L^p-Mapping Properties of a Class of Spherical Integral Operators

Abstract

In this paper, we study a class of spherical integral operators ${\mathcal{I}}_{\mathsf{\Omega }}f$. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ for some $1<p<2$ whenever $\mathsf{\Omega }$ belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on $\mathsf{\Omega }$ in order to obtain the $L^2$-boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}$. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators.

1. Introduction

For $d\ge 2$, let ${\mathbb{R}}^d$ be the d-dimensional Euclidean space and let ${\mathbb{S}}^{d-1}$ denote the unit sphere in ${\mathbb{R}}^d$ equipped with the normalized surface measure $\mathrm{d}\sigma$. Let $\mathsf{\Omega }\in L^1\left({\mathbb{S}}^{d-1}\right)$ be a homogeneous function of degree 0 on ${\mathbb{R}}^d$ which satisfies the cancellation property:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{S}}^{d-1}}\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)=0.\end{array}$$

(1)

For $f\in L^p\left({\mathbb{R}}^d\right)$ and $r>0$, we define the spherical averages:

$$\begin{array}{c}\hfill {\mathcal{A}}_r^{\mathsf{\Omega }}f\left(x\right)={\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right),\end{array}$$

and the corresponding maximal operator by:

$$\begin{array}{c}\hfill {\mathbb{A}}^{\mathsf{\Omega }}f\left(x\right)=\underset{r>0}{\sup }\left|{\mathcal{A}}_r^{\mathsf{\Omega }}f\left(x\right)\right|.\end{array}$$

An application of Minkowski’s inequality for integrals shows that the linear operators ${\mathcal{A}}_r$ are bounded on $L^p\left({\mathbb{R}}^d\right)$ with norms less than or equal to ${||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}$; hence, ${\mathcal{A}}_{r}f\left(x\right)$ is well-defined for almost every $x\in {\mathbb{R}}^d$ whenever $f\in L^p\left({\mathbb{R}}^d\right)$.

Observe that when $\mathsf{\Omega }\equiv 1$, ${\mathcal{A}}_r^{1}f\left(x\right)$ is a constant multiple of the average value of f on the sphere $S(x,r)$. Spherical means are used to reformulate PDE’s in the form of integral equations whose kernels are generalized functions [1]. Moreover, the spherical averages ${\mathcal{A}}_r^1$ play a crucial rule in solving certain types of partial differential equations that have several methods to solve, see [2,3]. For instance, in ${\mathbb{R}}^3$, the function:

$$\begin{array}{c}\hfill u(x,t)=\frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}f(x-ty)\mathrm{d}\sigma \left(y\right)\end{array}$$

is a solution of the Darboux equation:

$$\begin{array}{cc}\hfill {\Delta }_x\left(u\right)(x,t)& =\frac{{\partial }^{2}u}{\partial t^2}(x,t)+\frac{2}{t}\frac{\partial u}{\partial t}(x,t),\hfill \\ \hfill u(x,0)& =f\left(x\right),\hfill \\ \hfill \frac{\partial u}{\partial t}(x,0)& =0.\hfill \end{array}$$

The corresponding maximal function is given by:

$$\begin{array}{c}\hfill {\mathbb{A}}^{1}f\left(x\right)=\underset{r>0}{\sup }\left|{\mathcal{A}}_r^{1}f\left(x\right)\right|.\end{array}$$

Many authors studied the boundedness issue of the maximal operator ${\mathbb{A}}^1$ in order to attain the point-wise convergence of the spherical averages. For instance, Stein [4] proved that, for $d\ge 3$, the maximal spherical operator is bounded on $L^p\left({\mathbb{R}}^d\right)$ if and only if $p>\frac{d}{d-1}$. After several years, the two-dimensional case was established independently by Bourgain [5]. We mention here that different techniques were applied to conclude the theorems of Stein and Bourgain and others [6]. For the former, we refer the reader to [7,8], and different proofs for the latter result can be found in [9,10,11].

Chen and Lin [12] introduced the maximal operator:

$$\begin{array}{c}\hfill {\mathcal{M}}_{\mathsf{\Omega }}\left(f\right)\left(x\right)=\underset{h\in K}{\sup }|T_{\mathsf{\Omega },h}f\left(x\right)|,\end{array}$$

where K is the closed unit ball of the space $L^2({\mathbb{R}}^{+},\mathrm{d}r/r)$ and:

$$\begin{array}{c}\hfill T_{\mathsf{\Omega },h}f\left(x\right)=\underset{ϵ\to 0^{+}}{\lim }{\int }_{|y|\ge ϵ}f(x-y)\frac{\mathsf{\Omega }\left(y\right)}{{|y|}^d}h(|y|)\mathrm{d}y.\end{array}$$

The authors of [12] proved the boundedness of the maximal operator ${\mathcal{M}}_{\mathsf{\Omega }}f$ for $p>2d/2d-1$, whenever $\mathsf{\Omega }$ is a continuous function on the unit sphere of ${\mathbb{R}}^d$. Motivated by the observation that:

$$\begin{array}{c}\hfill |T_{\mathsf{\Omega },h}{f\left(x\right)|\le ||h||}_{L^2({\mathbb{R}}^{+},\mathrm{d}r/r)}{\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right),\end{array}$$

we study operators of the form:

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)& ={\left({\int }_0^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & :={\left({\int }_0^{\infty }{\left|{\mathcal{A}}_r^{\mathsf{\Omega }}f\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}},\hfill \end{array}$$

In particular, an inequality connecting this class of operators with some well-known Marcinkiewicz integral operators is proved by using the classical Hardy inequality. Then, the boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ is determined for some $1<p<2$ whenever $\mathsf{\Omega }$ belongs to a certain class of Lebesgue spaces. In addition, for the purpose of outlining the $L^2$-boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}$, a novel proof of the optimality condition is proposed on $\mathsf{\Omega }$.

In view of the previous results, it is natural to ask whether we can obtain the boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ for some $1<p<2$ whenever $\mathsf{\Omega }\in L^{\infty }\left({\mathbb{S}}^{d-1}\right)$, the space of bounded complex-valued measurable functions on ${\mathbb{S}}^{d-1}$. It is our goal in Section 2 to attain a partial answer to this inquiry by establishing several initial results including the boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ some $\frac{3}{2}<p<2$. In addition, by establishing the boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}f$, Al-Salman [13] improved the result of Chen and Lin and obtained the boundededness of ${\mathcal{M}}_{\mathsf{\Omega }}f$ for $p\ge 2$ whenever $\mathsf{\Omega }$ belongs to certain Zygmund classes. The relations among Zygmund classes and other Lebesgue space are clearly stated in Section 3. In addition, Al-Salman [13] proved that the condition $\mathsf{\Omega }\in L{\left(\log L\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$ is optimal in order to obtain the $L^2$-boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$. More precisely, he proved that there exists a function $\mathsf{\Omega }$ that lies in $L{\left(\log L\right)}^{\frac{1}{2}-ϵ}\left({\mathbb{S}}^{d-1}\right)$ for all $ϵ>0$ such that ${\mathcal{I}}_{\mathsf{\Omega }}f$ in not bounded on $L^2\left({\mathbb{R}}^d\right)$. However, in Section 3, we use the classical Hardy inequality to deduce the same result by proposing a new simple proof that uses the previously known results.

2. A First Look at ${\mathit{L}}^{\mathit{p}}$ Mapping Properties of ${\mathcal{I}}_{\mathsf{\Omega }}$

For a dense subspace in $L^p\left({\mathbb{R}}^d\right)$, we first prove that:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)<\infty \end{array}$$

for every $x\in {\mathbb{R}}^d$. More precisely, we have the following result.

Theorem 1. 

The operator ${\mathcal{I}}_{\mathsf{\Omega }}$ maps $\mathcal{S}\left({\mathbb{R}}^d\right)$ into the space of locally integrable functions on ${\mathbb{R}}^d$.

Proof. 

It is enough to show that for $f\in \mathcal{S}\left({\mathbb{R}}^d\right)$ there exists constants $\alpha ,\beta$ such that ${\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)\le \alpha +\beta |x|$ for all $x\in {\mathbb{R}}^d$. Firstly, by the cancellation property (1), we have:

$$\begin{array}{c}\hfill {\left({\int }_0^1{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\le {\left({\int }_0^1{\left({\int }_{{\mathbb{S}}^{d-1}}|f(x-ry)-f(x)||\mathsf{\Omega }(y)|\mathrm{d}\sigma (y)\right)}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}.\end{array}$$

By the mean value theorem and since f is a Schwartz function, there exists a constant $\theta \in (0,1)$ such that:

$$\begin{array}{cc}\hfill {\left({\int }_0^1{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}& \le {\left({\int }_0^1{\left({\int }_{{\mathbb{S}}^{d-1}}|\nabla f(x-\theta ry)·ry||\mathsf{\Omega }(y)|\mathrm{d}\sigma (y)\right)}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le {C||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}.\hfill \end{array}$$

(2)

On the other hand, the estimate:

$$\begin{array}{c}\hfill {(1+|x-ry|)}^{-1}\le (1+|x|){(1+r)}^{-1}\end{array}$$

gives:

$$\begin{array}{cc}\hfill {\left({\int }_1^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}& \le {C(1+|x|)||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}{\left({\int }_1^{\infty }\frac{1}{{(1+r)}^2}\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le {C||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}(1+|x|).\hfill \end{array}$$

(3)

Finally, by Equations (2) and (3) and the simple inequality $\sqrt{a+b}\le \sqrt{a}+\sqrt{b}$ such that $a,b\ge 0$, we obtain the required result. □

Remark 1. 

We notice that in case of Theorem 1, we have:

$$\begin{array}{c}\hfill {\mathcal{A}}_r^{\mathsf{\Omega }}f\left(x\right)\in L^2\left((0,\infty ),\frac{\mathrm{d}r}{r}\right)\end{array}$$

where f is a Schwartz function and $x\in {\mathbb{R}}^d$. So, by the properties of $L^p$-norms, we have:

$$\begin{array}{c}\hfill {\mathbb{A}}^{\mathsf{\Omega }}f\left(x\right)=\underset{q\to \infty }{\lim }{\left({\int }_0^{\infty }{\left|{\mathcal{A}}_r^{\mathsf{\Omega }}f\left(x\right)\right|}^q\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{q}}.\end{array}$$

Next, we wish to investigate for what finite values of $1<p<2$ the inequality:

$$\begin{array}{c}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}{f||}_{L^p\left({\mathbb{R}}^d\right)}\le C_p{||f||}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

holds for all $f\in L^p\left({\mathbb{R}}^d\right)$. In order to do this, we split the operator ${\mathcal{I}}_{\mathsf{\Omega }}$ into dyadic sub-operators as follows. Let $a_m=2^{m+1}$ for $m\ge 0$. Pick a radial smooth function ${\phi }_0$ such that ${\phi }_0\left(\xi \right)=1$ if $|\xi |\le 1$ and ${\phi }_0\left(\xi \right)=0$ if $|\xi |\ge a_m$. Now, for $k\ge 1$, define:

$$\begin{array}{c}\hfill {\phi }_{m,k}={\phi }_0\left(a_m^{-k}\xi \right)-{\phi }_0\left(a_m^{1-k}\xi \right).\end{array}$$

It can be readily seen that ${\phi }_{0,k}$ is supported in the annulus $2^{k-1}\le |\xi |\le 2^{k+1}$ and:

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{\phi }_{0,k}\left(\xi \right)=1\end{array}$$

for all $\xi \in {\mathbb{R}}^d$ with the convention ${\phi }_{0,0}={\phi }_0$. Let ${\mathsf{\Psi }}^{\left(k\right)}\left(x\right)=({\phi }_{0,k}\stackrel{ˇ}{}\ast \mathrm{d}\lambda )\left(x\right)=({\mathsf{\Phi }}_{2^{-k}}\ast f)\left(x\right)$ for a suitable Schwartz function $\mathsf{\Phi }$, where:

$$\begin{array}{c}\hfill \mathrm{d}\lambda \left(y\right)=\mathsf{\Omega }\left(y\right){\mathbb{1}}_{{\mathbb{S}}^{d-1}}\left(y\right)\mathrm{d}\sigma \left(y\right).\end{array}$$

For $k\ge 0$, define:

$$\begin{array}{c}\hfill {\mathcal{I}}_{\mathsf{\Omega }}^{k}f\left(x\right)={\left({\int }_0^{\infty }{\left|{\left({\mathsf{\Psi }}^{\left(k\right)}\right)}_r\ast f\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}},\end{array}$$

where:

$$\begin{array}{c}\hfill {\left({\mathsf{\Psi }}^{\left(k\right)}\right)}_r(·)=r^{-d}{\mathsf{\Psi }}^{\left(k\right)}\left(\frac{·}{r}\right).\end{array}$$

We shall need the following lemmas which can be found in [2] and references therein.

Lemma 1. 

Let $\mathsf{\Psi }\in L^1\left({\mathbb{R}}^d\right)$ that satisfies:

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^d}\mathsf{\Psi }\left(x\right)\mathrm{d}x=0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & |\mathsf{\Psi }\left(x\right)|\le B{(1+|x|)}^{-d-{ϵ}_1},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^d}|\mathsf{\Psi }(x-y)-\mathsf{\Psi }\left(x\right)|\mathrm{d}x\le {B|y|}^{{ϵ}_2},\hfill \end{array}$$

(6)

for some positive constants $B,{ϵ}_1,{ϵ}_2$ and for all $y\ne 0$. Then:

$$\begin{array}{c}\hfill {\left|\left|{\left({\int }_0^{\infty }{\left|{\mathsf{\Psi }}_r\ast f\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\right|\right|}_{L^p\left({\mathbb{R}}^d\right)}\le C_{d}B\max (p,{(p-1)}^{-1}){||f||}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

(7)

for $1<p<\infty$ and all $f\in L^p\left({\mathbb{R}}^d\right)$. Moreover, we have:

$$\begin{array}{c}\hfill {\left|\left|{\left({\int }_0^{\infty }{\left|{\mathsf{\Psi }}_r\ast f\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\right|\right|}_{L^{1,\infty }\left({\mathbb{R}}^d\right)}\le C_d{B||f||}_{L^1\left({\mathbb{R}}^d\right)},\end{array}$$

(8)

for all $f\in L^1\left({\mathbb{R}}^d\right)$.

Lemma 2. 

Let $N>M>d$. Then, the following inequality holds:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{S}}^{d-1}}\frac{2^{kd}}{(1+2^k{|x-y|)}^N}\mathrm{d}\sigma \left(y\right)\le C_{M,d}\frac{2^k}{{(1+|x|)}^M}.\end{array}$$

As a consequence of the previous results, we have the following lemma.

Lemma 3. 

Suppose that $\mathsf{\Omega }\in L^{\infty }\left({\mathbb{S}}^{d-1}\right)$ and satisfies the mean value zero condition (1). Then, ${\mathcal{I}}_{\mathsf{\Omega }}^{\left(k\right)}$ satisfies the conclusions (7) and (8) in Lemma 1.

Proof. 

It is clear that we need only to verify the assumptions of Lemma 1 for $k\ge 1$ to get the required result. The case $k=0$ can be treated in a similar manner. Obviously, ${\mathsf{\Psi }}^{\left(k\right)}\left(x\right)$ is integrable on ${\mathbb{R}}^d$. To prove (4), notice that:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^d}{\mathsf{\Psi }}^{\left(k\right)}\left(x\right)\mathrm{d}x& ={\int }_{{\mathbb{R}}^d}({\mathsf{\Phi }}_{2^{-k}}\ast \mathrm{d}\lambda )\left(x\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{S}}^{d-1}}{\mathsf{\Phi }}_{2^{-k}}(x-y)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_{{\mathbb{S}}^{d-1}}\mathsf{\Omega }\left(y\right){\int }_{{\mathbb{R}}^d}{\mathsf{\Phi }}_{2^{-k}}(x-y)\mathrm{d}x\mathrm{d}\sigma \left(y\right)\hfill \\ \hfill & ={\int }_{{\mathbb{S}}^{d-1}}\mathsf{\Omega }\left(y\right){\int }_{{\mathbb{R}}^d}{\mathsf{\Phi }}_{2^{-k}}\left(x\right)\mathrm{d}x\mathrm{d}\sigma \left(y\right)=0.\hfill \end{array}$$

Since $\mathsf{\Phi }$ is a Schwartz function, an application of Lemma 2 gives (5). More precisely, for $N>d+1$, we have:

$$\begin{array}{cc}\hfill |{\mathsf{\Psi }}^{\left(k\right)}\left(x\right)|& \le {\int }_{{\mathbb{S}}^{d-1}}|{\mathsf{\Phi }}_{2^{-k}}(x-y)||\mathsf{\Omega }\left(y\right)|\mathrm{d}\sigma \left(y\right)\hfill \\ \hfill & \le C_N{\int }_{{\mathbb{S}}^{d-1}}\frac{2^{kd}}{(1+2^k{|x-y|)}^N}|\mathsf{\Omega }\left(y\right)|\mathrm{d}\sigma \left(y\right)\hfill \\ \hfill & \le C_d{||\mathsf{\Omega }||}_{L^{\infty }\left({\mathbb{S}}^{d-1}\right)}\frac{2^k}{{(1+|x|)}^{d+1}}.\hfill \end{array}$$

Finally, by using the mean value theorem we get:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^d}|{\mathsf{\Psi }}^{\left(k\right)}(x-y)-{\mathsf{\Psi }}^{\left(k\right)}\left(x\right)|\mathrm{d}x& ={\int }_{{\mathbb{R}}^d}\left|{\int }_{{\mathbb{S}}^{d-1}}({\mathsf{\Phi }}_{2^{-k}}(x-y-z)-{\mathsf{\Phi }}_{2^{-k}}(x-z))\mathsf{\Omega }\left(z\right)\mathrm{d}\sigma \left(z\right)\right|\mathrm{d}x\hfill \\ \hfill & \le 2^k|y|{\int }_{{\mathbb{S}}^{d-1}}|\mathsf{\Omega }\left(z\right)|{\int }_{{\mathbb{R}}^d}{2}^{kd}|\nabla \mathsf{\Phi }\left(2^k(x-z-\theta y)\right)|\mathrm{d}x\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & ={||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}{||\nabla \mathsf{\Phi }||}_{L^1\left({\mathbb{R}}^d\right)}{2}^k|y|,\hfill \end{array}$$

and thus we have (6) from which the result follows. □

Lemma 4. 

Let Ω be as in Lemma 3. Then:

$$\begin{array}{c}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}^{\left(k\right)}{f||}_{L^2\left({\mathbb{R}}^d\right)}\le C{2}^{-\alpha k}{||f||}_{L^2\left({\mathbb{R}}^d\right)}\end{array}$$

for some $0<\alpha <\frac{1}{2}$ and all $f\in L^2\left({\mathbb{R}}^d\right)$.

Proof. 

Firstly, by using Plancherel and Fubini theorems, we have:

$$\begin{array}{cc}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}^{\left(k\right)}f{||}_{L^2\left({\mathbb{R}}^d\right)}^2& ={\int }_{{\mathbb{R}}^d}\left({\int }_0^{\infty }{\left|({\left({\mathsf{\Psi }}^{\left(k\right)}\right)}_r\ast f)\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^d}{\left|({\left({\mathsf{\Psi }}^{\left(k\right)}\right)}_r\ast f)\left(x\right)\right|}^2\mathrm{d}x\right)\frac{\mathrm{d}r}{r}\hfill \\ \hfill & ={\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^d}|{\phi }_k{\left(r\xi \right)|}^2|\widehat{\lambda }{\left(r\xi \right)|}^2{|\widehat{f}\left(\xi \right)|}^2\mathrm{d}\xi \right)\frac{\mathrm{d}r}{r}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^d}{|\widehat{f}\left(\xi \right)|}^2\left({\int }_{2^{k-1}/|\xi |}^{2^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|\widehat{\lambda }\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}\xi \hfill \end{array}$$

(9)

Now, an application of Van der Corput’s Lemma gives:

$$\begin{array}{cc}\hfill {\int }_{2^{k-1}/|\xi |}^{2^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|\widehat{\lambda }\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}& \le C{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}|\mathsf{\Omega }\left(y\right)||\overline{\mathsf{\Omega }\left(z\right)}|\left|{\int }_{2^{k-1}/|\xi |}^{2^{k+1}/|\xi |}{e}^{-ir<y-z,\xi >}\frac{\mathrm{d}r}{r}\right|\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & \le C{2}^{-\tilde{\alpha }k}{\int }_{{\mathbb{S}}^{d-1}\times {\mathbb{S}}^{d-1}}|{\mathsf{\Omega }}_m\left(y\right)||\overline{{\mathsf{\Omega }}_m\left(z\right)}|{\left|<y-z,{\xi }^{\prime }\right|}^{-\tilde{\alpha }}\mathrm{d}\sigma (y\otimes z)\hfill \\ \hfill & \le C{2}^{-\tilde{\alpha }k}\hfill \end{array}$$

(10)

for some $0<\tilde{\alpha }<1$. Finally, by combining (9) and (10) we deduce our result. □

Now, we are ready to conclude the main result of this section. More precisely, in [13] it was shown that the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ is bounded for $p\ge 2$. In the following Theorem, we attain the boundedness for $\frac{3}{2}<p<2$.

Theorem 2. 

Let Ω be as in Lemma 3. Then:

$$\begin{array}{c}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}{f||}_{L^p\left({\mathbb{R}}^d\right)}\le C_p{||f||}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

for $\frac{3}{2}<p<2$.

Proof. 

Notice that:

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)& ={\left({\int }_0^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & ={\left({\int }_0^{\infty }{\left|\left(\widehat{f}\left(\xi \right)\widehat{\lambda }\left(r\xi \right)\right){}^{ˇ}\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & ={\left({\int }_0^{\infty }{\left|{\left(\sum _{k=0}^{\infty }\widehat{f}\left(\xi \right){\phi }_k\left(r\xi \right)\widehat{\lambda }\left(r\xi \right)\right)}^{\stackrel{ˇ}{}}\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le {\mathcal{I}}_{\mathsf{\Omega }}^{\left(0\right)}f\left(x\right)+\sum _{k=1}^{\infty }{\mathcal{I}}_{\mathsf{\Omega }}^{\left(k\right))}f\left(x\right).\hfill \end{array}$$

(11)

Lemma 3 guarantees the boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}^{\left(0\right)}$ for all $1<p<\infty$. So, we need only to take care of the sum on most right-hand side in (11). Interpolating between the $L^2\to L^2$ and $L^1\to L^{1,\infty }$ estimates in Lemmas 3 and 4 yields:

$$\begin{array}{c}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}^{\left(k\right))}{f||}_{L^p\left({\mathbb{R}}^d\right)}\le C_d{2}^{(\frac{2+2\alpha }{p}-2\alpha -1)k}{||f||}_{L^p\left({\mathbb{R}}^d\right)}\end{array}$$

(12)

for $k\ge 1$ and $1<p<2$. Finally, combining the fact that $g\left(\alpha \right)=\frac{2\alpha +2}{2\alpha +1}$ is decreasing on $(0,1/2)$ with (11) and (12) ends the proof. □

3. ${\mathit{L}}^{\mathbf{2}}$-Boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}$

In this section, we introduce a new proof of the $L^2$-boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}$. The main feature of our proof is inequality (24) which reveals the relationship between the spherical operators and the well-known parametric Marcinkiewicz integral operators which we derive by using the classical Hardy inequality. We start this section by deriving a multiplier formula for ${\mathcal{I}}_{\mathsf{\Omega }}$ and as a consequence of this derivation, we find a necessary and sufficient condition on $\mathsf{\Omega }$ which guarantee the $L^2$-boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}f$. We emphasis here that the proof of the next result follows the same lines of the one in [13] but in great detail.

Theorem 3. 

Suppose that $\mathsf{\Omega }\in L^1\left({\mathbb{S}}^{d-1}\right)$ satisfies (1). Then, the operator ${\mathcal{I}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbb{R}}^d\right)$ iff:

$$\begin{array}{c}\hfill \underset{\xi \in {\mathbb{S}}^{d-1}}{\sup }\left|{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}\left(\log \frac{1}{|<y-z,\xi >|}\right)\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\right|<\infty .\end{array}$$

(13)

Proof. 

In this proof, we follow the same lines as the one in [14]. By Plancherel and Fubini theorems, we have:

$$\begin{array}{cc}\hfill ||{\mathcal{I}}_{\mathsf{\Omega }}f{||}_2^2& ={\int }_{{\mathbb{R}}^d}\left({\int }_0^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^d}{\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-ry)\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\mathrm{d}x\right)\frac{\mathrm{d}r}{r}\hfill \\ \hfill & ={\int }_0^{\infty }\left({\int }_{{\mathbb{R}}^d}{\left|\widehat{f}\left(\xi \right){\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\mathrm{d}\xi \right)\frac{\mathrm{d}r}{r}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^d}{|\widehat{f}\left(\xi \right)|}^2\left({\int }_0^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}\xi \hfill \\ \hfill & :={\int }_{{\mathbb{R}}^d}{|\widehat{f}\left(\xi \right)|}^2\mathcal{M}\left(\xi \right)\mathrm{d}\xi .\hfill \end{array}$$

(14)

On the other hand, consider:

$$\begin{array}{cc}\hfill {\left|{\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2& =\left({\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right)\left(\overline{{\int }_{{\mathbb{S}}^{d-1}}{e}^{-irz·\xi }\mathsf{\Omega }\left(z\right)\mathrm{d}\sigma \left(z\right)}\right)\hfill \\ \hfill & =\left({\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right)\left({\int }_{{\mathbb{S}}^{d-1}}{e}^{irz·\xi }\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(z\right)\right)\hfill \\ \hfill & ={\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}{e}^{-ir<y-z,\xi >}\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right).\hfill \end{array}$$

(15)

By combining (14) and (15) and applying Lebesgue dominated convergence theorem, we obtain:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\xi \right)& =\underset{\eta \to 0^{+},N\to \infty }{\lim }{\int }_{\eta }^N{\left|{\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}\hfill \\ \hfill & =\underset{\eta \to 0^{+},N\to \infty }{\lim }{\int }_{\eta }^N\left({\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}{e}^{-ir<y-z,\xi >}\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\right)\frac{\mathrm{d}r}{r}\hfill \\ \hfill & =\underset{\eta \to 0^{+},N\to \infty }{\lim }{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{\eta }^N{e}^{-ir<y-z,\xi >}\frac{\mathrm{d}r}{r}\right)\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & =\underset{\eta \to 0^{+},N\to \infty }{\lim }{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{\eta }^N{e}^{-ir<y-z,\xi >}-\cos (r|\xi |)\frac{\mathrm{d}r}{r}\right)\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & =\underset{\eta \to 0^{+},N\to \infty }{\lim }{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{\eta |\xi |}^{N|\xi |}{e}^{-ir<y-z,{\xi }^{\prime }>}-\cos \left(r\right)\frac{\mathrm{d}r}{r}\right)\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & ={\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}\left(\log \frac{1}{|<y-z,{\xi }^{\prime }>|}-i\frac{\pi }{2}\mathrm{sgn}(<y-z,{\xi }^{\prime }>)\right)\mathsf{\Omega }\left(y\right)\overline{\mathsf{\Omega }\left(z\right)}\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right).\hfill \end{array}$$

(16)

The sufficiency of (13) is obvious. Assume that ${\mathcal{I}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbb{R}}^d\right)$ and notice that:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d}|\widehat{f}{\left(\xi \right)|}^2(||{\mathcal{I}}_{\mathsf{\Omega }}{||}^2-\mathcal{M}\left(\xi \right))\mathrm{d}\xi \ge 0.\end{array}$$

The fact that Fourier transform is an isometry on $L^2\left({\mathbb{R}}^d\right)$ enables us to find a function $f\in L^2\left({\mathbb{R}}^d\right)$ such that $\widehat{f}\left(\xi \right)={\left(2r\right)}^{-d/2}{\mathsf{\Pi }}_{k=1}^d{\mathbb{1}}_{[-r,r]}\left({\xi }_k\right)$ and so an application of Lebesgue differentiation theorem proves the necessity of (13). □

Remark 2. 

The following inclusions among Zygmund classes and Lebesgue spaces hold and are proper. For $q>1$ and $0<\alpha <\beta$, we have:

$$\begin{array}{c}{L}^q\left({\mathbb{S}}^{d-1}\right)\subset L\left(\log L\right)\left({\mathbb{S}}^{d-1}\right)\subset L^1\left({\mathbb{S}}^{d-1}\right),\hfill \\ L{\left(\log L\right)}^{\beta }\left({\mathbb{S}}^{d-1}\right)\subset L{\left(\log L\right)}^{\alpha }\left({\mathbb{S}}^{d-1}\right).\hfill \end{array}$$

The following lemma can be found in [15].

Lemma 5. 

Let $\mathsf{\Omega }\in L\left(\log L\right)\left({\mathbb{S}}^{d-1}\right)$ be satisfy (1). Then:

$$\begin{array}{c}\hfill \underset{\xi \in {\mathbb{S}}^{d-1}}{\sup }{\int }_0^{\infty }{\left|{\int }_{{\mathbb{S}}^{d-1}}{e}^{-iry·\xi }\mathsf{\Omega }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^2\frac{\mathrm{d}r}{r}<\infty .\end{array}$$

Remark 3. 

Equation (14) and Lemma 5 show that ${\mathcal{I}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbb{R}}^d\right)$ whenever $\mathsf{\Omega }\in L\left(\log L\right)\left({\mathbb{S}}^{d-1}\right)$. Therefore, by the relations among Zygmund classes it is natural to ask what is the optimal α such that ${\mathcal{I}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbb{R}}^d\right)$ whenever $\mathsf{\Omega }\in L{\left(\log L\right)}^{\alpha }\left({\mathbb{S}}^{d-1}\right)$.

As a first step in answering the inquiry in Remark 3, we need to recall the $L{\left(\log L\right)}^{\alpha }\left({\mathbb{S}}^{n-1}\right)$ decomposition.

Lemma 6. 

Suppose that for some $\alpha >0$, $\mathsf{\Omega }\in L{\left(\log L\right)}^{\alpha }\left({\mathbb{S}}^{n-1}\right)$ and satisfies the mean value zero condition. Then there exists a subset $\mathbb{D}$ of $\mathbb{N}$, a sequence $\{{\lambda }_m:m\in \mathbb{N}\cup \left\{0\right\}\}$ of nonnegative real numbers, and a sequence of functions $\{{\mathsf{\Omega }}_m:m\in \mathbb{D}\cup \left\{0\right\}\}$ in $L^1\left({\mathbb{S}}^{n-1}\right)$ such that:

(i) 

${\int }_{{\mathbb{S}}^{n-1}}{\mathsf{\Omega }}_m\left(y^{\prime }\right)\mathrm{d}\sigma \left(y^{\prime }\right)=0$ for $m\in \mathbb{D}\cup \left\{0\right\}$,

(ii) 

${\sup }_{m\in \left\{0\right\}\cup \mathbb{D}}{||\mathsf{\Omega }||}_{L^1\left({\mathbb{S}}^{d-1}\right)}<\infty$,

(iii) 

$||{\mathsf{\Omega }}_m{||}_2\le C{a}_m^2$ for $m\in \mathbb{D}\cup \left\{0\right\}$,

(iv) 

${\displaystyle \sum _{m\in \mathbb{D}\cup \left\{0\right\}}}{(m+1)}^{\alpha }{\lambda }_m<\infty$,

(v) 

$\mathsf{\Omega }={\displaystyle \sum _{m\in \mathbb{D}\cup \left\{0\right\}}}{\lambda }_m{\mathsf{\Omega }}_m$.

Theorem 4. 

Let $\mathsf{\Omega }\in L{\left(\log L\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$ be satisfy (1). Then ${\mathcal{I}}_{\mathsf{\Omega }}$ is bounded on $L^2\left({\mathbb{R}}^d\right)$. Moreover, the condition $\mathsf{\Omega }\in L{\left(\log L\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$ is optimal for ${\mathcal{I}}_{\mathsf{\Omega }}$ to be $L^2$-bounded.

Proof. 

We follow the same proof of Theorem 2 with a slight modification. By Lemma 6, we have:

$$\begin{array}{cc}\hfill {\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)& \le {\lambda }_0{\mathcal{I}}_{{\mathsf{\Omega }}_0}f\left(x\right)+\sum _{m\in \mathbb{D}}{\lambda }_m{\mathcal{I}}_{{\mathsf{\Omega }}_m}f\left(x\right)\hfill \\ \hfill & \le {\lambda }_0{\mathcal{I}}_{{\mathsf{\Omega }}_0}f\left(x\right)+\sum _{m\in \mathbb{D}}{\lambda }_m({\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(0\right)}f\left(x\right)+\sum _{k=1}^{\infty }{\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(k\right)}f\left(x\right)).\hfill \end{array}$$

(17)

Let:

$$\begin{array}{c}\hfill {\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(k\right)}f\left(x\right)={\left({\int }_0^{\infty }{\left|({\left({\mathsf{\Psi }}_m^{\left(k\right)}\right)}_r\ast f)\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)}^{\frac{1}{2}},\end{array}$$

where:

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_m^{\left(k\right)}=\left({\phi }_{m,k}\right){}^{ˇ}\ast \mathrm{d}{\lambda }_m={\mathsf{\Phi }}_{a_m^{-k}}\ast \mathrm{d}{\lambda }_m,\end{array}$$

and:

$$\begin{array}{c}\hfill {\left({\mathsf{\Psi }}_m^{\left(k\right)}\right)}_r(·)=r^{-d}{\mathsf{\Psi }}_m^{\left(k\right)}\left(\frac{·}{r}\right).\end{array}$$

In view of Lemma 3 and Remarks 3 and 6, we need only to show that:

$$\begin{array}{c}\hfill {\left|\left|\sum _{m\in \mathbb{D}}{\lambda }_m\sum _{k=1}^{\infty }{\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(k\right)}f\right|\right|}_{L^2\left({\mathbb{R}}^d\right)}\le {C||f||}_{L^2\left({\mathbb{R}}^d\right)}.\end{array}$$

(18)

It is clear that:

$$\begin{array}{cc}\hfill ||{\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(k\right)}f{||}_{L^2\left({\mathbb{R}}^d\right)}^2& ={\int }_{{\mathbb{R}}^d}\left({\int }_0^{\infty }{\left|({\left({\mathsf{\Psi }}_m^{\left(k\right)}\right)}_r\ast f)\left(x\right)\right|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^d}{|\widehat{f}\left(\xi \right)|}^2\left({\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|{\widehat{\lambda }}_m\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}\right)\mathrm{d}\xi \hfill \end{array}$$

(19)

Now, interpolating between the simple estimates:

$$\begin{array}{c}\hfill {\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|{\widehat{\lambda }}_m\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}\le C(m+1)\end{array}$$

(20)

and:

$$\begin{array}{cc}\hfill {\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|{\widehat{\lambda }}_m\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}& \le C{\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}{|{\widehat{\lambda }}_m\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}\hfill \\ \hfill & \le C{\int }_{{\mathbb{S}}^{d-1}}{\int }_{{\mathbb{S}}^{d-1}}|{\mathsf{\Omega }}_m\left(y\right)||\overline{{\mathsf{\Omega }}_m\left(z\right)}|\left|{\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}{e}^{-ir<y-z,\xi >}\frac{\mathrm{d}r}{r}\right|\mathrm{d}\sigma \left(y\right)\mathrm{d}\sigma \left(z\right)\hfill \\ \hfill & \le C{a}_m^{\alpha (1-k)}(m+1){\int }_{{\mathbb{S}}^{d-1}\times {\mathbb{S}}^{d-1}}|{\mathsf{\Omega }}_m\left(y\right)||\overline{{\mathsf{\Omega }}_m\left(z\right)}|{\left|<y-z,{\xi }^{\prime }\right|}^{-\alpha }\mathrm{d}\sigma (y\otimes z)\hfill \\ \hfill & \le C{a}_m^{\alpha (1-k)+4}(m+1),\hfill \end{array}$$

(21)

yields:

$$\begin{array}{c}\hfill {\int }_{a_m^{k-1}/|\xi |}^{a_m^{k+1}/|\xi |}|{\phi }_k{\left(r\xi \right)|}^2{|{\widehat{\lambda }}_m\left(r\xi \right)|}^2\frac{\mathrm{d}r}{r}\le C{2}^{-\alpha k}(m+1),\end{array}$$

(22)

and:

$$\begin{array}{c}\hfill ||{\mathcal{I}}_{{\mathsf{\Omega }}_m}^{\left(k\right)}{f||}_{L^2\left({\mathbb{R}}^d\right)}\le C{2}^{-\alpha k/2}{(m+1)}^{1/2}{||f||}_{L^2\left({\mathbb{R}}^d\right)}.\end{array}$$

(23)

Therefore, (18) and (23) complete the first part of the proof. To prove the optimality of the condition $\mathsf{\Omega }\in L{\left(\log L\right)}^{1/2}\left({\mathbb{S}}^{d-1}\right)$, assume that $\mathsf{\Omega }$ is a homogeneous function of degree 0 and recall that the parametric Marcinkiewicz integral operator [16] is given by:

$$\begin{array}{cc}\hfill {\mathcal{M}}_{\mathsf{\Omega }}^{\rho }f\left(x\right)& ={\left({\int }_0^{\infty }{\left|t^{-\rho }{\int }_{|y|\le t}f(x-y)\frac{\mathsf{\Omega }\left(y\right)}{{|y|}^{d-\rho }}\mathrm{d}y\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2},\hfill \end{array}$$

for $\rho >0$. It can be easily seen that:

$$\begin{array}{c}\hfill {\mathcal{M}}_{\mathsf{\Omega }}^{\rho }f\left(x\right)\le \frac{1}{\rho }{\mathcal{I}}_{\mathsf{\Omega }}f\left(x\right)\end{array}$$

(24)

for almost every $x\in {\mathbb{R}}^d$. To see this, we need the following version of Hardy inequality [17]:

$$\begin{array}{c}\hfill {\left({\int }_0^{\infty }{\left(\frac{1}{x}{\int }_0^x|F\left(t\right)|dt\right)}^p{x}^{\eta }dx\right)}^{\frac{1}{p}}\le \frac{p}{p-1-\eta }{\left({\int }_0^{\infty }{|F\left(t\right)|}^p{t}^{\eta }dt\right)}^{\frac{1}{p}},\end{array}$$

(25)

where F is a measurable function, $1<p<\infty$ and $\eta <p-1$. Switching to polar coordinates gives:

$$\begin{array}{cc}\hfill {\mathcal{M}}_{\mathsf{\Omega }}^{\rho }f\left(x\right)& ={\left({\int }_0^{\infty }{\left|t^{-\rho }{\int }_{|y|\le t}f(x-y)\frac{\mathsf{\Omega }\left(y\right)}{{|y|}^{d-\rho }}\mathrm{d}y\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}\hfill \\ \hfill & \le {\left({\int }_0^{\infty }{\left[\frac{1}{t}{\int }_0^t\left|{\int }_{{\mathbb{S}}^{d-1}}f(x-r{y}^{\prime })\frac{\mathsf{\Omega }\left(y^{\prime }\right)}{r^{1-\rho }}\mathrm{d}\sigma \left(y^{\prime }\right)\right|\mathrm{d}r\right]}^2{t}^{1-2\rho }\mathrm{d}t\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Thus, we get (24) by applying (25) with:

$$\begin{array}{cc}\hfill & F(x,r)={\int }_{{\mathbb{S}}^{d-1}}f(x-r{y}^{\prime })\frac{\mathsf{\Omega }\left(y^{\prime }\right)}{r^{1-\rho }}\mathrm{d}\sigma \left(y^{\prime }\right),\hfill \\ \hfill & \eta =1-2\rho .\hfill \end{array}$$

Finally, Al-Salman [18] proved that there exists $\mathsf{\Omega }\in {\cap }_{ϵ>0}L{\left(\log L\right)}^{1/2-ϵ}\left({\mathbb{S}}^{d-1}\right)$ such that ${\mathcal{M}}_{\mathsf{\Omega }}^{\rho }$ could not be bounded on $L^2\left({\mathbb{R}}^d\right)$. Therefore, this fact and (24) complete the proof. □

4. Conclusions

In conclusion, it is important to declare that we have successfully reproved and extended several known results connected with a class of spherical integral operators. More precisely, an inequality has been properly proposed to describe the relation between such class of operators with some well-known Marcinkiewicz integral operators. As a consequence of this inequality, we have obtained the optimality of the condition on $\mathsf{\Omega }$ for the $L^2$-boundedness of ${\mathcal{I}}_{\mathsf{\Omega }}f$ differently from the one proposed in [13]. The boundedness of the operator ${\mathcal{I}}_{\mathsf{\Omega }}f$ has been proved for $\frac{3}{2}<p<2$ whenever $\mathsf{\Omega }\in L^{\infty }\left({\mathbb{S}}^{d-1}\right)$. In future work, we aim to answer the question whether this range of values of p is optimal or can be improved by using different techniques.