Boundedness of Generalized Parametric Marcinkiewicz Integrals Associated to Surfaces

Abstract

In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$ is considered. Under the condition that $\mathsf{\Omega }$ is a function in $L^q\left({\mathbf{S}}^{n-1}\right)$ with $q\in (1,2]$, appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to $L^p$ spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function $\mathsf{\Omega }$ belongs to the block space $B_q^{0,\nu -1}\left({\mathbf{S}}^{n-1}\right)$ or in the space $L{\left(\log L\right)}^{\nu }\left({\mathbf{S}}^{n-1}\right)$. Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

1. Introduction

Throughout this work, we assume that ${\mathbf{R}}^n$ ($n\ge 2$) is the n-dimensional Euclidean space and $x^{\prime }=x/\left|x\right|$ for $x\in {\mathbf{R}}^n\backslash \left\{0\right\}$. In addition, we assume that ${\mathbf{S}}^{n-1}$ is the unit sphere in ${\mathbf{R}}^n$, which is equipped with the normalized Lebesgue surface measure $d\sigma$.

For $\rho =\tau +i\upsilon$$(\tau ,\upsilon \in \mathbf{R}$ with $\tau >0$), let $K_{\mathsf{\Omega },h}$ be the kernel on ${\mathbf{R}}^n$ defined by

$$K_{\mathsf{\Omega },h}\left(u\right)={\left|u\right|}^{\rho -n}\mathsf{\Omega }\left(u^{\prime }\right)h\left(\left|u\right|\right),$$

where h is a measurable function on ${\mathbf{R}}^{+}$ and $\mathsf{\Omega }$ is a homogeneous function of degree zero on ${\mathbf{R}}^n$ with $\mathsf{\Omega }\in L^1\left({\mathbf{S}}^{n-1}\right)$ and

$${\int }_{{\mathbf{S}}^{n-1}}\mathsf{\Omega }\left(u\right)d\sigma \left(u\right)=0.$$

(1)

For a suitable function $\varphi :{\mathbf{R}}^{+}\to \mathbf{R}$, we consider the generalized parametric Marcinkiewicz integral operator ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$ given by

$${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)\left(x\right)={\left({\int }_0^{\infty }{\left|\frac{1}{t^{\rho }}{\int }_{\left|u\right|\le t}f(x-\varphi (\left|u\right|)u^{\prime })K_{\mathsf{\Omega },h}\left(u\right)du\right|}^r\frac{dt}{t}\right)}^{1/r},$$

where $r>1$ and $f\in \mathcal{S}\left({\mathbf{R}}^n\right)$.

If $\varphi \left(t\right)=t$, $h=1$, $\rho =1$, and $r=2$, then the operator ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$, denoted by ${\mathcal{M}}_{\mathsf{\Omega }}$, reduces to the classical Marcinkiewicz integral operator. The operator ${\mathcal{M}}_{\mathsf{\Omega }}$ was introduced by Stein in [1] in which Stein established the $L^p$ $(1<p\le 2)$ boundedness of ${\mathcal{M}}_{\mathsf{\Omega }}$ provided that $\mathsf{\Omega }\in Li{p}_{\alpha }\left({\mathbf{S}}^{n-1}\right)$ with $0<\alpha \le 1$. This result was discussed and improved by many mathematicians. For example, the authors of [2] proved that, if $\mathsf{\Omega }\in C^1\left({\mathbf{S}}^{n-1}\right)$, then the $L^p$ boundedness of ${\mathcal{M}}_{\mathsf{\Omega }}$ is satisfied for all $p\in (1,\infty )$. Later on, Al-Qassem and Al-Salman found in [3] that ${\mathcal{M}}_{\mathsf{\Omega }}$ is bounded on $L^p\left({\mathbf{R}}^n\right)$ for $1<p<\infty$ whenever $\mathsf{\Omega }\in B_q^{(0,-1/2)}\left({\mathbf{S}}^{n-1}\right)$ with $q>1$. Moreover, they proved that the condition $\mathsf{\Omega }\in B_q^{(0,-1/2)}\left({\mathbf{S}}^{n-1}\right)$ is optimal in the sense that the operator ${\mathcal{M}}_{\mathsf{\Omega }}$ may lose the $L^2$ boundedness when $\mathsf{\Omega }$ belongs to the space $\mathsf{\Omega }\in B_q^{(0,-\frac{1}{2}-\epsilon )}\left({\mathbf{S}}^{n-1}\right)$ for some $0<\epsilon <1/2$. Walsh in [4] obtained that ${\mathcal{M}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbf{R}}^n\right)$ if $\mathsf{\Omega }\in L{\left(\log L\right)}^{1/2}\left({\mathbf{S}}^{n-1}\right)$. Furthermore, he established the optimality of the condition $\mathsf{\Omega }\in L{(logL)}^{1/2}\left({\mathbf{S}}^{n-1}\right)$ in the sense that the exponent $1/2$ in $L{\left(\log L\right)}^{1/2}\left({\mathbf{S}}^{n-1}\right)$ cannot be replaced by any smaller number.

Hörmander in [5] started studying the parametric Marcinkiewicz integral operator ${\mathcal{M}}_{\mathsf{\Omega },t,1,\rho }^{\left(2\right)}$. In fact, he proved the $L^p\left({\mathbf{R}}^n\right)$$(1<p<\infty )$ boundedness of ${\mathcal{M}}_{\mathsf{\Omega },t,1,\rho }^{\left(2\right)}$ provided that $\rho >0$ and $\mathsf{\Omega }\in Li{p}_{\alpha }\left({\mathbf{S}}^{n-1}\right)$ with $\alpha >0$. Subsequently, the investigation of the $L^p$ boundedness of the parametric Marcinkiewicz integrals under very various conditions on $\mathsf{\Omega }$, $\varphi$, and h has attracted the attention of many authors. For a sampling of studies of such operators, the readers are referred to [6,7,8,9,10,11,12,13,14] and the references therein.

Although some open problems related to the boundedness of the operators ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(2\right)}$ remain open, the investigation to determine the boundedness of the generalized parametric Marcinkiewicz integrals has been started. Historically, the operator ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$ was introduced by Chen, Fan and Ying in [15]; they showed that, if $h\equiv 1$, $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $q>1$ and $1<r<\infty$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },t,h,1}^{\left(r\right)}f∥}_{L^p\left({\mathbf{R}}^n\right)}\le C{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

(2)

holds for all $1<p<\infty$. However, Le in [16] improved this result. As a matter of fact, he found that the last result is still true for all $p\in (1,\infty )$ under the conditions that $\mathsf{\Omega }\in L(logL)\left({\mathbf{S}}^{n-1}\right)$, $1<r<\infty$ and $h\in {\mathsf{\Gamma }}_{max\{r^{\prime },2\}}\left({\mathbf{R}}^{+}\right)$, where ${\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ is the collection of all measurable functions $h:[0,\infty )\to \mathbf{C}$ satisfying

$${\parallel h\parallel }_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}={\displaystyle \underset{k\in \mathbf{Z}}{sup}}{\left({\int }_{2^k}^{2^{k+1}}{\left|h\left(t\right)\right|}^{{}^s}\frac{dt}{t}\right)}^{1/s}<\infty .$$

For the significance and recent advances on the study of such operators, readers may consult [14,17,18,19,20].

For $s\ge 1$, we let ${\mathfrak{L}}^s\left({\mathbf{R}}^{+}\right)$ denote the set of all measurable functions $h:[0,\infty )\to \mathbf{C}$ that satisfy the condition

$$L_s\left(h\right)={\displaystyle {\displaystyle \underset{k\in \mathbf{Z}}{sup}}}\left({\int }_{2^k}^{2^{k+1}}\left|h\left(t\right)\right|{\left(log(2+\left|h\left(t\right)\right|)\right)}^s\frac{dt}{t}\right)<\infty .$$

In addition, we let ${\mathcal{N}}^s\left({\mathbf{R}}^{+}\right)$ denote the set of all measurable functions $h:[0,\infty )\to \mathbf{C}$ that satisfy the condition

$$N_s\left(h\right)={\displaystyle \sum _{k=1}^{\infty }}{2}^k{k}^s{d}_k\left(h\right)<\infty ,$$

where $d_k\left(h\right)={\displaystyle \underset{j\in \mathbf{Z}}{sup}}{2}^{-j}\left|E(j,k)\right|$ with $E(j,k)=\left\{t\in (2^j,2^{j+1}]:2^{k-1}<\left|h\left(t\right)\right|\le 2^k\right\}$ for $k\ge 2$ and $E(j,1)=\left\{t\in (2^j,2^{j+1}]:\left|h\left(t\right)\right|\le 2\right\}$.

It is obvious that ${\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)\subset {\mathcal{N}}^{\beta }\left({\mathbf{R}}^{+}\right)\subset {\mathfrak{L}}^{\beta }\left({\mathbf{R}}^{+}\right)$ for any $s\ge 1$, $\beta >0$; and also ${\mathfrak{L}}^{s+\beta }\left({\mathbf{R}}^{+}\right)\subset {\mathcal{N}}^{\beta }\left({\mathbf{R}}^{+}\right)$ for all $s>1$, $\beta >0$.

For $\nu >0$, let $L{\left(\log L\right)}^{\nu }\left({\mathbf{S}}^{n-1}\right)$ denote the space of all measurable functions $\mathsf{\Omega }$ on ${\mathbf{S}}^{n-1}$ that satisfy

$${∥\mathsf{\Omega }∥}_{L{\left(\log L\right)}^{\nu }\left({\mathbf{S}}^{n-1}\right)}={\int }_{{\mathbf{S}}^{n-1}}\left|\mathsf{\Omega }\left(w\right)\right|\left({log}^{\nu }(2+\left|\mathsf{\Omega }\left(w\right)\right|\right)d\sigma \left(w\right)<\infty .$$

It is worth mentioning that $B_q^{(0,\delta )}\left({\mathbf{S}}^{n-1}\right)$ (for $q>1$ and $\delta >-1$) is denoted for the special class of the block spaces, which was introduced by Jiang and Lu in [21].

Let us recall the definition of the Triebel–Lizorkin spaces. For $\alpha \in \mathbf{R}$ and $1<p,r\le \infty$ with $(p\ne \infty )$, the homogeneous Triebel–Lizorkin space ${\stackrel{.}{F}}_{p,r}^{\alpha }\left({\mathbf{R}}^n\right)$ is defined by

$${\stackrel{.}{F}}_{p,r}^{\alpha }\left({\mathbf{R}}^n\right)=\left\{f\in {\mathcal{S}}^{\prime }\left({\mathbf{R}}^n\right):{∥f∥}_{{\stackrel{.}{F}}_{p,r}^{\alpha }\left({\mathbf{R}}^n\right)}={∥{\left({\displaystyle \sum _{j\in \mathbf{Z}}}{2}^{j\alpha r}{\left|{\mathsf{\Psi }}_j\ast f\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}<\infty \right\},$$

where ${\mathcal{S}}^{\prime }$ denotes the tempered distribution class on ${\mathbf{R}}^n$, $\widehat{{\mathsf{\Psi }}_j}\left(\zeta \right)=\mathsf{\Phi }\left(2^{-j}\zeta \right)$ for $j\in \mathbf{Z}$ and $\mathsf{\Phi }$ is a radial function satisfying the following conditions:

(a)

$0\le \mathsf{\Phi }\le 1$;

(b)

$supp\mathsf{\Phi }\subset \left\{\zeta :\frac{1}{2}\le \left|\zeta \right|\le 2\right\}$;

(c)

$\mathsf{\Phi }\left(\zeta \right)\ge c>0$ if $\frac{3}{5}\le \left|\zeta \right|\le \frac{5}{3}$;

(d)

${\displaystyle {\displaystyle \sum _{j\in \mathbf{Z}}}}\mathsf{\Phi }\left(2^{-j}\zeta \right)=1$$(\zeta \ne 0)$.

The following properties of the Triebel–Lizorkin space are well known:

(i)

${\mathcal{S}}^{\prime }\left({\mathbf{R}}^n\right)$ is dense in ${\stackrel{.}{F}}_{p,r}^{\alpha }\left({\mathbf{R}}^n\right)$;

(ii)

${\stackrel{.}{F}}_{p,2}^0\left({\mathbf{R}}^n\right)=L^p\left({\mathbf{R}}^n\right)$ for $1<p<\infty$, and ${\stackrel{.}{F}}_{\infty ,2}^0\left({\mathbf{R}}^n\right)=$ BMO;

(iii)

${\stackrel{.}{F}}_{p,r_1}^{\alpha }\left({\mathbf{R}}^n\right)\subset {\stackrel{.}{F}}_{p,r_2}^{\alpha }\left({\mathbf{R}}^n\right)$ if $r_1<r_2$;

(iv)

${\left({\stackrel{.}{F}}_{p,r}^{\alpha }\left({\mathbf{R}}^n\right)\right)}^{\ast }={\stackrel{.}{F}}_{p^{\prime },r^{\prime }}^{-\alpha }\left({\mathbf{R}}^n\right)$.

In this work, we let ${\mathcal{H}}_d$ ($d\ne 0$) to be the class of all smooth functions $\varphi :(0,\infty )\to \mathbf{R}$ satisfying the following growth conditions:

$$\left|\varphi \left(t\right)\right|\le C_1{t}^d,|{\varphi }^{″}\left(t\right)|\le C_2{t}^{d-2},C_3{t}^{d-1}\le \left|{\varphi }^{\prime }\left(t\right)\right|\le C_4{t}^{d-1}$$

for $t\in (0,\infty )$, where the positive constants $C_1$, $C_2$, $C_3$, and $C_4$ are independent of the variable t.

It is worth mentioning that, when $d=0$, the class ${\mathcal{H}}_d$ is empty. Some model examples for the class ${\mathcal{H}}_d$ are $t^d$ with $d>0$ and $t^l$ with $l<0$.

Here, and henceforth, we let $p^{\prime }$ denote the conjugate index of p defined by $1/p+1/p^{\prime }=1$.

Our main results are formulated as follows:

Theorem 1.

Let $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $1<q\le 2$ satisfy the condition (1), and $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $1<s\le 2$. Suppose that $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$. Then, for any $f\in {\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)$, there exists a positive constant $C_p$ (independent of Ω, ϕ, h, r, s, and q) such that

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p{(q-1)}^{-1}{(s-1)}^{-1}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

(3)

for $1<p<r$; and

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p{(q-1)}^{-1/r}{(s-1)}^{-1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

(4)

for $r\le p<\infty$.

Theorem 2.

Assume that ϕ and Ω are given as in Theorem 1. Suppose that $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $s>2$. Then, there is a constant $C_p>0$ such that

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p{(q-1)}^{-1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

(5)

for $1<p<r$ with $r\le s^{\prime }$ and $2<s<\infty$; and

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p{(q-1)}^{-1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

(6)

for $s^{\prime }<p<\infty$ with $r>s^{\prime }$ and $2<s\le \infty$.

By the conclusions in Theorems 1 and 2 and the extrapolation arguments used in [18,22,23], we get the following results.

Theorem 3.

Assume that $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$ and Ω satisfies (1).

$\left(i\right)$ If $\mathsf{\Omega }\in B_q^{(0,\frac{1}{r}-1)}\left({\mathbf{S}}^{n-1}\right)$ for some $q>1$ and $h\in {\mathcal{N}}^{1/r}\left({\mathbf{R}}^{+}\right)$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{B_q^{(0,\frac{1}{r}-1)}\left({\mathbf{S}}^{n-1}\right)}\right)\left(1+N_{1/r}\left(h\right)\right){∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $r\le p<\infty$;

$\left(ii\right)$ If $\mathsf{\Omega }\in B_q^{(0,0)}\left({\mathbf{S}}^{n-1}\right)$ for some $q>1$ and $h\in {\mathcal{N}}^1\left({\mathbf{R}}^{+}\right)$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{B_q^{(0,0)}\left({\mathbf{S}}^{n-1}\right)}\right)\left(1+N_1\left(h\right)\right){∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $1<p<r$;

$\left(iii\right)$ If $\mathsf{\Omega }\in L{(logL)}^{1/r}\left({\mathbf{S}}^{n-1}\right)$ and $h\in {\mathcal{N}}^{1/r}\left({\mathbf{R}}^{+}\right)$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{L{(logL)}^{1/r}\left({\mathbf{S}}^{n-1}\right)}\right)\left(1+N_{1/r}\left(h\right)\right){∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $r\le p<\infty$;

$\left(iv\right)$ If $\mathsf{\Omega }\in L(logL)\left({\mathbf{S}}^{n-1}\right)$ and $h\in {\mathcal{N}}^1\left({\mathbf{R}}^{+}\right)$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{L(logL)\left({\mathbf{S}}^{n-1}\right)}\right)\left(1+N_1\left(h\right)\right){∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $1<p<r$, where $C_p$ is a bounded positive constant independent of h, $\mathsf{\Omega }$ and ϕ.

Theorem 4.

Let $\mathsf{\Omega }$ satisfy the condition (1), $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $s>2$ and $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$.

$\left(i\right)$ If $\mathsf{\Omega }\in B_q^{(0,\frac{1}{r}-1)}\left({\mathbf{S}}^{n-1}\right)$ for some $q>1$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{B_q^{(0,\frac{1}{r}-1)}\left({\mathbf{S}}^{n-1}\right)}\right){∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $1<p<r$ with $r\le s^{\prime }$ and $2<s<\infty$; and for $s^{\prime }<p<\infty$ with $r>s^{\prime }$ and $2<s\le \infty$.

$\left(ii\right)$ If $\mathsf{\Omega }\in L{(logL)}^{1/r}\left({\mathbf{S}}^{n-1}\right)$, then

$${∥{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\le C_p\left(1+{∥\mathsf{\Omega }∥}_{L{(logL)}^{1/r}\left({\mathbf{S}}^{n-1}\right)}\right){∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}$$

for $1<p<r$ with $r\le s^{\prime }$ and $2<s<\infty$; and for $s^{\prime }<p<\infty$ with $r>s^{\prime }$ and $2<s\le \infty$.

We point out that our results generalize what Al-Qassem found in [18]; and also extend and improve ([24] Theorems 1 and 2). Precisely, the results in [18] are acheived when we take $\varphi \left(t\right)=t$ in our results. However, when we take $r=2$, we directly obtain the results in [24].

2. Preparation

In this section, we establish some lemmas used in the proof of our results. Let us start this section by introducing some notations. Let $\theta \ge 2$. For a suitable mapping $\varphi :{\mathbf{R}}^{+}\to \mathbf{R}$, $\mathsf{\Omega }:{\mathbf{S}}^{n-1}\to \mathbf{R}$ and a measurable function $h:{\mathbf{R}}^{+}\to \mathbf{C}$; the family of measures $\{{\sigma }_{\mathsf{\Omega },\varphi ,h,t}:t\in {\mathbf{R}}^{+}\}$ and the corresponding maximal operators ${\sigma }_{\mathsf{\Omega },\varphi ,h}^{\ast }$ and $M_{\mathsf{\Omega },\varphi ,h,\theta }$ on ${\mathbf{R}}^n$ are defined by

$$\begin{array}{ccc}\hfill {\int }_{{\mathbf{R}}^n}fd{\sigma }_{\mathsf{\Omega },\varphi ,h,t}& =& t^{-\rho }{\int }_{t/2\le \left|u\right|\le t}f(\varphi (\left|u\right|)u^{\prime })K_{\mathsf{\Omega },h}\left(u\right)du,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\sigma }_{\mathsf{\Omega },\varphi ,h}^{\ast }\left(f\right)& =& {\displaystyle \underset{t\in {\mathbf{R}}^{+}}{sup}}\left|\right|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}|\ast f|,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill M_{\mathsf{\Omega },\varphi ,h,\theta }f\left(u\right)& =& {\displaystyle \underset{k\in \mathbf{Z}}{sup}}{\int }_{{\theta }^k}^{{\theta }^{k+1}}\left|\right|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}|\ast f\left(u\right)|\frac{dt}{t},\hfill \end{array}$$

where $|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}|$ is defined in the same way as ${\sigma }_{\mathsf{\Omega },\varphi ,h,t}$, but with replacing $\mathsf{\Omega }$ by $\left|\mathsf{\Omega }\right|$ and h by $\left|h\right|$. We write $r^{\pm \gamma }=min\left\{r^{\gamma },r^{-\gamma }\right\}$ and $∥{\sigma }_{\mathsf{\Omega },\varphi ,h,t}∥$ for the total variation of ${\sigma }_{\mathsf{\Omega },\varphi ,h,t}.$

We shall need the following lemma which can be derived by applying the same arguments (with only minor modifications) used in the proof of ([24], Lemma 4).

Lemma 1.

Let $\theta \ge 2$, $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $s>1$ and $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $q>1$. Suppose that $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$. Then, there exist constants C and a with $0<2a{q}^{\prime }<1$ such that, for all $k\in \mathbf{Z}$,

$$\begin{array}{c}∥{\sigma }_{\mathsf{\Omega },\varphi ,h,t}∥\le C,\hfill \end{array}$$

(7)

$$\begin{array}{c}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\widehat{\sigma }}_{\mathsf{\Omega },\varphi ,h,t}\left(\zeta \right)\right|}^2\frac{dt}{t}\le C(ln\theta ){\left|\zeta {\theta }^{kd}\right|}^{\pm \frac{2a}{ln\theta }}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^2{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^2,\hfill \end{array}$$

(8)

where the constant C is independent of ζ, k and ϕ.

By using ([9], Lemma 2.4) and following the same approaches employed in ([8], Lemmas 2.4 and 2.5), we immediately get the following lemma.

Lemma 2.

Let $\theta \ge 2$, $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$, $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $1<q\le 2$, and $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $s>1$. Then, there is a constant $C_p$ such that

$$\parallel M_{\mathsf{\Omega },\varphi ,h,\theta }{\left(f\right)\parallel }_{L^p\left({\mathbf{R}}^n\right)}\le C_p(ln\theta ){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{\parallel f\parallel }_{L^p\left({\mathbf{R}}^n\right)},$$

(9)

$$\parallel {\sigma }_{\mathsf{\Omega },\varphi ,h}^{\ast }{\left(f\right)\parallel }_{L^p\left({\mathbf{R}}^n\right)}\le C_p{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{\parallel f\parallel }_{L^p\left({\mathbf{R}}^n\right)}$$

(10)

for all $1<p\le \infty$ with $1<s\le 2$; and

$$\parallel {\sigma }_{\mathsf{\Omega },\varphi ,h}^{\ast }{\left(f\right)\parallel }_{L^p\left({\mathbf{R}}^n\right)}\le C_p{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{\parallel f\parallel }_{L^p\left({\mathbf{R}}^n\right)}$$

(11)

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

By applying the same procedures (with only minor modifications) as those in [18], we obtain the following:

Lemma 3.

Let $\theta \ge 2$, $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $1<q\le 2$ and $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $1<s\le 2$. Let $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$ and $r>1$ be a real number. Then, there is a positive constant $C_p$ such that the inequalities

$$\begin{array}{ll}& {∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_p{(ln\theta )}^{1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}forr\le p<\infty \end{array}$$

(12)

and

$$\begin{array}{ll}& {∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_p(ln\theta ){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}for1<p<r\end{array}$$

(13)

hold for arbitrary functions $\{g_k(·),k\in \mathbf{Z}\}$ on ${\mathbf{R}}^n$.

Proof. 

Let us first prove the inequality (12). On one hand, if $p=r$, then Hölder’s inequality and (9) lead us to

$$\begin{array}{ll}& {∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}^r\le C{∥h∥}_{{\mathsf{\Gamma }}_1\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })}\\ \times & {\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\displaystyle \underset{\frac{1}{2}t}{\overset{t}{\int }}}{\displaystyle \underset{{\mathbf{S}}^{n-1}}{\int }}{\left|g_k(x-\varphi \left(l\right)u)\right|}^r\left|\mathsf{\Omega }\left(u\right)\right|\left|h\left(l\right)\right|d\sigma \left(u\right)\frac{dl}{l}\frac{dt}{t}dx\\ \le & C(ln\theta ){∥h∥}_{{\mathsf{\Gamma }}_1\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })+1}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })+1}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\left(x\right)\right|}^{r}dx\right)}^{p/r}.\end{array}$$

(14)

Hence, (12) is true for the case $p=r$. On the other hand, if $p>r$, then, by duality, there exists a non-negative function $\mathsf{\Lambda }\in L^{{(p/r)}^{\prime }}\left({\mathbf{R}}^n\right)$ with ${∥\mathsf{\Lambda }∥}_{L^{{(p/r)}^{\prime }}\left({\mathbf{R}}^n\right)}\le 1$ such that

$$\begin{array}{c}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}^r={\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle \sum _{k\in \mathbf{Z}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\left(x\right)\right|}^r\frac{dt}{t}\mathsf{\Lambda }\left(x\right)dx.\hfill \end{array}$$

(15)

By Hölder’s inequality, we obtain

$$\begin{array}{c}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\left(x\right)\right|}^r\le C{∥h∥}_{{\mathsf{\Gamma }}_1\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })}\underset{\frac{1}{2}t}{\overset{t}{\int }}{\displaystyle \underset{{\mathbf{S}}^{n-1}}{\int }}{\left|g_k(x-\varphi \left(l\right)u)\right|}^r\left|\mathsf{\Omega }\left(u\right)\right|\left|h\left(l\right)\right|d\sigma \left(u\right)\frac{dl}{l}.\hfill \end{array}$$

Thus, by a change of variable, Hölder’s inequality and (9), we reach that

$$\begin{array}{ll}& {∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}^r\\ \le & C{∥h∥}_{{\mathsf{\Gamma }}_1\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\left(x\right)\right|}^r\right)M_{\left|\mathsf{\Omega }\right|,\varphi ,\left|h\right|,\theta }\tilde{\mathsf{\Lambda }}(-x)dx\\ \le & C{∥h∥}_{{\mathsf{\Gamma }}_1\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })}{∥{\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r∥}_{L^{(p/r)}\left({\mathbf{R}}^n\right)}{∥M_{\left|\mathsf{\Omega }\right|,\varphi ,\left|h\right|,\theta }\tilde{\mathsf{\Lambda }}∥}_{L^{{(p/r)}^{\prime }}\left({\mathbf{R}}^n\right)}\\ \le & C_p(ln\theta ){∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{(r/r^{\prime })+1}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^{(r/r^{\prime })+1}{∥{\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r∥}_{L^{(p/r)}\left({\mathbf{R}}^n\right)}{∥\tilde{\mathsf{\Lambda }}∥}_{L^{{(p/r)}^{\prime }}\left({\mathbf{R}}^n\right)},\end{array}$$

where $\tilde{\mathsf{\Lambda }}(-x)=\mathsf{\Lambda }\left(x\right)$. Therefore, (12) is satisfied.

Now, consider the case $1<p<r$ which gives $r^{\prime }<p^{\prime }$. Again, by the duality, there exist functions $\zeta ={\zeta }_k(x,t)$ defined on ${\mathbf{R}}^n\times {\mathbf{R}}^{+}$ with ${∥{∥\parallel {\zeta }_k{\parallel }_{L^{r^{\prime }}([{\theta }^k,{\theta }^{k+1}],\frac{dt}{t})}∥}_{l^{r^{\prime }}}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}\le 1$ such that

$$\begin{array}{c}\hfill {∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}\underset{{\theta }^k}{\overset{{\theta }^k}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}={\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}\left({\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\left(x\right)\right){\zeta }_k(x,t)\frac{dt}{t}dx.\end{array}$$

(16)

Let $\mathsf{{\rm Y}}\left(\zeta \right)$ be given by

$$\mathsf{{\rm Y}}\left(\zeta \right)\left(x\right)={\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast {\zeta }_k(x,t)\right|}^{r^{\prime }}\frac{dt}{t}.$$

As $(p^{\prime }/r^{\prime })>1$, we conclude that there is a function $\vartheta \in L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)$ such that

$$\begin{array}{ll}& {∥{\left(\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{1/r^{\prime }}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}^{r^{\prime }}={\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast {\zeta }_k(x,t)\right|}^{r^{\prime }}\frac{dt}{t}\vartheta \left(x\right)dx\\ \le & C{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{(r^{\prime }/r)}{∥{{\sigma }^{\ast }}_{\left|\mathsf{\Omega }\right|,\varphi ,\left|h\right|}\left(\vartheta \right)∥}_{L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}{∥\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\zeta }_k(·,t)\right|}^{r^{\prime }}\frac{dt}{t}\right)∥}_{L^{(p^{\prime }/r^{\prime })}\left({\mathbf{R}}^n\right)}\\ \le & C_p(ln\theta ){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)+1}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{(r^{\prime }/r)+1}{∥\vartheta ∥}_{L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}.\end{array}$$

Hence, by Hölder’s inequality and (16), we obtain that

$$\begin{array}{ll}& {∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_{p}ln{\left(\theta \right)}^{1/r}{∥{\left(\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{1/r^{\prime }}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}{∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_p(ln\theta ){∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥{\left({\displaystyle {\displaystyle \sum _{k\in \mathbf{Z}}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}\end{array}$$

(17)

for all $1<p<r$. Therefore, the proof of Lemma 3 is complete. □

In the same manner, we obtain the following:

Lemma 4.

Let $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $2\le s<\infty$; and let Ω, θ, ϕ, and r be given as in Lemma 3. Then, a positive constant $C_p$ exists such that

(i) If $r\le s^{\prime }$, we have

$$\begin{array}{ll}& {∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_p{(ln\theta )}^{1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}for1<p<r.\end{array}$$

(18)

(ii) If $r>s^{\prime }$, we have

$$\begin{array}{ll}& {∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C_p{(ln\theta )}^{1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}for{s}^{\prime }<p<\infty ,\end{array}$$

(19)

where $\{g_k(·),k\in \mathbf{Z}\}$ are arbitrary functions on ${\mathbf{R}}^n$.

Proof. 

Let us first consider the case $1<p<r$ with $r\le s^{\prime }$. As above, by the duality, there are functions $\psi ={\psi }_k(x,t)$ defined on ${\mathbf{R}}^n\times {\mathbf{R}}^{+}$ with ${∥{∥\parallel {\psi }_k{\parallel }_{L^{r^{\prime }}([{\theta }^k,{\theta }^{k+1}],\frac{dt}{t})}∥}_{l^{r^{\prime }}}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}\le 1$ such that

$$\begin{array}{ll}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\theta }^k}{\overset{{\theta }^k}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)}=& {\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle \sum _{k\in \mathbf{Z}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}\left({\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\left(x\right)\right){\psi }_k(x,t)\frac{dt}{t}dx\end{array}$$

$$\begin{array}{cc}\le & C_{p}ln{\left(\theta \right)}^{1/r}{∥{\left(\mathsf{\Theta }\left(\psi \right)\right)}^{1/r^{\prime }}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^r\right)}^{1/r}∥}_{L^p\left({\mathbf{R}}^n\right)},\end{array}$$

(20)

where

$$\mathsf{\Theta }\left(\psi \right)\left(x\right)={\displaystyle \sum _{k\in \mathbf{Z}}\underset{{\theta }^k}{\overset{{\theta }^k}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast {\psi }_k(x,t)\right|}^{r^{\prime }}\frac{dt}{t}.$$

As $r\le s^{\prime }\le s$, then, by Hölder’s inequality, we have that

$$\begin{array}{lll}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast {\psi }_k(x,t)\right|}^{r^{\prime }}& \le & C{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)}{∥h∥}_{{\mathsf{\Gamma }}_r\left({\mathbf{R}}^{+}\right)}^{r^{\prime }}{\displaystyle {\int }_{{\theta }^k}^{{\theta }^{k+1}}{\int }_{{\mathbf{S}}^{n-1}}}\left|\mathsf{\Omega }\left(u\right)\right|\\ & \times & {\left|{\psi }_k(x-\varphi \left(l\right)u,t)\right|}^{r^{\prime }}d\sigma \left(u\right)\frac{dl}{l}\\ & \le & C{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{r^{\prime }}{\displaystyle {\int }_{{\theta }^k}^{{\theta }^{k+1}}{\int }_{{\mathbf{S}}^{n-1}}}\left|\mathsf{\Omega }\left(u\right)\right|\\ & \times & {\left|{\psi }_k(x-\varphi \left(l\right)u,t)\right|}^{r^{\prime }}d\sigma \left(u\right)\frac{dl}{l}.\end{array}$$

(21)

Again, since $(p^{\prime }/r^{\prime })>1$, we deduce that there is a function $\nu \in L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)$ such that

$$\begin{array}{c}{∥{\left(\mathsf{\Theta }\left(\psi \right)\right)}^{1/r^{\prime }}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}^{r^{\prime }}={\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast {\psi }_k(x,t)\right|}^{r^{\prime }}\frac{dt}{t}\nu \left(x\right)dx.\hfill \end{array}$$

Hence, by a simple change of variables, Hölder’s inequality, ([9], Lemma 2.5) and (21), we get that

$$\begin{array}{lll}{∥{\left(\mathsf{\Theta }\left(\psi \right)\right)}^{1/r^{\prime }}∥}_{L^{p^{\prime }}\left({\mathbf{R}}^n\right)}^{r^{\prime }}& \le & C{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{r^{\prime }}{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)}{∥{{\sigma }^{\ast }}_{\left|\mathsf{\Omega }\right|,\varphi ,1}\left(\nu \right)∥}_{L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}\\ & \times & {∥\left({\displaystyle \sum _{k\in \mathbf{Z}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\psi }_k(·,t)\right|}^{r^{\prime }}\frac{dt}{t}\right)∥}_{L^{(p^{\prime }/r^{\prime })}\left({\mathbf{R}}^n\right)}\\ & \le & C_p{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(r^{\prime }/r)+1}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{r^{\prime }}{∥\nu ∥}_{L^{{(p^{\prime }/r^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}.\end{array}$$

Therefore, by (20) and the last inequality, we reach (18) for any $1<p<r$ with $r\le s^{\prime }$. Now, we consider the case $s^{\prime }<p<\infty$ with $s^{\prime }<r$. Thanks to (11), we get that

$$\begin{array}{lll}{∥{\displaystyle \underset{k\in \mathbf{Z}}{sup}}{\displaystyle \underset{t\in [1,\theta ]}{sup}}\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\right|∥}_{L^p\left({\mathbf{R}}^n\right)}& \le & {∥{\sigma }_{\mathsf{\Omega },\varphi ,h}^{\ast }\left({\displaystyle \underset{k\in \mathbf{Z}}{sup}}\left|g_k\right|\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\\ & \le & C_p{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\displaystyle \underset{k\in \mathbf{Z}}{sup}}\left|g_k\right|∥}_{L^p\left({\mathbf{R}}^n\right)}\end{array}$$

(22)

for all $s^{\prime }<p<\infty$ and $s\ge 2$. This implies

$$\begin{array}{lll}{∥{∥\parallel {\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k{\parallel }_{L^{\infty }([1,\theta ],\frac{dt}{t})}∥}_{l^{{}^{\infty }}\left(\mathbf{Z}\right)}∥}_{L^p\left({\mathbf{R}}^n\right)}& \le & C_p{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}\\ & \times & {∥{∥g_k∥}_{l^{{}^{\infty }}\left(\mathbf{Z}\right)}∥}_{L^p\left({\mathbf{R}}^n\right)}.\end{array}$$

(23)

Here, we follow the same above procedure; by Hölder’s inequality, we get

$$\begin{array}{lll}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\left(x\right)\right|}^{s^{\prime }}& \le & C{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(s^{\prime }/s)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{s^{\prime }}{\displaystyle {\int }_{{\theta }^{k}t/2}^{{\theta }^{k}t}{\int }_{{\mathbf{S}}^{n-1}}}\left|\mathsf{\Omega }\left(u\right)\right|\\ & \times & {\left|g_k(x-\varphi \left(l\right)u)\right|}^{s^{\prime }}d\sigma \left(u\right)\frac{dl}{l}.\end{array}$$

By duality, there is a function $\phi \in L^{{(p/s^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)$ with ${∥\phi ∥}_{L^{{(p/s^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}\le 1$ such that

$$\begin{array}{l}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}\underset{1}{\overset{\theta }{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\right|}^{s^{\prime }}\frac{dt}{t}\right)}^{\frac{1}{s^{\prime }}}∥}_{L^p\left({\mathbf{R}}^n\right)}^{s^{\prime }}={\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle \sum _{k\in \mathbf{Z}}\underset{1}{\overset{\theta }{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\left(x\right)\right|}^{r^{\prime }}\frac{dt}{t}\phi \left(x\right)dx\\ \le C{∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(s^{\prime }/s)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{s^{\prime }}{\displaystyle \underset{{\mathbf{R}}^n}{\int }}{\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\left(x\right)\right|}^{s^{\prime }}{\sigma }_{\mathsf{\Omega },\varphi ,1}^{\ast }\overline{\phi }(-x)dx\\ \le Cln\left(\theta \right){∥\mathsf{\Omega }∥}_{L^1\left({\mathbf{S}}^{n-1}\right)}^{(s^{\prime }/s)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{s^{\prime }}{∥{\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^{s^{\prime }}∥}_{{}_{L^{(p/s^{\prime })}\left({\mathbf{R}}^n\right)}}{∥{\sigma }_{\mathsf{\Omega },\varphi ,1}^{\ast }\overline{\phi }∥}_{{}_{L^{{(p/s^{\prime })}^{\prime }}\left({\mathbf{R}}^n\right)}}\\ \le Cln\left(\theta \right){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^{(s^{\prime }/s)+1}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^{s^{\prime }}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^{s^{\prime }}\right)}^{\frac{1}{s^{\prime }}}∥}_{{}_{L^p\left({\mathbf{R}}^n\right)}}^{s^{\prime }},\end{array}$$

(24)

where $\overline{\phi }\left(x\right)=\phi (-x)$. Thus, when we define the linear operator H on any function $\omega =g_k\left(x\right)$ by $H\left(g_k\left(x\right)\right)={\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\left(x\right)$, then, by interpolation (23) and (24), we directly obtain that

$$\begin{array}{l}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}\underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\le {∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}\underset{1}{\overset{\theta }{\int }}}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,{\theta }^{k}t}\ast g_k\right|}^r\frac{dt}{t}\right)}^{\frac{1}{r}}∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le C_p{(ln\theta )}^{1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥{\left({\displaystyle \sum _{k\in \mathbf{Z}}}{\left|g_k\right|}^r\right)}^{\frac{1}{r}}∥}_{{}_{L^p\left({\mathbf{R}}^n\right)}}\end{array}$$

for all $s^{\prime }<p<\infty$ and $s\ge 2$. This ends the proof of Lemma 4. □

3. Proof of the Main Results

Proof of Theorem 1.

The proof of this theorem depends on the arguments used in [9,18]. Let us first assume that $\varphi \in {\mathcal{H}}_d$ for some $d\ne 0$, $\mathsf{\Omega }\in L^q\left({\mathbf{S}}^{n-1}\right)$ for some $q\in (1,2]$ and $h\in {\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)$ for some $s\in (1,2]$. Thanks to Minkowski’s inequality, we have that

$$\begin{array}{lll}{\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}\left(f\right)\left(x\right)& \le & {\displaystyle \sum _{k=0}^{\infty }}{\left({\displaystyle {\int }_0^{\infty }}{\left|t^{-\rho }{\displaystyle {\int }_{2^{-k-1}t<\left|u\right|\le 2^{-k}t}}f(x-\varphi (\left|u\right|)u^{\prime })K_{\mathsf{\Omega },h}\left(u\right)du\right|}^r\frac{dt}{t}\right)}^{1/r}\\ & =& \frac{2^{\tau }}{2^{\tau }-1}{\left({\displaystyle {\int }_0^{\infty }}{\left|{\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast f\left(x\right)\right|}^r\frac{dt}{t}\right)}^{1/r}.\end{array}$$

(25)

Let $\theta =2^{q^{\prime }{s}^{\prime }}$. For $k\in \mathbf{Z}$, let ${\left\{{\mathsf{\Phi }}_k\right\}}_{-\infty }^{\infty }$ be a smooth partition of unity in $(0,\infty )$ adapted to the interval ${\mathcal{I}}_{k,\theta }=[{\theta }^{-kd-\left|d\right|},{\theta }^{-kd+\left|d\right|}]$. In fact, we require the following:

$$\begin{array}{lll}& & 0\le {\mathsf{\Phi }}_k\le 1,{\displaystyle \sum _k}{\mathsf{\Phi }}_k\left(t\right)=1,\\ \mathrm{supp} {\mathsf{\Phi }}_k& \subseteq & {\mathcal{I}}_{k,\theta },\mathrm{and}\left|\frac{d^s{\mathsf{\Phi }}_k\left(t\right)}{d{t}^s}\right|\le \frac{C_s}{t^s}.\end{array}$$

Let $\widehat{{\mathsf{\Psi }}_k}\left(\zeta \right)={\mathsf{\Phi }}_k\left(\left|\zeta \right|\right)$. Then, for $f\in \mathcal{S}\left({\mathbf{R}}^n\right)$, one can deduce that

$${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}f\left(x\right)\le \frac{2^{\tau }}{2^{\tau }-1}{\displaystyle \sum _{j\in \mathbf{Z}}}{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}\left(f\right),$$

(26)

where

$${\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}f\left(x\right)={\left(\underset{0}{\overset{\infty }{\int }}{\left|{\mathcal{F}}_{\mathsf{\Omega },\varphi ,h,j,\theta }(x,t)\right|}^r\frac{dt}{t}\right)}^{1/r},$$

$${\mathcal{F}}_{\mathsf{\Omega },\varphi ,h,j,\theta }(x,t)={\displaystyle \sum _{k\in \mathbf{Z}}}({\mathsf{\Psi }}_{k+j}\ast {\sigma }_{\mathsf{\Omega },\varphi ,h,t}\ast f)\left(x\right){\chi }_{{}_{[{\theta }^k,{\theta }^{k+1})}}\left(t\right).$$

Notice that, we prove Theorem 1 for the case $s\in (1,2]$ once we show that

$$\begin{array}{ll}& {∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C{2}^{-\epsilon \left|j\right|}{\left(q-1\right)}^{-1/r}{\left(s-1\right)}^{-1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}\end{array}$$

(27)

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

$$\begin{array}{ll}& {∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C{2}^{-\epsilon \left|j\right|}{\left(q-1\right)}^{-1}{\left(s-1\right)}^{-1}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}\end{array}$$

(28)

for $1<p<r$ and for some $0<\epsilon <1$.

Let us prove the inequality (27). First, we consider the case $p=r=2$. In this case, we have ${∥f∥}_{{\stackrel{.}{F}}_{2,2}^0\left({\mathbf{R}}^n\right)}={∥f∥}_{L^2\left({\mathbf{R}}^n\right)}$. Thus, by Plancherel’s theorem, (8), and the fact $ln\theta \le C{\left(s-1\right)}^{-1}{\left(q-1\right)}^{-1}$ with $s,q\in (1,2]$, we get that

$$\begin{array}{lll}{∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(2\right)}\left(f\right)∥}_{L^2\left({\mathbf{R}}^n\right)}^2& \le & {\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\mathcal{B}}_{k+j,\theta }}{\int }}\left({\displaystyle \underset{{\theta }^k}{\overset{{\theta }^{k+1}}{\int }}}{\left|{\widehat{\sigma }}_{\mathsf{\Omega },\varphi ,h,t}\left(\zeta \right)\right|}^2\frac{dt}{t}\right){\left|\widehat{f}\left(\zeta \right)\right|}^{2}d\zeta \\ & \le & C(ln\theta ){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^2{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^2{\displaystyle \sum _{k\in \mathbf{Z}}}{\displaystyle \underset{{\mathcal{B}}_{k+j,\theta }}{\int }}\left({\left|{\theta }^{kd}\zeta \right|}^{\pm \frac{2a}{q^{\prime }{s}^{\prime }}}\right){\left|\widehat{f}\left(\zeta \right)\right|}^{2}d\zeta \\ & \le & C(ln\theta ){∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^2{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^2{2}^{-\eta \left|j\right|}{\displaystyle \sum _{k\in \mathbf{Z}}{\int }_{{\mathcal{B}}_{k+j,\theta }}}{\left|\widehat{f}\left(\zeta \right)\right|}^{2}d\zeta \\ & \le & C{\left(s-1\right)}^{-1}{\left(q-1\right)}^{-1}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}^2{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}^2{2}^{-\eta \left|j\right|}{∥f∥}_{L^2\left({\mathbf{R}}^n\right)}^2,\end{array}$$

where ${\mathcal{B}}_{k,\theta }=\left\{\zeta \in {\mathbf{R}}^n:\left|\zeta \right|\in {\mathcal{I}}_{k,\theta }\right\}$ and $0<\eta <1$. Therefore,

$$\begin{array}{ll}& {∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(2\right)}\left(f\right)∥}_{L^2\left({\mathbf{R}}^n\right)}\\ \le & C{2}^{-\frac{\eta }{2}\left|j\right|}{\left(s-1\right)}^{-1/2}{\left(q-1\right)}^{-1/2}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{2,2}^0\left({\mathbf{R}}^n\right)}.\end{array}$$

(29)

On the other hand, by Lemma 3, we directly get that

$$\begin{array}{ll}& {∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C{\left(q-1\right)}^{-1/r}{\left(s-1\right)}^{-1/r}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_{p,r}^0\left({\mathbf{R}}^n\right)}\end{array}$$

(30)

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

$$\begin{array}{ll}& {∥{\mathcal{G}}_{\mathsf{\Omega },\varphi ,h,j}^{\left(r\right)}\left(f\right)∥}_{L^p\left({\mathbf{R}}^n\right)}\\ \le & C{\left(q-1\right)}^{-1}{\left(s-1\right)}^{-1}{∥\mathsf{\Omega }∥}_{L^q\left({\mathbf{S}}^{n-1}\right)}{∥h∥}_{{\mathsf{\Gamma }}_s\left({\mathbf{R}}^{+}\right)}{∥f∥}_{{\stackrel{.}{F}}_p^{0,r}\left({\mathbf{R}}^n\right)}\end{array}$$

(31)

for $1<p<r$. Consequently, interpolating (29) with (30) and (31), we achieve (27) and (28). □

Proof of Theorem 2.

The proof of Theorem 2 can be obtained by applying the above approaches except we need to invoke $\theta =2^{q^{\prime }}$ instead of $\theta =2^{q^{\prime }{s}^{\prime }}$, and Lemma 4 instead of Lemma 3. □