,{\displaystyle \frac{\eta -\alpha }{\eta \alpha }})}\left({\mathbb{B}}^{s-1}\right)$ with $q>1$, then for $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{B_q^{(0,{\displaystyle \frac{\eta -\alpha }{\eta \alpha }})}\left({\mathbb{B}}^{s-1}\right)}\right).$$

(vi)

If $\Phi \in B_q^{(0,{\displaystyle \frac{1+\alpha -2\eta \alpha }{\eta \alpha }})}\left({\mathbb{B}}^{s-1}\right)$ with $q>1$, then for $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$,

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{B_q^{(0,{\displaystyle \frac{1+\alpha -2\eta \alpha }{\eta \alpha }})}\left({\mathbb{B}}^{s-1}\right)}\right).$$

Theorem 4.

Let $\mathcal{P}$, Ψ, and ζ be given as in the Theorem 2.

(i)

If $\Phi \in L{(logL)}^{1/{\eta }^{\prime }}\left({\mathbb{B}}^{s-1}\right)$ and $\alpha >{\eta }^{\prime }$, then the estimate

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{L{(logL)}^{1/{\eta }^{\prime }}\left({\mathbb{B}}^{s-1}\right)}\right)$$

holds for $p\in ({\eta }^{\prime },\infty )$.

(ii)

If $\Phi \in L(logL)\left({\mathbb{B}}^{s-1}\right)$ and $\alpha \le {\eta }^{\prime }$, then the inequality

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{L(logL)\left({\mathbb{B}}^{s-1}\right)}\right)$$

holds for $p\in (1,\alpha )$.

(iii)

If $\Phi \in B_q^{(0,-1/{\eta }^{\prime })}\left({\mathbb{B}}^{s-1}\right)$ and ${\eta }^{\prime }<\alpha$, then we have

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{q^{(0,-1/{\eta }^{\prime })}\left({\mathbb{B}}^{s-1}\right)}\right)$$

for $p\in ({\eta }^{\prime },\infty )$.

(iv)

If $\Phi \in B_q^{(0,0)}\left({\mathbb{B}}^{s-1}\right)$ and $\alpha \le {\eta }^{\prime }$, then we have

$${∥{\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_p{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\left(1+{∥\Phi ∥}_{q^{(0,0)}\left({\mathbb{B}}^{s-1}\right)}\right)$$

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

Remarks

(i)

For the case $\alpha =2$, $s=d$, $\mathcal{P}\equiv I$, $\Psi \equiv 0$, $\zeta \equiv 1$, and $a=1$, Theorem 4 gives the $L^p$ boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ for all $p\in (1,\infty )$ under the condition $\Phi \in L(logL)\left({\mathbb{B}}^{s-1}\right)\cup B_q^{(0,-1)}\left({\mathbb{B}}^{s-1}\right)\supset Li{p}_{ϵ}\left({\mathbb{B}}^{s-1}\right)\supset C^1\left({\mathbb{B}}^{s-1}\right)$. Therefore, our results improve and extend the findings in [1,2].

(ii)

When $\alpha =2$, $s=d$, $\mathcal{P}\equiv I$, $\Psi \equiv 0$, $a=1$, $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$, and $\Phi \in L(logL)\left({\mathbb{B}}^{s-1}\right)$, the authors of [12] obtained only $L^2$ boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$. Thus, our findings generalize and improve the findings in [12].

(iii)

The operator ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ was discussed in [21] whenever $\mathcal{P}\equiv I$, $\Psi \equiv 0$, and $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ under the rough assumptions $\Phi \in L^q\left({\mathbb{B}}^{s-1}\right)$ which is stronger than the conditions on $\Phi$ in this paper.

(iv)

This work was studied in [10] only for the case $\alpha =2$. Moreover, the conditions $\Phi \in L{(logL)}^{1/2}\left({\mathbb{B}}^{s-1}\right)$ and $\Phi \in B_q^{(0,-1/2)}\left({\mathbb{B}}^{s-1}\right)$ in Theorem 3 (i) and (iv) are the weakest conditions among their respective spaces, see [3,5].

(v)

When $\zeta \equiv 1$, we get the full range for p which is $p\in (1,\infty )$.

(vi)

In Theorem 3, the space $L{(logL)}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}$ is the best space among the spaces $\left(i\right)$-$\left(iii\right)$, and the space $B_q^{(0,{\displaystyle \frac{\alpha -2\alpha \eta +1}{\eta \alpha }})}\left({\mathbb{B}}^{s-1}\right)$ is the best space among the spaces in $\left(iv\right)$-$\left(vi\right)$. However, the range of $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$ is better than the range of $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$.

(vii)

As $L{(logL)}^{1/{\eta }^{\prime }}\left({\mathbb{B}}^{s-1}\right)\supset L(logL)\left({\mathbb{B}}^{s-1}\right)$ and $B_q^{(0,-1/{\eta }^{\prime })}\left({\mathbb{B}}^{s-1}\right)\supset B_q^{(0,0)}\left({\mathbb{B}}^{s-1}\right)$, then in Theorem 4, the classes in $\left(i\right)$ and $\left(iii\right)$ are larger than the classes in $\left(ii\right)$ and $\left(iv\right)$.

(viii)

A model example about the surfaces considered in this work is ${\mathbf{S}}_{\mathcal{P},\Psi }\left(v\right)=(v_1^2,v_2^2,\dots ,v_d^2,{\left|v\right|}^4)$.

2. Principle Lemmas

In this part, we stablish some subsidiary lemmas that are needed to prove our main results. Let $\tau \ge 2$. For compatible mappings $\Phi :{\mathbb{B}}^s\to \mathbb{R}$, $\mathcal{P}:{\mathbb{R}}^s\to {\mathbb{R}}^d$, $\Psi :{\mathbb{R}}_{+}\to \mathbb{R}$, and $\zeta :{\mathbb{R}}_{+}\to \mathbb{C}$, we consider the set of measures $\{{\mho }_{\Phi ,\mathcal{P},\Psi ,\zeta ,t}:={\mho }_{\zeta ,t}:t\in {\mathbb{R}}_{+}\}$ and its maximal operators $M_{\tau }$ and ${\mho }_{\zeta }^{\ast }$ on ${\mathbb{R}}^{d+1}$ by

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{R}}^{d+1}}\mathcal{V}d{\mho }_{\zeta ,t}& =& {\displaystyle \frac{1}{t^a}}{\int }_{t/2\le \left|v\right|\le t}\mathcal{V}(\mathcal{P}\left(v\right),\Psi (\left|v\right|\left)\right){\displaystyle \frac{\Phi \left(v^{\prime }\right)\zeta \left(\left|v\right|\right)}{{\left|v\right|}^{s-a}}}dv,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_{\tau }\left(\mathcal{V}\right)& =& \underset{j\in \mathbb{Z}}{sup}{\int }_{{\tau }^j}^{{\tau }^{j+1}}\left|\right|{\mho }_{\zeta ,t}|\ast \mathcal{V}|{\displaystyle \frac{dt}{t}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mho }_{\zeta }^{\ast }\left(\mathcal{V}\right)& =& \underset{t\in {\mathbb{R}}_{+}}{sup}\left|\right|{\mho }_{\zeta ,t}|\ast \mathcal{V}|.\hfill \end{array}$$

Here, we define $|{\mho }_{\zeta ,t}|$ similarly as the definition of ${\mho }_{\zeta ,t}$ with substituting $\Phi \zeta$ by $|\Phi \zeta |$.

Let us present the following lemma which is due to the authors of [10].

Lemma 1.

Let $\Phi ,\mathcal{P}$, and Ψ be given as in Theorems 1 and 2. Assume that $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta >1$. Then, the following estimates

$$\parallel M_{\tau }{\left(\mathcal{V}\right)\parallel }_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }(ln\tau ){\parallel \mathcal{V}\parallel }_{L_p\left({\mathbb{R}}^{d+1}\right)}$$

(2)

and

$$\parallel {\mho }_{\zeta }^{\ast }{\left(\mathcal{V}\right)\parallel }_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{\parallel \mathcal{V}\parallel }_{L_p\left({\mathbb{R}}^{d+1}\right)}$$

(3)

hold for all $p>{\eta }^{\prime }$.

To prove our main results, we need to establish the following lemmas.

Lemma 2.

Let $\Phi ,\mathcal{P}$, and Ψ be given as in Theorems 1 and 2. Assume that $\tau \ge 2$, $\alpha >1$, and $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta \in (2,\infty )$. Then for arbitrary set of functions $\{{\mathcal{H}}_j,j\in \mathbb{Z}\}$ on ${\mathbb{R}}^{d+1}$, we have the following:

(a) 

If $\alpha >{\eta }^{\prime }$, then for all $p\in ({\eta }^{\prime },\infty )$,

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(b) 

If $\alpha \le {\eta }^{\prime }$, then for all $p\in (1,\alpha )$,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }(ln\tau ){∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

Proof. 

By (3), it is easy to see that

$$\begin{array}{ccc}\hfill {∥\underset{j\in \mathbb{Z}}{sup}\underset{t\in [{\tau }^j,{\tau }^{j+1}]}{sup}\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}& \le & {∥{\mho }_{\zeta }^{\ast }\left(\underset{j\in \mathbb{Z}}{sup}\left|{\mathcal{H}}_j\right|\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & \le & C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{∥\underset{j\in \mathbb{Z}}{sup}\left|{\mathcal{H}}_j\right|∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\hfill \end{array}$$

for all $p\in ({\eta }^{\prime },\infty )$. This leads to

$$\begin{array}{ccc}\hfill {∥{∥\parallel {\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j{\parallel }_{L^{\infty }([1,\tau ],{\displaystyle \frac{dt}{t}})}∥}_{L_{\infty }\left(\mathbb{Z}\right)}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}& \le & C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{∥{∥{\mathcal{H}}_j∥}_{L_{\infty }\left(\mathbb{Z}\right)}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(4)

As $p>{\eta }^{\prime }$, then by duality, there is $f\in L_{{(p/{\eta }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)$ such that ${∥f∥}_{L_{{(p/{\eta }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\le 1$ and

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{1}{\overset{\tau }{\int }}}{\left|{\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j\right|}^{{\eta }^{\prime }}{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{{\eta }^{\prime }}}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}^{{\eta }^{\prime }}={\int }_{{\mathbb{R}}^{d+1}}\sum _{j\in \mathbb{Z}}{\int }_1^{\tau }{\left|{\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j\left(\dot{x}\right)\right|}^{{\eta }^{\prime }}{\displaystyle \frac{dt}{t}}f\left(\dot{x}\right)d\dot{x}\hfill \\ & & \le C{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\eta }^{\prime }/\eta )}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{{\eta }^{\prime }}{\int }_{{\mathbb{R}}^{d+1}}\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\left(\dot{x}\right)\right|}^{{\eta }^{\prime }}{\mho }_{\zeta }^{\ast }\tilde{f}(-\dot{x})d\dot{x}\hfill \\ & & \le C{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\eta }^{\prime }/\eta )}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{{\eta }^{\prime }}{∥\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{{\eta }^{\prime }}∥}_{L_{\left({\displaystyle \frac{p}{{\eta }^{\prime }}}\right)}\left({\mathbb{R}}^{d+1}\right)}{∥{\mho }_{\zeta }^{\ast }\left(\tilde{f}\right)∥}_{L_{{\left({\displaystyle \frac{p}{{\eta }^{\prime }}}\right)}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & & \le C_p(ln\tau ){∥\Phi ∥}_{L_q\left({\mathbb{B}}^{s-1}\right)}^{({\eta }^{\prime }/\eta )+1}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}^{+}\right)}^{{\eta }^{\prime }}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{{\eta }^{\prime }}\right)}^{{\displaystyle \frac{1}{{\eta }^{\prime }}}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}^{{\eta }^{\prime }},\hfill \end{array}$$

(5)

where $\tilde{f}\left(\dot{x}\right)=f(-\dot{x})$. Therefore,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{1}{\overset{\tau }{\int }}}{\left|{\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j\right|}^{{\eta }^{\prime }}{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{{\eta }^{\prime }}}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{{\eta }^{\prime }}\right)}^{{\displaystyle \frac{1}{{\eta }^{\prime }}}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(6)

Define the linear operator $\mathcal{T}$ on ${\mathcal{H}}_j$ by $\mathcal{T}\left({\mathcal{H}}_j\right)={\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j$, and then interpolate (4) with (6), so we conclude for $\alpha >{\eta }^{\prime }$,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le {∥{\left(\sum _{j\in \mathbb{Z}}{\int }_1^{\tau }{\left|{\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & & \le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/{\eta }^{\prime }}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{{\alpha }^{\prime }}\right)}^{{\displaystyle \frac{1}{{\alpha }^{\prime }}}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\hfill \end{array}$$

(7)

for all $p\in ({\eta }^{\prime },\infty )$, which proves part (a).

Let us now prove part (b). By invoking the duality, there is a mapping $g_j(\dot{x},t)$ on ${\mathbb{R}}^{d+1}\times {\mathbb{R}}_{+}$ such that ${∥{∥\parallel g_j{\parallel }_{L_{{\alpha }^{\prime }}([{\tau }^j,{\tau }^{j+1}],{\displaystyle \frac{dt}{t}})}∥}_{l_{{\alpha }^{\prime }}}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\le 1$ and

$$\begin{array}{ccc}\hfill {∥{\left(\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^j}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{s+1}\right)}=& & {\int }_{{\mathbb{R}}^{d+1}}\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}\left({\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right)g_j(\dot{x},t){\displaystyle \frac{dt}{t}}d\dot{x}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & C_p{(ln\tau )}^{1/\alpha }{∥{\left(\mathcal{F}\left(g_j\right)\right)}^{1/{\alpha }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)},\hfill \end{array}$$

(8)

where

$$\mathcal{F}\left(g_j\right)\left(\dot{x}\right)=\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^j}{\left|{\mho }_{\zeta ,t}\ast g_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}{\displaystyle \frac{dt}{t}}.$$

Thanks to Hölder’s inequality, we deduce that

$$\begin{array}{ccc}\hfill {\left|{\mho }_{\zeta ,t}\ast g_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}& \le & C{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\alpha }^{\prime }/\alpha )}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{{\alpha }^{\prime }}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\int }_{{\mathbb{B}}^{s-1}}\left|\Phi \left(y\right)\right|\hfill \\ & \times & {\left|g_j(x-\mathcal{P}\left(ry\right),x_{d+1}-\Psi \left(r\right),t)\right|}^{{\alpha }^{\prime }}d{\sigma }_s\left(y\right){\displaystyle \frac{dr}{r}}.\hfill \end{array}$$

(9)

By using duality, we have a function $h\in L_{{(p^{\prime }/{\alpha }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)$ such that

$$\begin{array}{ccc}& & {∥{\left(\mathcal{F}\left(g_j\right)\right)}^{1/{\alpha }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}^{{\alpha }^{\prime }}=\sum _{j\in \mathbb{Z}}{\int }_{{\mathbb{R}}^{d+1}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\left|{\mho }_{\zeta ,t}\ast g_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}{\displaystyle \frac{dt}{t}}h\left(\dot{x}\right)d\dot{x}.\hfill \end{array}$$

Therefore, by Hölder’s inequality together with (3) and (9), we obtain

$$\begin{array}{ccc}\hfill {∥{\left(\mathcal{F}\left(g_j\right)\right)}^{1/{\alpha }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}^{{\alpha }^{\prime }}& \le & C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{{\alpha }^{\prime }}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\alpha }^{\prime }/\alpha )}{∥{{\mho }^{\ast }}_{\left|\zeta \right|}\left(h\right)∥}_{L_{{\left({\displaystyle \frac{p^{\prime }}{{\alpha }^{\prime }}}\right)}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & \times & {∥\left(\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\left|g_j(\dot{v},t)\right|}^{{\alpha }^{\prime }}{\displaystyle \frac{dt}{t}}\right)∥}_{L_{{\left({\displaystyle \frac{p^{\prime }}{{\alpha }^{\prime }}}\right)}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & \le & C_p(ln\tau ){∥\Phi ∥}_{L_q\left({\mathbb{B}}^{s-1}\right)}^{({\alpha }^{\prime }/\alpha )+1}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{{\alpha }^{\prime }}{∥h∥}_{L_{{\left({\displaystyle \frac{p^{\prime }}{{\alpha }^{\prime }}}\right)}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(10)

Consequently, by the estimates (8) and (10), we complete the proof of part (b). □

Lemma 3.

Let τ, α, $\Phi ,\mathcal{P}$, Ψ, and ${\mathcal{H}}_j$ be given as in Lemma 2. Assume that $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta \in (1,2]$. Then, we have the following:

(i) 

For all $p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/\alpha }{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(ii) 

For all $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{{\displaystyle \frac{\eta \alpha -\alpha +\eta }{\eta \alpha }}}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}_{d+1}\right)}.\hfill \end{array}$$

(iii) 

For all $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$,

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{{\displaystyle \frac{\eta \alpha -\alpha -1}{\eta \alpha }}}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

Proof. 

To prove part $\left(\mathbf{i}\right)$, we need to consider two cases:

Case 1. $p\in (\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$. By duality, there is a function $\vartheta$ which is non-negative and belonging to $L_{{(p/\alpha )}^{\prime }}\left({\mathbb{R}}^{d+1}\right)$ with norm less than or equal one such that

$$\begin{array}{ccc}\hfill & & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}^{\alpha }={\int }_{{\mathbb{R}}^{d+1}}\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\left(\dot{x}\right)\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\vartheta \left(\dot{x}\right)d\dot{x}.\hfill \end{array}$$

(11)

It is easy to get, by using Hölder’s inequality, that

$$\begin{array}{ccc}\hfill {\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\left(\dot{v}\right)\right|}^{\alpha }& \le & C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })}{\int }_{t/2}^t{\int }_{{\mathbb{B}}^{s-1}}{\left|{\mathcal{H}}_j(x-\mathcal{P}\left(ry\right),x_{d+1}-\Psi \left(r\right))\right|}^{\alpha }\hfill \\ & \times & \left|\Phi \left(y\right)\right|{\left|\zeta \left(r\right)\right|}^{\alpha -{\displaystyle \frac{\alpha \eta }{{\alpha }^{\prime }}}}d{\sigma }_s\left(y\right){\displaystyle \frac{dr}{r}}.\hfill \end{array}$$

(12)

Hence, Lemma 1 and Hölder’s inequality along with (11)-(12) give

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}^{\alpha }\hfill \\ & \le & C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })}{\int }_{{\mathbb{R}}^{d+1}}\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\left(\dot{v}\right)\right|}^{\alpha }\right)M_{{\left|\zeta \right|}^{\alpha -{\displaystyle \frac{\alpha \eta }{{\alpha }^{\prime }}}}}\tilde{\vartheta }(-\dot{x})d\dot{x}\hfill \\ & \le & C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })}{∥\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }∥}_{L_{(p/\alpha )}\left({\mathbb{R}}^{d+1}\right)}{∥M_{{\left|\zeta \right|}^{{\displaystyle \frac{\alpha ({\alpha }^{\prime }-\eta )}{{\alpha }^{\prime }}}}}\left(\tilde{\vartheta }\right)∥}_{L_{{(p/\alpha )}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & \le & C(ln\tau )C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })+1}{∥\Phi ∥}_{L_q\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })+1}{∥\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }∥}_{L_{(p/\alpha )}\left({\mathbb{R}}^{d+1}\right)}{∥\tilde{\vartheta }∥}_{L_{{(p/\alpha )}^{\prime }}\left({\mathbb{R}}^{d+1}\right)},\hfill \end{array}$$

where $\tilde{\vartheta }\left(\dot{x}\right)=\vartheta (-\dot{x})$. Consequently, by taking the ${\alpha }^{th}$ root, we finish the prove of (i) for the case $p\in (\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$.

Case 2. $p=\alpha$. By Hölder’s inequality, (12), and then (2), we get

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}^{\alpha }\le C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })}\hfill \\ & \times & \sum _{j\in \mathbb{Z}}{\int }_{{\mathbb{R}}^{d+1}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\int }_{t/2}^t{\int }_{{\mathbb{B}}^{s-1}}{\left|{\mathcal{H}}_j(x-\mathcal{P}\left(ry\right),x_{d+1}-\Psi \left(r\right))\right|}^{\alpha }\left|\Phi \left(y\right)\right|{\left|\zeta \left(r\right)\right|}^{\alpha -{\displaystyle \frac{\alpha \eta }{{\alpha }^{\prime }}}}d{\sigma }_s\left(y\right){\displaystyle \frac{dr}{r}}{\displaystyle \frac{dt}{t}}d\dot{x}\hfill \\ & \le & C(ln\tau ){∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })+1}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })+1}{\int }_{{\mathbb{R}}^{d+1}}{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\left(\dot{x}\right)\right|}^{\alpha }d\dot{x}\right)}^{p/\alpha }\hfill \\ & \le & C(ln\tau ){∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{(\alpha /{\alpha }^{\prime })+1}{∥\Phi ∥}_{L_q\left({\mathbb{B}}^{s-1}\right)}^{(\alpha /{\alpha }^{\prime })+1}{∥\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }∥}_{L_1\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{1/\alpha }{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(13)

To prove part $\left(\mathbf{ii}\right)$, we use duality. So we have a set of mappings $\left\{{\rho }_j(\dot{v},t)\right\}$ on ${\mathbb{R}}^{d+1}\times {\mathbb{R}}_{+}$ such that ${∥{∥\parallel {\rho }_j{\parallel }_{L_{{\alpha }^{\prime }}([{\tau }^j,{\tau }^{j+1}],{\displaystyle \frac{dt}{t}})}∥}_{l_{{\alpha }^{\prime }}}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\le 1$ and

$$\begin{array}{c}\hfill {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^j}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}={\int }_{{\mathbb{R}}^{d+1}}\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}\left({\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right){\rho }_j(\dot{x},t){\displaystyle \frac{dt}{t}}d\dot{x}.\end{array}$$

(14)

Let

$$\mathcal{A}\left({\rho }_j\right)\left(\dot{x}\right)=\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^j}{\left|{\mho }_{\zeta ,t}\ast {\rho }_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}{\displaystyle \frac{dt}{t}}.$$

Then again by invoking the duality, there is function $Y\in L_{{(p^{\prime }/{\alpha }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)$ with norm 1 and

$$\begin{array}{ccc}\hfill & & {∥{\left(\mathcal{A}\left({\rho }_j\right)\right)}^{1/{\alpha }^{\prime }}∥}_{L_{p^{\prime }}\left({\mathbb{R}}^{d+1}\right)}^{{\alpha }^{\prime }}=\sum _{j\in \mathbb{Z}}{\int }_{{\mathbb{R}}^{d+1}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\left|{\mho }_{\zeta ,t}\ast {\rho }_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}Y\left(\dot{x}\right){\displaystyle \frac{dt}{t}}d\dot{x}\hfill \\ & \le & C{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{({\alpha }^{\prime }/\alpha )}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\alpha }^{\prime }/\alpha )}{∥{{\mho }^{\ast }}_{{\left|\zeta \right|}^{{\displaystyle \frac{{\alpha }^{\prime }(\alpha -\eta )}{\alpha }}}}\left(Y\right)∥}_{L_{{(p^{\prime }/{\alpha }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}{∥\left(\sum _{j\in \mathbb{Z}}{\int }_{{\tau }^j}^{{\tau }^{j+1}}{\left|{\rho }_j(\dot{x},t)\right|}^{{\alpha }^{\prime }}{\displaystyle \frac{dt}{t}}\right)∥}_{L_{(p^{\prime }/{\alpha }^{\prime })}\left({\mathbb{R}}^{d+1}\right)}\hfill \\ & \le & C_p{(ln\tau )}^{1/{\left({\displaystyle \frac{\alpha \eta }{\alpha -\eta }}\right)}^{\prime }}{∥\zeta ∥}_{{\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)}^{({\alpha }^{\prime }/\alpha )+1}{∥\Phi ∥}_{L_1\left({\mathbb{B}}^{s-1}\right)}^{({\alpha }^{\prime }/\alpha )+1}{∥Y∥}_{L_{{(p^{\prime }/{\alpha }^{\prime })}^{\prime }}\left({\mathbb{R}}^{d+1}\right)}\hfill \end{array}$$

(15)

for all $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$. Therefore, by Hölder’s inequality together with (14)-(15), we reach that

$$\begin{array}{ccc}& & {∥{\left(\sum _{j\in \mathbb{Z}}{\displaystyle \underset{{\tau }^j}{\overset{{\tau }^{j+1}}{\int }}}{\left|{\mho }_{\zeta ,t}\ast {\mathcal{H}}_j\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{\alpha }}}∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{(ln\tau )}^{{\displaystyle \frac{\eta \alpha -\alpha +\eta }{\eta \alpha }}}{∥{\left(\sum _{j\in \mathbb{Z}}{\left|{\mathcal{H}}_j\right|}^{\alpha }\right)}^{1/\alpha }∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(16)

We prove part $\left(\mathbf{iii}\right)$ as follows: let $\mathcal{T}$ be the linear operator $\mathcal{T}$ defined on ${\mathcal{H}}_j$ by $\mathcal{T}\left({\mathcal{H}}_j\right)={\mho }_{\zeta ,{\tau }^{j}t}\ast {\mathcal{H}}_j$. Then $\mathcal{T}$ satisfies

$$\begin{array}{c}\hfill {∥{∥{∥\mathcal{T}\left({\mathcal{H}}_j\right)∥}_{L_1(1,\tau ),{\displaystyle \frac{dt}{t}}}∥}_{l_1\left(\mathbb{Z}\right)}∥}_{L_1\left({\mathbb{R}}^{d+1}\right)}\le Cln\left(\tau \right){∥\left(\sum _{j\in \mathbb{Z}}\left|{\mathcal{H}}_j\right|\right)∥}_{L_1\left({\mathbb{R}}^{d+1}\right)}.\end{array}$$

(17)

Consequently, by interpolating between (4) and (17), the statement $\left(\mathbf{iii}\right)$ holds. □

3. Proof of the Main Results

Let $d_1,d_2,\dots ,d_K$ be non-negative integers with $d_{\ell }<d_{\ell +1}$ for all $\ell \in \{1,2,\dots ,K-1\}$. Then, any polynomial $\mathcal{P}$ can be written in the form

$$\mathcal{P}\left(v\right)={\displaystyle \sum _{m=1}^K}{\mathcal{P}}^{\left(m\right)}\left(v\right)+{\mathcal{R}}^{\left(m\right)}\left(\right|v\left|\right),$$

where $v\in {\mathbb{R}}^s$, ${\mathcal{P}}^{\left(m\right)}\left(v\right)=(P_{1,m}\left(v\right),P_{2,m}\left(v\right),\dots ,P_{d,m}\left(v\right))$, $\{P_{i,m}\left(v\right):1\le i\le d,1\le m\le K\}$ are real-valued homogeneous polynomials of degree $d_m$ with ${\left|v\right|}^{d_m}\notin span\{P_{1,m},\dots ,P_{d,m}\}$, ${\mathcal{R}}^{\left(m\right)}\left(t\right)=({\mathcal{R}}_1^{\left(m\right)}\left(t\right),{\mathcal{R}}_2^{\left(m\right)}\left(t\right),\dots ,{\mathcal{R}}_d^{\left(m\right)}\left(t\right))$, and $\{{\mathcal{R}}_i^{\left(m\right)}\left(t\right):1\le i\le d,1\le m\le K\}$ be polynomials on $\mathbb{R}$ of degree less than $d_m$. Let ${\gamma }_m$ indicate to the number of elements of $\{\alpha =({\alpha }_1,{\alpha }_2\dots ,{\alpha }_s)\in {\left(\mathbb{N}\cup \left\{0\right\}\right)}^s:\left|\alpha \right|=d_m\}=\{\alpha \left(1\right),\alpha \left(2\right),\dots ,\alpha \left({\gamma }_m\right)\}$. Then $P_{i,m}$ can be written in the form $P_{i,m}\left(v\right)={\displaystyle \sum _{n=1}^{{\gamma }_m}}{c}_{n,i}{v}^{\alpha \left(n\right)}$. Define the linearfunction $L_m:{\mathbb{R}}^d\to {\mathbb{R}}^{{\gamma }_m}$ by

$$L_m(\zeta )=\left({\displaystyle \sum _{i=1}^d}{c}_{1,i}^m{\zeta }_i,\dots ,{\displaystyle \sum _{i=1}^d}{c}_{{\gamma }_m,i}^m{\zeta }_i\right),$$

and set ${\mathbf{P}}_m\left(v\right)={\displaystyle \sum _{i=1}^m}{\mathcal{P}}^{\left(i\right)}\left(v\right)+\mathcal{W}\left(\right|v\left|\right)$ and ${\mathbf{P}}_0\left(v\right)=\mathcal{W}\left(\right|v\left|\right)$. This leads to ${\mathbf{P}}_K\left(v\right)=\mathcal{P}\left(v\right)$. For $1\le m\le K$, we let ${\mho }_{\zeta ,t}^{\left(m\right)}={\mho }_{\Phi ,{\mathbf{P}}_m,\Psi ,\zeta ,t}$ and ${\mho }_{\zeta }^{\left(m\right)\ast }\left(\mathcal{V}\right)=\underset{t\in {\mathbb{R}}_{+}}{sup}\left|\right|{\mho }_{\zeta ,t}^{\left(m\right)}|\ast \mathcal{V}|$.

Now, we are ready to prove our main results.

• Proof of Theorem 1. Let $\Phi ,\mathcal{P},\Psi$, and $\zeta$ be given as in Theorem 1. Set $\tau =2^{{\eta }^{\prime }{q}^{\prime }}$. By Minkowski’s inequality, we reach

  $$\begin{array}{ccc}\hfill & & {\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)\left(\dot{x}\right)\hfill \\ & \le & {\displaystyle \sum _{j=0}^{\infty }}{\left({\int }_{{\mathbb{R}}_{+}}{\left|{\displaystyle \frac{1}{t^a}}{\int }_{2^{-j-1}l<\left|v\right|\le 2^{-j}l}\mathcal{V}(x-\mathcal{P}\left(v\right),x_{d+1}-\Psi \left(\right|v\left|\right)){\displaystyle \frac{\Phi \left(v\right)\zeta \left(\left|v\right|\right)}{{\left|v\right|}^{s-a}}}dv\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }\hfill \\ & \le & C{\left({\int }_{{\mathbb{R}}_{+}}{\left|{\mho }_{\zeta ,t}^{\left(K\right)}\ast \mathcal{V}\left(\dot{x}\right)\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }.\hfill \end{array}$$

  (18)

Let ${\left\{{\phi }_j\right\}}_{j\in \mathbb{Z}}$ be a set of functions satisfying

$$\begin{array}{ccc}\hfill & & {\phi }_j\in C^{\infty }(0,\infty ),{\phi }_j\in [0,1],\sum _j{\phi }_j\left(t\right)=1,\hfill \\ \hfill \mathrm{supp}{\phi }_j& \subseteq & [{\tau }^{-1-j},{\tau }^{1-j}],and\left|{\displaystyle \frac{d^k{\phi }_j\left(t\right)}{d{t}^k}}\right|\le {\displaystyle \frac{C_k}{t^k}}.\hfill \end{array}$$

Define the multiplier operator $\widehat{{\Lambda }_j\mathcal{V}}\left(\dot{\xi }\right)={\phi }_j\left(\left|\xi \right|\right)\widehat{\mathcal{V}}\left(\dot{\xi }\right)$. Thus, by (18), we get that

$${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }\left(\mathcal{V}\right)\le CK{\left({\int }_{{\mathbb{R}}_{+}}{\left|{\mho }_{\zeta ,t}^{\left(K\right)}\ast \mathcal{V}\left(\dot{x}\right)\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }\le C\sum _{j\in \mathbb{Z}}{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right),$$

(19)

where

$${\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)\left(\dot{x}\right)={\left({\int }_{{\mathbb{R}}_{+}}{\left|{\Gamma }_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}(\dot{x},t)\right|}^{\alpha }{\displaystyle \frac{dt}{t}}\right)}^{1/\alpha }$$

and

$${\Gamma }_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}(\dot{x},t)=\sum _{k\in \mathbb{Z}}({\Lambda }_{k+j}\ast {\mho }_{\zeta ,t}^{\left(M\right)}\ast \mathcal{V})\left(\dot{x}\right){\chi }_{[{\tau }^k,{\tau }^{k+1})}\left(t\right).$$

Thus, to finish the proof of Theorem 1, it is enough to find a positive number $\delta <1$ such that

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le 2^{-\delta \left|j\right|}{C}_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{-1/\alpha }{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)};\hfill \end{array}$$

(20)

for $p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le 2^{-\delta \left|j\right|}{C}_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha -\eta }{\eta \alpha }}}{∥\mathcal{V})∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\hfill \end{array}$$

(21)

for $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha );$ and

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le 2^{-\delta \left|j\right|}{C}_{p,\Phi ,\zeta }{\left[(\eta -1)(q-1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}\hfill \end{array}$$

(22)

for $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$.

Let us first consider the case $p=\alpha =2$. Thanks to Plancherel’s theorem and the estimates of Lemma 2 in [10] we get

$$\begin{array}{ccc}& & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_2\left({\mathbb{R}}^{d+1}\right)}^2\hfill \\ & \le & \sum _{\mu \in \mathbb{Z}}{\int }_{{\Delta }_{\tau ,j+\mu }}\left({\int }_{{\tau }^{\mu }}^{{\tau }^{\mu +1}}{\left|{\widehat{\mho }}_{\zeta ,t}^{\left(K\right)}\left(\dot{\xi }\right)\right|}^2{\displaystyle \frac{dt}{t}}\right){\left|\widehat{\mathcal{V}}\left(\dot{\xi }\right)\right|}^{2}d\dot{\xi }\hfill \\ & \le & (ln\tau )C_{2,\Phi ,\zeta }^2\sum _{\mu \in \mathbb{Z}}{\int }_{{\Delta }_{\tau ,j+\mu }}\left(min\left\{{\left|{\tau }^{j{d}_K}\xi \right|}^{-{\displaystyle \frac{\delta }{ln\tau }}},{\left|{\tau }^{j{d}_K}\xi \right|}^{{\displaystyle \frac{\delta }{ln\tau }}}\right\}\right){\left|\widehat{\mathcal{V}}\left(\dot{\xi }\right)\right|}^{2}d\dot{\xi }\hfill \\ & \le & (ln\tau )C_{2,\Phi ,\zeta }^2{2}^{-\delta \left|j\right|}\sum _{\mu \in \mathbb{Z}}{\int }_{{\Delta }_{\tau ,j+\mu }}{\left|\widehat{\mathcal{V}}\left(\dot{\xi }\right)\right|}^{2}d\dot{\xi }\hfill \\ & \le & (ln\tau )C_{2,\Phi ,\zeta }^2{2}^{-\delta \left|j\right|}{∥\mathcal{V}∥}_{L^2\left({\mathbb{R}}^{d+1}\right)}^2=(ln\tau )C_{2,\Phi ,\zeta }^2{2}^{-\delta \left|j\right|}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_2^{0,2}\left({\mathbb{R}}^{d+1}\right)}^2,\hfill \end{array}$$

where ${\Delta }_{\tau ,\mu }=\left\{\dot{\xi }\in {\mathbb{R}}^{d+1}:\left|\dot{\xi }\right|\in [{\tau }^{-1-j},{\tau }^{1-j}]\right\}$ and $0<\delta <1$. So,

$$\begin{array}{c}\hfill {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_2\left({\mathbb{R}}^{d+1}\right)}\le {\left[(\eta -1)(q-1)\right]}^{-1/2}{C}_{2,\Phi ,\zeta }{2}^{-\delta \left|j\right|/2}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_2^{0,2}\left({\mathbb{R}}^{d+1}\right)}.\end{array}$$

(23)

By invoking Lemma 3, we get

(i) For all $p\in [\alpha ,{\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }}]$,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(q-1)(\eta -1)\right]}^{-1/\alpha }{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(24)

(ii) For all $p\in ({\displaystyle \frac{\eta \alpha }{\eta \alpha -\alpha +\eta }},\alpha )$,

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(q-1)(\eta -1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha -\eta }{\eta \alpha }}}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(25)

(iii) For all $p\in ({\displaystyle \frac{\eta {\alpha }^{\prime }}{{\alpha }^{\prime }-\eta }},\alpha )$

$$\begin{array}{ccc}\hfill & & {∥{\mathcal{B}}_{\Phi ,\mathcal{P},\Psi ,\zeta ,j}^{\alpha }\left(\mathcal{V}\right)∥}_{L_p\left({\mathbb{R}}^{d+1}\right)}\le C_{p,\Phi ,\zeta }{\left[(q-1)(\eta -1)\right]}^{{\displaystyle \frac{\alpha -\eta \alpha +1}{\eta \alpha }}}{∥\mathcal{V}∥}_{{\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)}.\hfill \end{array}$$

(26)

Therefore, by interpolating between (23) and (24)-(26), we obtain (20)-(22), which gives by (18)-(19) that the conclusions of Theorem 1 hold.

• Proof of Theorem 2. In fact, we can prove this theorem by employing the above arguments with employing Lemma 2 instead of Lemma 3 and replacing $\tau =2^{{\eta }^{\prime }{q}^{\prime }}$ by $\tau =2^{q^{\prime }}$.

4. Conclusions

This research paper investigated the boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ from the space ${\stackrel{.}{F}}_p^{0,\alpha }\left({\mathbb{R}}^{d+1}\right)$ to the space $L_p\left({\mathbb{R}}^{d+1}\right)$. For a polynomial mapping $\mathcal{P}:{\mathbb{R}}^s\to {\mathbb{R}}^d$ and a function $\Psi :{\mathbb{R}}_{+}\to \mathbb{R}$, where $\Psi \left(t\right)={\psi }_1\left(t\right)+{\psi }_2\left(t\right)$, ${\psi }_1$ is a polynomial, and ${\psi }_2$ is positive non-decreasing on ${\mathbb{R}}_{+}$, we proved the boundedness of ${\mathbf{G}}_{\Phi ,\mathcal{P},\Psi ,\zeta }^{\alpha }$ on $L_p\left({\mathbb{R}}^{d+1}\right)$ provided that $\zeta \in {\Theta }_{\eta }\left({\mathbb{R}}_{+}\right)$ for some $\eta >1$ and $\Phi \in L{(logL)}^{\nu }\left({\mathbb{B}}^{s-1}\right)\cup B_q^{(0,\nu -1)}\left({\mathbb{B}}^{s-1}\right)$ for some $\nu$. The results of this paper are essential refinements, improvements, or extensions to several past known findings as those in [1,2,3,4,5,7,8,9,10,21,22,23].