Stability of the Comparison Problem for the Spherical Radon Transform

Abstract

This paper addresses the comparison problem for the spherical Radon transform, which was posed by Koldobsky, Roysdon, and Zvavitch. By applying Fourier analytic techniques, we derive linear stability results for both the affirmative and negative solutions to this problem. Furthermore, we investigate the linear separation in this framework.

1. Introduction

For a continuous function $f:S^{n-1}\to \mathbb{R}$, its spherical Radon transform $\mathcal{R}f$ is a bounded linear operator defined by

$$\mathcal{R}f\left(u\right)={\int }_{S^{n-1}\cap u^{\perp }}f\left(v\right)dv,\forall u\in S^{n-1},$$

where $S^{n-1}$ denotes the unit sphere, i.e., the surface of the unit ball $B_2^n$ in ${\mathbb{R}}^n$, and $u^{\perp }$ denotes the hyperplane through the origin orthogonal to the unit vector u. An interesting comparison problem for the spherical Radon transform asks: given two even, positive, and continuous functions f and g on $S^{n-1}$ with $n⩾3$ such that

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right),\forall u\in S^{n-1},$$

does it follow that

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}$$

(1)

holds for $p>0$? Here, ${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}$ denotes the $L^p$-norm of the function f on $S^{n-1}$ (see Section 2). This problem was raised by Koldobsky, Roysdon, and Zvavitch [1]. It was verified that the answer is affirmative for $p=1$ in [1], using an integration formula on $S^{n-1}$ (see, e.g., [2] (the formula (2.22))) and applying the Fubini’s theorem. However, the above problem does not necessarily hold for $p\ne 1$. In particular, by choosing $f={\parallel ·\parallel }_K^{-n+1}$, $g={\parallel ·\parallel }_L^{-n+1}$, and $p={\displaystyle \frac{n}{n-1}}$, problem (1) reduces to the classical Busemann–Petty problem in convex geometry, proposed in 1956 (see [3]). Here, ${\parallel ·\parallel }_K$ and ${\parallel ·\parallel }_L$ denote the Minkowski functionals of the origin-symmetric convex bodies K and L in ${\mathbb{R}}^n$ (see (4) for details), respectively. Specifically, does

$$V_{n-1}(K\cap u^{\perp })⩽V_{n-1}(L\cap u^{\perp }),\forall u\in S^{n-1}$$

(2)

imply $V\left(K\right)⩽V\left(L\right)$ for any origin-symmetric convex bodies K and L in ${\mathbb{R}}^n$? Here $V_{n-1}(·)$ and $V(·)$ denote the $(n-1)$-dimensional volume and n-dimensional volume, respectively. It has been shown that the answer to this problem is positive for $n⩽4$ and negative for $n⩾5$ (see, e.g., [4,5,6,7,8,9]). In fact, Lutwak [10] introduced the notion of intersection bodies, and showed that problem (2) is always true when K is an intersection body. Therefore, determining in which dimensions all origin-symmetric convex bodies are intersection bodies became the approach used in its solution. Hence, it is interesting to consider answers of problem (1) which can be viewed as a further extension of the Busemann–Petty problem. Based on the research ideas of Lutwak in the Busemann–Petty problem, a positive answer and a negative answer to the spherical comparison problem (1) were given by Koldobsky, Roysdon, and Zvavitch [1] through the use of Fourier analysis.

Simultaneously, slicing inequalities related to functions were also established in [1] based on the positive answer of problem (1). Moreover, this result reflects a close connection with Bourgain’s slicing problem (see [1,11,12]) concerning intersection bodies. Recently, Bourgain’s slicing problem was resolved by Guan [13] and subsequently by Klartag and Lehec [14] using Guan’s bound. The significant progress with respect to the slicing problems can also be found in [15,16,17,18,19,20,21,22], and the references therein. In addition, the slicing inequalities in [1] are closely related to the famous Oberlin–Stein type estimates (see, e.g., [23,24,25,26]) for the Radon transform. These findings sufficiently demonstrate that studying the spherical comparison problem (1) is a very meaningful work.

In [27], Koldobsky explored the linear stability and separation for the Busemann–Petty problem (2), which may be known as the stability and separation results of Lutwak’s connection (see, e.g., [10] (Theorem 10.1)). For more stable results of the comparison problem involving the volumes, one can refer to [28,29,30,31,32,33]. Perhaps of equal significance as the stability of the Busemann–Petty problem is to consider the stability of the comparison problem (1). Inspired by the work of Koldobsky [27], the main aim of this paper is to study the following linear stability problem: Let f and g be even, positive, and continuous functions on $S^{n-1}$ with $n⩾3$. For $\epsilon >0$ small enough and $p>0$, whether there exists a positive constant C such that

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}$$

imply

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}+C\epsilon ?$$

(3)

We mainly employ Fourier analytic techniques to investigate the linear stability problem (3). For convenience, we use $M_g$ and $m_g$ to denote the maximum and the minimum of the function g on $S^{n-1}$, respectively. And we use ${\omega }_n=\left(2{\pi }^{n/2}\right)/\left(\mathsf{\Gamma }(n/2)\right)$ to denote the surface area of the unit ball $B_2^n$, where $\mathsf{\Gamma }(·)$ is the Gamma function. In accordance with the Fourier transform, we first establish a stability result of the affirmative answer for the comparison problem (1) as follows:

Theorem 1.

Assume that f and g are even, positive, and continuous functions on the sphere $S^{n-1}$ such that for small enough $\epsilon >0$,

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}.$$

(i) 

If the function ${\left|x\right|}^{-1}{f}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$, for $p>1$, is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}+C_1\epsilon ,$$

where the constant $C_1>0$ depends only on n, i.e., $C_1=\sqrt{\pi }\mathsf{\Gamma }\left({\displaystyle \frac{n-1}{2}}\right)/\phantom{\sqrt{\pi }\mathsf{\Gamma }\left({\displaystyle \frac{n-1}{2}}\right)\mathsf{\Gamma }\left({\displaystyle \frac{n}{2}}\right)}\mathsf{\Gamma }\left({\displaystyle \frac{n}{2}}\right).$

(ii) 

If the function ${\left|x\right|}^{-1}{g}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$, for $0<p<1$, is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}+C_2\epsilon ,$$

where the constant $C_2>0$ depends on n, p and g, i.e., $C_2={\omega }_n^{1/\phantom{1p}p}{M}_g/\phantom{{\omega }_n^{1/\phantom{1p}p}{M}_g\left({\omega }_{n-1}{m}_g\right)}\left({\omega }_{n-1}{m}_g\right).$

Furthermore, we also prove the following stability result of the negative answer for comparison problem (1).

Theorem 2.

The following statements are true:

(i) 

Assume that $g^{p-1}$ with $p>1$ is an infinitely smooth strictly positive even function on $S^{n-1}$. If the function ${\left|x\right|}^{-1}{g}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$ is not a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then for small enough $\epsilon >0$, there is an infinitely smooth even function f on $S^{n-1}$ such that

$$\mathcal{R}g\left(u\right)⩽\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1},$$

but

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩾{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}+C_3\epsilon ,$$

where the constant $C_3>0$ depends on n, p, and g, i.e., $C_3=\left({\omega }_n^{1/\phantom{1p}p}/\phantom{{\omega }_n^{1/\phantom{1p}p}{\omega }_{n-1}}{\omega }_{n-1}\right){\left(m_g/\phantom{m_g{M}_g}{M}_g\right)}^{p-1}.$

(ii) 

Assume that $f^{p-1}$ with $0<p<1$ is an infinitely smooth strictly positive even function on $S^{n-1}$. If the function ${\left|x\right|}^{-1}{f}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$ is not a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then for small enough $\epsilon >0$, there is an infinitely smooth even function g on $S^{n-1}$ such that

$$\mathcal{R}g\left(u\right)⩽\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1},$$

but

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩾{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}+C_4\epsilon ,$$

where the constant $C_4>0$ depends on n and p, i.e., $C_4={\omega }_n^{1/\phantom{1p}p}/\phantom{{\omega }_n^{1/\phantom{1p}p}{\omega }_{n-1}}{\omega }_{n-1}.$

2. Background and Notations

In this section, we collect some basic concepts and preliminaries from convex geometry and harmonic analysis that are very important tools for this paper (see, e.g., [2,34,35]).

As we mentioned, we operate in the n-dimensional Euclidean space ${\mathbb{R}}^n$ equipped with the standard inner product $⟨·,·⟩$ and the norm $|·|$. The set $B_2^n=\{x\in {\mathbb{R}}^n:\left|x\right|⩽1\}$ is the unit ball and the set $S^{n-1}=\{x\in {\mathbb{R}}^n:\left|x\right|=1\}$ is the unit sphere. A convex body is a convex compact subset of ${\mathbb{R}}^n$ with a nonempty interior. Moreover, $K\subset {\mathbb{R}}^n$ is an origin-symmetric convex body if and only if the convex body K satisfies $K=-K$. A subset $M\subset {\mathbb{R}}^n$ is a star-shaped set with respect to the origin if every straight line passing through the origin crosses the boundary of M at exactly two points. We say that $M\subset {\mathbb{R}}^n$ is a star body if M is a compact star-shaped set about the origin and has a positive continuous radial function, ${\rho }_M\left(x\right):=max\{\lambda ⩾0:\lambda x\in M\}$, for any $x\in {\mathbb{R}}^n\setminus \left\{0\right\}$. It means that a star body M contains the origin in its interior. The Minkowski functional ${\parallel ·\parallel }_M:{\mathbb{R}}^n\setminus \left\{0\right\}\to (0,\infty )$ of a star body M is defined by

$${\parallel x\parallel }_M:=max\{\lambda ⩾0:x\in \lambda M\}.$$

(4)

It is easy to check that ${\rho }_M\left(x\right)={\parallel x\parallel }_M^{-1}$ for any $x\in {\mathbb{R}}^n\setminus \left\{0\right\}$. Obviously, the convex body containing the origin in its interior must be a star body.

For a measure metric space $(X,d,\mu )$ and $p\ne 0$, if a function $f:X\to \mathbb{R}$ satisfies

$${\int }_X{\left|f\left(x\right)\right|}^{p}d\mu \left(x\right)<\infty ,$$

then the function f belongs to $L^p\left(X\right)$. Thus the $L^p\left(X\right)$-norm of a function $f:X\to \mathbb{R}$ is defined by

$${\parallel f\parallel }_{L^p\left(X\right)}:={\left({\int }_X{\left|f\left(x\right)\right|}^{p}d\mu \left(x\right)\right)}^{{\displaystyle \frac{1}{p}}}.$$

For a function $\varphi \in L^1\left({\mathbb{R}}^n\right)$, its the Fourier transform $\widehat{\varphi }$ is defined by

$$\widehat{\varphi }\left(x\right):={\int }_{{\mathbb{R}}^n}\varphi \left(y\right)e^{-i⟨x,y⟩}dy.$$

In the following, we will review some basic properties for the Fourier transform of distributions (see [2,36,37]). We use $\mathfrak{D}\left({\mathbb{R}}^n\right)$ to denote the space of rapidly decreasing infinitely differentiable functions (test functions) on ${\mathbb{R}}^n$, and ${\mathfrak{D}}^{\prime }\left({\mathbb{R}}^n\right)$ to denote the space of distributions over $\mathfrak{D}\left({\mathbb{R}}^n\right)$. Every locally integrable real-valued function f on ${\mathbb{R}}^n$ with power growth at infinity represents a distribution acting by integration:

$$⟨f,\varphi ⟩={\int }_{{\mathbb{R}}^n}f\left(x\right)\varphi \left(x\right)dx,\mathrm{for}\mathrm{every}\varphi \in \mathfrak{D}\left({\mathbb{R}}^n\right).$$

The Fourier transform of a distribution $f\in {\mathfrak{D}}^{\prime }\left({\mathbb{R}}^n\right)$ is defined by $⟨\widehat{f},\varphi ⟩=⟨f,\widehat{\varphi }⟩$ for every test function $\varphi \in \mathfrak{D}\left({\mathbb{R}}^n\right)$. For any even test function $\varphi$, one has ${\left(\widehat{\varphi }\right)}^{\wedge }={\left(2\pi \right)}^n\varphi$. Moreover, it follows that for an even $\varphi \in \mathfrak{D}\left({\mathbb{R}}^n\right)$ and $f\in {\mathfrak{D}}^{\prime }\left({\mathbb{R}}^n\right)$, $⟨\widehat{f},\widehat{\varphi }⟩={\left(2\pi \right)}^n⟨f,\varphi ⟩$, which can be regarded as Parseval’s formula on ${\mathbb{R}}^n$. It is well-known that for a real-valued even test function $\varphi$, its Fourier transform $\widehat{\varphi }$ is also an even real-valued function. A distribution f is called positive definite if its Fourier transform $\widehat{f}$ is a positive distribution, i.e., $⟨\widehat{f},\varphi ⟩\ge 0$ for each non-negative test function $\varphi$.

Let $C\left(S^{n-1}\right)$ be the set of all continuous functions on $S^{n-1}$ and $C_e^{\infty }\left(S^{n-1}\right)$ be the set of all infinitely smooth even functions in $C\left(S^{n-1}\right)$. For $f\in C\left(S^{n-1}\right)$ and $0<p<n$, the extension $f·r^{-p}$, with $r>0$, of f is a homogeneous continuous function of degree $-p$ on ${\mathbb{R}}^n\setminus \left\{0\right\}$, that is,

$$(f·r^{-p})\left(x\right)={\left|x\right|}^{-p}f\left({\displaystyle \frac{x}{\left|x\right|}}\right),\forall x\in {\mathbb{R}}^n\setminus \left\{0\right\}.$$

Clearly, this function $f·r^{-p}$ is locally integrable on ${\mathbb{R}}^n$ and represents a distribution due to $0<p<n$. By Lemma 3.16 in [2], if f is an infinitely smooth function, then there exists a function $g\in C_e^{\infty }\left(S^{n-1}\right)$ such that

$${(f·r^{-p})}^{\wedge }=g·r^{-n+p}.$$

(5)

This fact implies that the Fourier transform of $f·r^{-p}$ is an infinitely smooth function on ${\mathbb{R}}^n\setminus \left\{0\right\}$.

In addition, the connection between the Fourier transform and the spherical Radon transform about homogeneous functions has been proven as follows (see, e.g., [2] (Lemma 3.7)): Suppose that $f\in C\left(S^{n-1}\right)$ has an even homogeneous extension $f·r^{-n+1}$ of degree $-n+1$ on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then the Fourier transform of $f·r^{-n+1}$ is an even homogeneous continuous function of degree $-1$ on ${\mathbb{R}}^n\setminus \left\{0\right\}$ such that

$$\mathcal{R}f\left(u\right)={\int }_{S^{n-1}\cap u^{\perp }}f\left(v\right)dv={\displaystyle \frac{1}{\pi }}{(f·r^{-n+1})}^{\wedge }\left(u\right),\forall u\in S^{n-1}.$$

(6)

3. The Stability and Separation Results in the Affirmative Direction

In this section, we will mainly prove a stability result of the affirmative answer for the comparison problem related to the spherical Radon transform, i.e., Theorem 1. In order to achieve this goal, we present some important results as follows: Firstly, we list a Parseval’s formula with respect to continuous functions according to Corollary 3.23 of [2], which can also be found in [1].

Lemma 1.

Suppose that $g\in C\left(S^{n-1}\right)$ and $g·r^{-1}$ with $r>0$ is a positive definite distribution. Then there exists a non-negative finite Borel measure ${\mu }_g$ on $S^{n-1}$ such that for each $f\in C\left(S^{n-1}\right)$,

$${\int }_{S^{n-1}}{(f·r^{-n+1})}^{\wedge }\left(u\right)d{\mu }_g\left(u\right)={\left(2\pi \right)}^n{\int }_{S^{n-1}}f\left(\theta \right)g\left(\theta \right)d\theta .$$

Furthermore, the following result will be useful in proving Theorem 1, as stated in [38] (Theorem 1).

Lemma 2.

Let M be an origin-symmetric star body. Then for any $u\in S^{n-1}$,

$$V_{n-1}(M\cap u^{\perp })={\displaystyle \frac{1}{\pi (n-1)}}{(\parallel x\parallel }_M^{-n+1}{)}^{\wedge }\left(u\right),\forall x\in {\mathbb{R}}^n\setminus \left\{0\right\}.$$

Note that the Minkowski functional ${\parallel ·\parallel }_M$ is homogeneous of degree 1 by definition (4), i.e., ${\parallel \lambda x\parallel }_M=\lambda {\parallel x\parallel }_M$ for any $x\in {\mathbb{R}}^n\setminus \left\{0\right\}$ and $\lambda >0$. Thus, by Lemma 2 and the fact that ${\parallel u\parallel }_{B_2^n}=1$ for any $u\in S^{n-1}$, we can verify that for any $x\in {\mathbb{R}}^n\setminus \left\{0\right\}$ and any $u\in S^{n-1}$, if we choose $r=\left|x\right|>0$, then the following identity holds:

$${\left(r^{-n+1}\right)}^{\wedge }\left(u\right)=\pi (n-1)V_{n-1}(B_2^n\cap u^{\perp })=\pi (n-1)V_{n-1}\left(B_2^{n-1}\right)={\displaystyle \frac{2{\pi }^{\frac{n+1}{2}}}{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}}.$$

(7)

Based on the above facts, we now prove Theorem 1.

Proof of Theorem 1. 

Let us start with the proof of part (i). Since $f,g$ are even and continuous functions on $S^{n-1}$, we can respectively extend f and g to even homogeneous functions, denoted as $f·r^{-n+1}$ and $g·r^{-n+1}$, of degree $-n+1$ on ${\mathbb{R}}^n\setminus \left\{0\right\}$, where $r>0$. Then by (6), we have that for any $u\in S^{n-1}$,

$${\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)=\pi \mathcal{R}f\left(u\right)\mathrm{and}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)=\pi \mathcal{R}g\left(u\right).$$

Hence, for small enough $\epsilon >0$, the assumption

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}$$

is equivalent to

$${\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)⩽{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)+\pi \epsilon ,\forall u\in S^{n-1}.$$

(8)

Because the function $f^{p-1}·r^{-1}$ with $p>1$ is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, Lemma 1 gives that there is a non-negative finite Borel measure ${\mu }_{f^{p-1}}$ on $S^{n-1}$ such that

$${∥f∥}_{L^p\left(S^{n-1}\right)}^p={\int }_{S^{n-1}}f\left(\theta \right)·f^{p-1}\left(\theta \right)d\theta ={\left(2\pi \right)}^{-n}{\int }_{S^{n-1}}{\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{f^{p-1}}\left(u\right).$$

Thus, combining with (8), (7), and Lemma 1 again, we have

$$\begin{array}{cc}\hfill {∥f∥}_{L^p\left(S^{n-1}\right)}^p⩽& {\left(2\pi \right)}^{-n}{\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{f^{p-1}}\left(u\right)\hfill \\ \hfill & +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}·{\left(2\pi \right)}^{-n}{\int }_{S^{n-1}}{\left(r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{f^{p-1}}\left(u\right)\hfill \\ \hfill =& {\int }_{S^{n-1}}g\left(\theta \right)f^{p-1}\left(\theta \right)d\theta +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta .\hfill \end{array}$$

(9)

On the other hand, according to the Hölder’s inequality, we have that for $p>1$,

$$\begin{array}{cc}\hfill {\int }_{S^{n-1}}g\left(\theta \right)f^{p-1}\left(\theta \right)d\theta ⩽& {\left({\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ \hfill =& {∥g∥}_{L^p\left(S^{n-1}\right)}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1},\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta ⩽& {\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}{\left({\int }_{S^{n-1}}d\theta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ \hfill =& {\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}{\left({\displaystyle \frac{2{\pi }^{\frac{n}{2}}}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}\right)}^{{\displaystyle \frac{1}{p}}}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}\hfill \\ \hfill ⩽& {\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}{\displaystyle \frac{2{\pi }^{\frac{n}{2}}}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}\epsilon \hfill \\ \hfill =& {\displaystyle \frac{\sqrt{\pi }\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}\epsilon ,\hfill \end{array}$$

(11)

where we have used $p>1$ and a fact that $(2{\pi }^{{\displaystyle \frac{n}{2}}})/\mathsf{\Gamma }({\displaystyle \frac{n}{2}})>1$ is the surface area of the unit ball $B_2^n$. Therefore, substituting (10) and (11) into (9), we have

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{\sqrt{\pi }\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}\epsilon .$$

We next prove part (ii). Since $g^{p-1}·r^{-1}$ with $0<p<1$ is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$ with $r>0$, we apply Lemma 1 and obtain that there exists a non-negative finite Borel measure ${\mu }_{g^{p-1}}$ on $S^{n-1}$ satisfying

$${\int }_{S^{n-1}}{\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{g^{p-1}}\left(u\right)={\left(2\pi \right)}^n{\int }_{S^{n-1}}f\left(\theta \right)g^{p-1}\left(\theta \right)d\theta ,$$

$${\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{g^{p-1}}\left(u\right)={\left(2\pi \right)}^n{\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta ,$$

and

$${\int }_{S^{n-1}}d{\mu }_{g^{p-1}}\left(u\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n+1}{2}}}}{\int }_{S^{n-1}}{\left(r^{-n+1}\right)}^{\wedge }\left(u\right)d{\mu }_{g^{p-1}}\left(u\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n+1}{2}}}}{\left(2\pi \right)}^n{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta ,$$

where we also used the formula (7). Hence, integrating both sides of the inequality (8) over $S^{n-1}$ with respect to the measure ${\mu }_{g^{p-1}}$, we have

$${\int }_{S^{n-1}}f\left(\theta \right)g^{p-1}\left(\theta \right)d\theta ⩽{\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta +\pi \epsilon ·{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n+1}{2}}}}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta .$$

(12)

On the other hand, the Hölder’s inequality gives that for $0<p<1$,

$${\int }_{S^{n-1}}f\left(\theta \right)g^{p-1}\left(\theta \right)d\theta ⩾{\left({\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{p-1}{p}}}.$$

Combining with (12) and ${\omega }_{n-1}=\left(2{\pi }^{(n-1)/2}\right)/\left(\mathsf{\Gamma }((n-1)/2)\right)$, it follows that

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{\epsilon }{{\omega }_{n-1}}}{∥g∥}_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta .$$

(13)

We now estimate the second term in (13). Because of $g>0$ and the continuity of the function g on $S^{n-1}$, we have that there exist the maximum $M_g$ and the minimum $m_g$ of g on $S^{n-1}$ such that $0<m_g⩽g⩽M_g$. Then

$$\begin{array}{cc}\hfill {∥g∥}_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta & ={\left({\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1-p}{p}}}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta \hfill \\ \hfill & ⩽M_g^{1-p}{\left({\int }_{S^{n-1}}d\theta \right)}^{{\displaystyle \frac{1-p}{p}}}{m}_g^{p-1}{\int }_{S^{n-1}}d\theta \hfill \\ \hfill & ={\left({\displaystyle \frac{M_g}{m_g}}\right)}^{1-p}{\left({\int }_{S^{n-1}}d\theta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ⩽{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}{M}_g}{m_g}}.\hfill \end{array}$$

This, together with (13), gives

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}{M}_g}{{\omega }_{n-1}{m}_g}}\epsilon .$$

Consequently, we complete the proof of Theorem 1. □

In [27], Koldobsky also considered the linear separation in the well-known Busemann–Petty problem (2). The comparison problem (1) as an analytical version of the problem (2), naturally motivates the study of linear separation related to problem (1). We state a linear separation result as follows.

Theorem 3.

Assume that $f,g$ are even, continuous, and positive functions on $S^{n-1}$ such that for small enough $\epsilon >0$,

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)-\epsilon ,\forall u\in S^{n-1}.$$

(i) 

If ${\left|x\right|}^{-1}{f}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$ with $p>1$ is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}-C_5\epsilon ,$$

where the constant $C_5>0$ depends on n, p and f, i.e., $C_5=\left({\omega }_n^{1/\phantom{1p}p}/\phantom{{\omega }_n^{1/\phantom{1p}p}{\omega }_{n-1}}{\omega }_{n-1}\right){\left(m_f/\phantom{m_f{M}_f}{M}_f\right)}^{p-1}.$

(ii) 

If ${\left|x\right|}^{-1}{g}^{p-1}\left({\displaystyle \frac{x}{\left|x\right|}}\right)$ with $0<p<1$ is a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, then

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}-C_1\epsilon .$$

Proof. 

The proof goes along the same lines as that of Theorem 1. For case (i), we have that for $p>1$,

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}^p⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}{\parallel f\parallel }_{L^p\left(S^{n-1}\right)}^{p-1}-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)\epsilon }{2{\pi }^{\frac{n-1}{2}}}}{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta .$$

Thus,

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}-{\displaystyle \frac{\epsilon }{{\omega }_{n-1}}}{\parallel f\parallel }_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta .$$

Since $f>0$ on $S^{n-1}$ and $f\in C\left(S^{n-1}\right)$, there exist the maximum $M_f$ and the minimum $m_f$ of f on $S^{n-1}$ such that $0<m_f⩽f⩽M_f$. Then for $p>1$,

$$\begin{array}{cc}\hfill {∥f∥}_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta & ={\left({\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1-p}{p}}}{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta \hfill \\ \hfill & ⩾{\left({\displaystyle \frac{m_f}{M_f}}\right)}^{p-1}{\left({\int }_{S^{n-1}}d\theta \right)}^{{\displaystyle \frac{1}{p}}}={\left({\displaystyle \frac{m_f}{M_f}}\right)}^{p-1}{\omega }_n^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Hence,

$${\parallel f\parallel }_{L^p\left(S^{n-1}\right)}⩽{\parallel g\parallel }_{L^p\left(S^{n-1}\right)}-{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}}{{\omega }_{n-1}}}{\left({\displaystyle \frac{m_f}{M_f}}\right)}^{p-1}\epsilon .$$

For case (ii), we have that for $0<p<1$,

$${∥f∥}_{L^p\left(S^{n-1}\right)}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}^p-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta .$$

Meanwhile, the Hölder’s inequality gives

$$\begin{array}{cc}\hfill {\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta & ⩾{\left({\int }_{S^{n-1}}d\theta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{p-1}{p}}}\hfill \\ \hfill & ⩾{\displaystyle \frac{2{\pi }^{\frac{n}{2}}}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}.\hfill \end{array}$$

Then

$${∥f∥}_{L^p\left(S^{n-1}\right)}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}^p-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}{\displaystyle \frac{2{\pi }^{\frac{n}{2}}}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}\epsilon .$$

Hence

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩽{∥g∥}_{L^p\left(S^{n-1}\right)}-{\displaystyle \frac{\sqrt{\pi }\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{\mathsf{\Gamma }\left(\frac{n}{2}\right)}}\epsilon .$$

This completes the proof of Theorem 3. □

4. The Stability and Separation Results in the Negative Direction

In this section, we will consider the linear stability and separation related to a negative answer of comparison problem (1) in [1]. Firstly, we introduce the following spherical version of Parseval’s formula by applying Lemma 3.22 of [2], which can also be found in [1].

Lemma 3.

Let $0<p<n$. If $f,g\in C^{\infty }\left(S^{n-1}\right)$ are even functions, then for $r>0$,

$${\int }_{S^{n-1}}{\left(f·r^{-p}{)}^{\wedge }\left(u\right)(g·r^{-n+p}\right)}^{\wedge }\left(u\right)du={\left(2\pi \right)}^n{\int }_{S^{n-1}}f\left(\theta \right)g\left(\theta \right)d\theta .$$

We next provide the proof of the stability result in Theorem 2.

Proof of Theorem 2. 

Let us start with the proof of part (i). Assume that $g^{p-1}$ with $p>1$ is an even infinitely smooth strictly positive function on $S^{n-1}$. By (5), there exists a function $h\in C_e^{\infty }\left(S^{n-1}\right)$ such that

$${(g^{p-1}·r^{-1})}^{\wedge }=h·r^{-n+1},\mathrm{with}r>0.$$

Since $g^{p-1}·r^{-1}$ is not a positive definite distribution on ${\mathbb{R}}^n\setminus \left\{0\right\}$, there is a symmetric open set $\Omega \subset S^{n-1}$ such that $h<0$ on $\Omega$. Let $\varphi$ be an infinitely smooth even non-negative function on $S^{n-1}$ supported in $\Omega$. Then, by (5) again, we have that for some function $\psi \in C_e^{\infty }\left(S^{n-1}\right)$,

$${\left(\varphi ·r^{-1}\right)}^{\wedge }=\psi ·r^{-n+1}.$$

Then

$${\left(\psi ·r^{-n+1}\right)}^{\wedge }={\left({\left(\varphi ·r^{-1}\right)}^{\wedge }\right)}^{\wedge }={\left(2\pi \right)}^n(\varphi ·r^{-1}).$$

(14)

We now construct an even function f on $S^{n-1}$ defined by

$$f\left(\theta \right)=g\left(\theta \right)-\psi \left(\theta \right)\delta +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ,\forall \theta \in S^{n-1},$$

(15)

where $\delta ,\epsilon >0$ are small enough such that $f>0$ everywhere on $S^{n-1}$. Clearly, $f\in C^{\infty }\left(S^{n-1}\right)$. We extend the functions $f,g,\psi$ to ${\mathbb{R}}^n\setminus \left\{0\right\}$ such that they have the homogeneity of degree $-n+1$. Then combining with the Formula (15), we have

$$f·r^{-n+1}=g·r^{-n+1}-\left(\psi ·r^{-n+1}\right)\delta +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}{r}^{-n+1}\epsilon .$$

It follows that for any $u\in S^{n-1}$,

$$\begin{array}{cc}\hfill {\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)& ={\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)-{\left(2\pi \right)}^n\varphi \left(u\right)r^{-1}·\delta +\pi \epsilon \hfill \\ \hfill & ⩽{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)+\pi \epsilon ,\hfill \end{array}$$

(16)

where we have used (14), (7), and $\varphi \left(u\right)⩾0$. Hence by (6), we have

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}.$$

(17)

Moreover, the Formula (16) also gives

$${(f·r^{-n+1})}^{\wedge }\left(u\right)⩾{(g·r^{-n+1})}^{\wedge }\left(u\right)-{\left(2\pi \right)}^n\varphi \left(u\right)r^{-1}\delta .$$

In this case, let $\delta \to 0^{+}$, then using (6) again, we have

$$\mathcal{R}f\left(u\right)⩾\mathcal{R}g\left(u\right),\forall u\in S^{n-1}.$$

(18)

Therefore, the formulas (17) and (18) give that for small enough $\epsilon >0$,

$$\mathcal{R}g\left(u\right)⩽\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}.$$

On the other hand, multiplying (16) by ${(g^{p-1}·r^{-1})}^{\wedge }$ and then integrating the equality over $S^{n-1}$, we have

$$\begin{array}{cc}\hfill & {\int }_{S^{n-1}}{\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill =& {\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du-{\left(2\pi \right)}^n{r}^{-1}\delta {\int }_{S^{n-1}}\varphi \left(u\right){\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill & +\pi \epsilon {\int }_{S^{n-1}}{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill ⩾& {\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du+\pi \epsilon {\int }_{S^{n-1}}{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill =& {\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill & +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{\left(r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(g^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du,\hfill \end{array}$$

where we have used the Formula (7). This, together with Lemma 3, gives

$${\int }_{S^{n-1}}f\left(\theta \right)g^{p-1}\left(\theta \right)d\theta ⩾{\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta .$$

(19)

Moreover, by using Hölder’s inequality, we have that for $p>1$,

$${\int }_{S^{n-1}}f\left(\theta \right)g^{p-1}\left(\theta \right)d\theta ⩽{\left({\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{S^{n-1}}{g}^p\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{p-1}{p}}}.$$

(20)

Hence, by (19) and (20), we have

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩾{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {∥g∥}_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta .$$

(21)

Since for $p>1$,

$${∥g∥}_{L^p\left(S^{n-1}\right)}^{1-p}{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta ⩾{\left({\displaystyle \frac{m_g}{M_g}}\right)}^{p-1}{\omega }_n^{{\displaystyle \frac{1}{p}}},$$

this, together with (21), gives that for small enough $\epsilon >0$,

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩾{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}}{{\omega }_{n-1}}}{\left({\displaystyle \frac{m_g}{M_g}}\right)}^{p-1}\epsilon .$$

Now, we prove part (ii). The proof follows along the same lines of the proof of part (i). Since $f^{p-1}·r^{-1}$, with $0<p<1$ and $r>0$, is not positive definite on ${\mathbb{R}}^n\setminus \left\{0\right\}$, there is a symmetric open set $\Omega \subset S^{n-1}$ such that ${\left(f^{p-1}·r^{-1}\right)}^{\wedge }<0$ on $\Omega$. Choose the functions $\varphi$ and $\psi$ as defined in the proof of part (i). Then we define an even function g on $S^{n-1}$ by

$$g\left(\theta \right)=f\left(\theta \right)+\psi \left(\theta \right)\delta -{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ,\forall \theta \in S^{n-1},$$

where $\delta ,\epsilon >0$ are small enough such that $g>0$ everywhere on $S^{n-1}$.

Based on the above facts and similar to the proof of (i), we have that for small enough $\epsilon >0$,

$$\mathcal{R}g\left(u\right)⩽\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)+\epsilon ,\forall u\in S^{n-1}.$$

Moreover,

$$\begin{array}{cc}\hfill & {\int }_{S^{n-1}}{\left(g·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill =& {\int }_{S^{n-1}}{\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du+{\left(2\pi \right)}^n{r}^{-1}\delta {\int }_{S^{n-1}}\varphi \left(u\right){\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill & -{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{\left(r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill ⩽& {\int }_{S^{n-1}}{\left(f·r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du\hfill \\ \hfill & -{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{\left(r^{-n+1}\right)}^{\wedge }\left(u\right)·{\left(f^{p-1}·r^{-1}\right)}^{\wedge }\left(u\right)du.\hfill \end{array}$$

Hence, by Lemma 3, we have

$${\int }_{S^{n-1}}g\left(\theta \right)f^{p-1}\left(\theta \right)d\theta ⩽{\int }_{S^{n-1}}{f}^p\left(\theta \right)d\theta -{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon {\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta .$$

(22)

Therefore, applying the Hölder’s inequality and the Formula (22), we infer that for $0<p<1$,

$${∥g∥}_{L^p\left(S^{n-1}\right)}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}⩽{∥f∥}_{L^p\left(S^{n-1}\right)}^p-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}{\omega }_n^{{\displaystyle \frac{1}{p}}}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}\epsilon .$$

Consequently,

$${∥f∥}_{L^p\left(S^{n-1}\right)}⩾{∥g∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}}{{\omega }_{n-1}}}\epsilon .$$

Thus, we complete the proof of Theorem 2. □

Furthermore, we also establish the following linear separation result of a negative answer related to problem (1) in [1].

Theorem 4.

The following statements are true:

(i) 

Under the assumptions of (i) of Theorem 2, there exists a function $f\in C_e^{\infty }\left(S^{n-1}\right)$ such that for small enough $\epsilon >0$,

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)-\epsilon ,\forall u\in S^{n-1},$$

but

$${∥g∥}_{L^p\left(S^{n-1}\right)}⩽{∥f∥}_{L^p\left(S^{n-1}\right)}+C_6\epsilon ,withp>1,$$

where $C_6={\omega }_n^{1/\phantom{1p}p}/\phantom{{\omega }_n^{1/\phantom{1p}p}{\omega }_{n-1}}{\omega }_{n-1}.$

(ii) 

Under the assumptions of (ii) of Theorem 2, there exists a function $g\in C_e^{\infty }\left(S^{n-1}\right)$ such that for small enough $\epsilon >0$,

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)-\epsilon ,\forall u\in S^{n-1},$$

but

$${∥g∥}_{L^p\left(S^{n-1}\right)}⩽{∥f∥}_{L^p\left(S^{n-1}\right)}+C_7\epsilon ,with0<p<1,$$

where $C_7=\left({\omega }_n^{1/\phantom{1p}p}/\phantom{{\omega }_n^{1/\phantom{1p}p}{\omega }_{n-1}}{\omega }_{n-1}\right){\left(M_f/\phantom{M_f{m}_f}{m}_f\right)}^{1-p}.$

Proof. 

Note that we define the functions $\varphi$ and $\psi$ in the same manner as in the proof of Theorem 2. For case (i), we can construct a function $f\in C_e^{\infty }\left(S^{n-1}\right)$ defined by

$$f\left(\theta \right)=g\left(\theta \right)-\psi \left(\theta \right)\delta -{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ,\forall \theta \in S^{n-1}$$

for small enough $\delta ,\epsilon >0$. Similar to the proof of Theorem 2, it follows that

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)-\epsilon ,\forall u\in S^{n-1}.$$

Moreover, by the Hölder’s inequality, we have that for $p>1$,

$$\begin{array}{cc}\hfill {∥f∥}_{L^p\left(S^{n-1}\right)}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}& ⩾{∥g∥}_{L^p\left(S^{n-1}\right)}^p-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ·{\int }_{S^{n-1}}{g}^{p-1}\left(\theta \right)d\theta \hfill \\ \hfill & ⩾{∥g∥}_{L^p\left(S^{n-1}\right)}^p-{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}}{{\omega }_{n-1}}}{∥g∥}_{L^p\left(S^{n-1}\right)}^{p-1}\epsilon .\hfill \end{array}$$

Hence

$${∥g∥}_{L^p\left(S^{n-1}\right)}⩽{∥f∥}_{L^p\left(S^{n-1}\right)}+C_6\epsilon .$$

For case (ii), we define a function $g\in C_e^{\infty }\left(S^{n-1}\right)$ as

$$g\left(\theta \right)=f\left(\theta \right)+\psi \left(\theta \right)\delta +{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ,\forall \theta \in S^{n-1}$$

for small enough $\delta ,\epsilon >0$. Then, according to the idea of the proof for Theorem 2, we have

$$\mathcal{R}f\left(u\right)⩽\mathcal{R}g\left(u\right)-\epsilon ,\forall u\in S^{n-1},$$

and for $0<p<1$,

$${∥g∥}_{L^p\left(S^{n-1}\right)}{∥f∥}_{L^p\left(S^{n-1}\right)}^{p-1}⩽{∥f∥}_{L^p\left(S^{n-1}\right)}^p+{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}\epsilon ·{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta .$$

Hence, for $0<p<1$,

$$\begin{array}{cc}\hfill {∥g∥}_{L^p\left(S^{n-1}\right)}& ⩽{∥f∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{2{\pi }^{\frac{n-1}{2}}}}{∥f∥}_{L^p\left(S^{n-1}\right)}^{1-p}\epsilon ·{\int }_{S^{n-1}}{f}^{p-1}\left(\theta \right)d\theta \hfill \\ & ⩽{∥f∥}_{L^p\left(S^{n-1}\right)}+{\displaystyle \frac{{\omega }_n^{\frac{1}{p}}}{{\omega }_{n-1}}}{\left({\displaystyle \frac{M_f}{m_f}}\right)}^{1-p}\epsilon .\hfill \end{array}$$

This completes the proof. □