A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Abstract

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

1. Introduction and Preliminaries

The fractional Laplacian [1,2], for $0<s<2$ is defined as:

$${(-\mathrm{\Delta })}^{{\displaystyle \frac{s}{2}}}f\left(x\right)=\gamma (n,s)\underset{\epsilon \to 0}{lim}{\int }_{{\mathbb{R}}^n\setminus B(0,\epsilon )}{\displaystyle \frac{f\left(x\right)-f(x+y)}{{\left|y\right|}^{n+s}}}dy,$$

where $B(0,\epsilon )$ denotes the ball in ${\mathbb{R}}^n$ centered at the origin with radius $\epsilon$ and

$$\mathrm{\Delta }={\displaystyle \frac{{\partial }^2}{\partial x_1^2}}+\cdots +{\displaystyle \frac{{\partial }^2}{\partial x_n^2}}$$

is the Laplacian in ${\mathbb{R}}^n$. The constant $\gamma (n,s)$ is a normalization factor that ensures proper scaling of the operator, given explicitly by:

$$\gamma (n,s)={\left({\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1-cos{y}_1}{{\left|y\right|}^{n+s}}}dy\right)}^{-1}={\displaystyle \frac{2^s\mathrm{\Gamma }\left(\frac{n+s}{2}\right)}{{\pi }^{n/2}\left|\mathrm{\Gamma }\left(-\frac{s}{2}\right)\right|}}.$$

The fractional Laplacian is fundamental in modeling nonlocal diffusion and long-range interactions, which are observed in phenomena such as anomalous diffusion [3,4,5,6,7] and $\alpha$-stable Lévy processes [8,9,10]. There are more than ten equivalent definitions of the fractional Laplacian, depending on the function space under consideration [11]. For instance, it can also be defined as a Fourier multiplier:

$${(-\mathrm{\Delta })}^{{\displaystyle \frac{s}{2}}}f\left(x\right)={\mathcal{F}}^{-1}{\left(\right|\xi |}^s\mathcal{F}f)\left(x\right),$$

where $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ denote the Fourier transform and its inverse.

A particularly useful equivalent definition of the fractional Laplacian involves the spherical mean-value operator, expressed as ([2], Proposition 2.11)

$${(-\mathrm{\Delta })}^{{\displaystyle \frac{s}{2}}}f\left(x\right)={\sigma }_{n-1}\gamma (n,s){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}_{\tau }f\left(x\right)}{{\tau }^{1+s}}}d\tau ,$$

(1)

where the spherical mean-value operator ${\mathcal{M}}_{t}f\left(x\right)$ is defined by [12]

$${\mathcal{M}}_{t}f\left(x\right)={\displaystyle \frac{1}{{\sigma }_{n-1}}}{\int }_{{\mathbb{S}}^{n-1}}f(x-t\sigma )d\sigma ,t>0.$$

(2)

Here, ${\mathbb{S}}^{n-1}$ is the unit sphere in ${\mathbb{R}}^n$ with surface area:

$${\sigma }_{n-1}={\displaystyle \frac{2{\pi }^{n/2}}{\mathrm{\Gamma }(n/2)}},$$

and $d\sigma$ represents the $(n-1)$-dimensional Lebesgue measure on ${\mathbb{S}}^{n-1}$.

The integral representation (1) is especially useful for developing efficient numerical schemes aimed at discretizing the fractional Laplacian. Given its nonlocal nature, an algorithm based on spherical means was proposed in [13] to efficiently discretize the multidimensional fractional Laplacian. Furthermore, using the spherical surface operator ${\mathcal{M}}_{y}f\left(x\right)$, we can define the local Blaschke-Privalov Laplacian as:

$$\tilde{\mathrm{\Delta }}f\left(x\right)=2n\underset{r\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}_{r}f\left(x\right)}{r^2}},$$

whenever the limit on the right exists. The notation $\tilde{\mathrm{\Delta }}f$ is used to distinguish this operator from the standard Laplacian $\mathrm{\Delta }$. For $f\in C^2\left(\mathbb{R}\right)$, we have:

$$\tilde{\mathrm{\Delta }}f=\mathrm{\Delta }f\mathrm{on}{\mathbb{R}}^n.$$

In this case, the Blaschke-Privalov Laplacian coincides with the standard Laplacian. However, unlike the standard Laplacian, this operator can be applied to functions that are not necessarily smooth. For further details, see [14,15].

For $n\ge 2$, the generalized spherical mean-value operator in ${\mathbb{R}}^n$ is given by [16]

$${\mathcal{M}}_t^{\alpha }f\left(x\right)={\mathcal{M}}^{\alpha }f(x,t)={\mathcal{F}}^{-1}\left(m_{\alpha }\left(t\right|·\left|\right)\mathcal{F}f\right)\left(x\right),t>0,$$

where $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ denote the Fourier transform and its inverse in ${\mathbb{R}}^n$, and the radial multiplier is:

$$m_{\alpha }\left(r\right)={\mathcal{J}}_{\alpha +n/2-1}\left(r\right),$$

with ${\mathcal{J}}_{\alpha +n/2-1}\left(r\right)$ being the normalized Bessel function defined in Section 2. The parameter $\alpha$ can generally be a complex number, except for the cases $\alpha =-{\displaystyle \frac{n}{2}},-{\displaystyle \frac{n}{2}}-1,-{\displaystyle \frac{n}{2}}-2,\cdots$. For $\Re \left(\alpha \right)>0$, the operator takes the integral form [17,18]

$${\mathcal{M}}_t^{\alpha }f\left(x\right)={\mathcal{M}}^{\alpha }f(x,t)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}f(x-ty)dy.$$

(3)

In the special case $\alpha =0$, ${\mathcal{M}}_t^{\alpha }u$ reduces to the spherical mean-value operator as defined in Equation (2). The generalized spherical mean-value operator ${\mathcal{M}}_t^{\alpha }$ has attracted considerable attention and has been extensively studied by several authors, including Miyachi [19], Peral [20], Stein [16], and Strichartz [21]. Notably, if $1<p<\infty$, the operator extends as a bounded linear operator from $L^p\left({\mathbb{R}}^n\right)$ to itself. More general results, including extensions of the operator acting from $L^p\left({\mathbb{R}}^n\right)$ to $L^q\left({\mathbb{R}}^n\right)$ for different p and q, are provided in [17,22].

A natural question arises: what kind of local and nonlocal operators involving the generalized spherical mean-value operator ${\mathcal{M}}_t^{\alpha }f\left(x\right)$ could extend the fractional Laplacian operator?

The aim of this work is to derive a new pointwise formula for the fractional Laplacian operator ${(-\mathrm{\Delta })}^{{\displaystyle \frac{s}{2}}}$ analogous to the representation in (1), but involving the generalized mean-value operator ${\mathcal{M}}_t^{\alpha }$ defined in (3). Specifically, we first establish the following result:

$$\mathrm{\Delta }f\left(x\right)=-2(n+2\alpha )\underset{t\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^2}},x\in \mathbb{R}.$$

This demonstrates that the operator ${\mathcal{M}}_t^{\alpha }f\left(x\right)$ can be used to generalize the Blaschke-Privalov operator [14]. In the nonlocal setting, we show that for $0<s<2$ and $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$, the fractional Laplacian can be expressed as:

$$\begin{array}{cc}\hfill {(-\mathrm{\Delta })}^{s/2}f\left(x\right)& ={\gamma }_{\alpha }(n,s){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}_{\tau }^{\alpha }f\left(x\right)}{{\tau }^{1+s}}}d\tau ,\hfill \end{array}$$

(4)

where the normalization constant ${\gamma }_{\alpha }(n,s)$ is given by:

$${\gamma }_{\alpha }(n,s)={\displaystyle \frac{2^{1+s}\mathrm{\Gamma }\left(\alpha +\frac{s+n}{2}\right)}{\mathrm{\Gamma }\left(\alpha +\frac{n}{2}\right)\left|\mathrm{\Gamma }\left(-\frac{s}{2}\right)\right|}}.$$

The primary motivation for this work is that the representation (4) offers an alternative equivalent definition of the fractional Laplacian. Additionally, similar to the spherical mean-value operator, the generalized spherical mean-value operator is particularly valuable for developing efficient numerical schemes for discretizing the fractional Laplacian. Specifically, we show that generalized spherical means can be used to derive formulas that reduce any n-dimensional fractional Laplacian to a one-dimensional operator, making the computation of the fractional Laplacian more straightforward and efficient.

This paper is structured as follows:

• Section 2 provides an overview of the notations used and recalls the definitions and properties of Bessel functions.
• Section 3 presents our primary research outcomes, summarizing the significant findings.
• Section 4 offers comprehensive proofs of the core results.
• Section 5 presents additional results where we propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

Notations

In this context, ${\mathbb{R}}^n$ denotes the real n-dimensional Euclidean space and ${\mathbb{S}}^{n-1}\subset {\mathbb{R}}^n$ represents the $(n-1)$-dimensional unit sphere, equipped with the normalized surface area measure $d\sigma$. We introduce the following notations:

• ${\mathcal{C}}_{\mathrm{even}}^{\infty }\left(\mathbb{R}\right)$ denotes the space of even $C^{\infty }$-functions on $\mathbb{R}$. This space is endowed with the topology of uniform convergence on compact sets, for both the functions and their derivatives.
• $\mathcal{S}\left({\mathbb{R}}^n\right)$, the Schwartz space, consists of $C^{\infty }$-functions on ${\mathbb{R}}^n$ whose derivatives, along with the functions themselves, decay rapidly at infinity.

The Fourier transform $\mathcal{F}f$ and the inverse Fourier transform ${\mathcal{F}}^{-1}f$ of a function $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$ are defined as follows:

$$\mathcal{F}f\left(\xi \right)=\widehat{f}\left(\xi \right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}f\left(x\right)e^{-ix·\xi }dx,\xi \in {\mathbb{R}}^n,$$

$${\mathcal{F}}^{-1}f\left(\xi \right)=\widehat{f}(-\xi )={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}f\left(x\right)e^{ix·\xi }dx,\xi \in {\mathbb{R}}^n,$$

where $x·\xi =x_1{\xi }_1+\cdots +x_n{\xi }_n$ denotes the inner product between x and $\xi$.

2. The Bessel Function

The normalized Bessel function is given by:

$${\mathcal{J}}_{\nu }\left(x\right):=\mathrm{\Gamma }(\nu +1){\left({\displaystyle \frac{2}{x}}\right)}^{\nu }{J}_{\nu }\left(x\right),\nu >-1,$$

(5)

where $\mathrm{\Gamma }(\nu +1)$ is the Gamma function [23] and $J_{\nu }\left(x\right)$ represents the Bessel function of the first kind [24]. The Bessel function $J_{\nu }\left(x\right)$ is defined as follows:

$$J_{\nu }\left(x\right)={\left({\displaystyle \frac{x}{2}}\right)}^{\nu }\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-\frac{1}{4}{x}^2\right)}^k}{\mathrm{\Gamma }(\nu +1+k)k!}},\nu >-1.$$

The normalized Bessel function ${\mathcal{J}}_{\nu }\left(x\right)$ solves an eigenvalue problem associated with the Bessel equation. Specifically, ${\mathcal{J}}_{\nu }\left(\lambda x\right)$ satisfies the following eigenvalue problem [24,25]:

$$\left\{\begin{array}{c}{\mathcal{L}}_{\nu }u\left(x\right)=-{\lambda }^{2}u\left(x\right),\hfill \\ u\left(0\right)=1,u^{\prime }\left(0\right)=0,\hfill \end{array}\right.$$

where ${\mathcal{L}}_{\nu }$ is the Bessel operator:

$${\mathcal{L}}_{\nu }:={\displaystyle \frac{d^2}{d{x}^2}}+{\displaystyle \frac{2\nu +1}{x}}{\displaystyle \frac{d}{dx}},\nu \ge -{\displaystyle \frac{1}{2}}.$$

(6)

The function ${\mathcal{J}}_{\nu }\left(x\right)$ is even and entire, and for specific values of $\nu$, it simplifies to well-known trigonometric functions such as:

$${\mathcal{J}}_{-1/2}\left(x\right)=cosx,{\mathcal{J}}_{1/2}\left(x\right)={\displaystyle \frac{sinx}{x}}.$$

For $\nu >-{\displaystyle \frac{1}{2}}$, the Riemann–Liouville integral transform [25,26] is defined as:

$${\mathcal{R}}_{\nu }f\left(x\right)={\displaystyle \frac{2\mathrm{\Gamma }(\nu +1)x^{-2\nu }}{\sqrt{\pi }\mathrm{\Gamma }(\nu +\frac{1}{2})}}{\int }_0^x{(x^2-t^2)}^{\nu -{\displaystyle \frac{1}{2}}}f\left(t\right)dt,$$

where f is a continuous function on $[0,\infty )$. This operator was extensively studied by Trimeche [26], and shown to be a transmutation operator for the Bessel operator ${\mathcal{L}}_{\nu }$. Specifically, the Riemann–Liouville transform ${\mathcal{R}}_{\nu }$ is the unique automorphism on ${\mathcal{C}}_{\mathrm{even}}^{\infty }\left(\mathbb{R}\right)$ that satisfies:

$${\mathcal{R}}_{\nu }\circ {\mathcal{L}}_{\nu }\left(f\right)={\displaystyle \frac{d^2}{d{x}^2}}\circ {\mathcal{R}}_{\nu }\left(f\right),$$

with

$${\mathcal{R}}_{\nu }\left(f\right)\left(0\right)=f\left(0\right).$$

This demonstrates that ${\mathcal{R}}_{\nu }$ plays a crucial role in mapping solutions of the Bessel equation to simpler second-order differential equations.

3. Statement of Mains Results

In this section, we present the principal findings of this paper.

Theorem 1.

For $0<s<2$, and for $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$, the following expression holds:

$$\begin{array}{cc}\hfill {(-\mathrm{\Delta })}^{s/2}f\left(x\right)& ={\gamma }_{\alpha }(n,s){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}_{\tau }^{\alpha }f\left(x\right)}{{\tau }^{1+s}}}d\tau ,\hfill \end{array}$$

(7)

where the normalization constant ${\gamma }_{\alpha }(n,s)$ is given by:

$${\gamma }_{\alpha }(n,s)={\displaystyle \frac{2^{1+s}\mathrm{\Gamma }(\alpha +(s+n)/2)}{\mathrm{\Gamma }(\alpha +n/2)\left|\mathrm{\Gamma }(-s/2)\right|}}.$$

(8)

When $\alpha =1$, the generalized spherical mean-value operator coincides ${\mathcal{M}}_t^{\alpha }f$ with the solid mean-value operator ${\mathcal{A}}_{t}f$, defined as [2]

$${\mathcal{A}}_{t}f\left(x\right)={\displaystyle \frac{\mathrm{\Gamma }(1+n/2)}{{\pi }^{n/2}}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}f(x-ty)dy.$$

From Theorem 1, for $\alpha =1$, we derive the following corollary for the fractional Laplacian.

Corollary 1.

Let $0<s<2$. For $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$, the fractional Laplacian of f can be expressed in terms of the solid mean-value operator ${\mathcal{A}}_{t}f\left(x\right)$ as:

$$\begin{array}{cc}\hfill {(-\mathrm{\Delta })}^{s/2}f\left(x\right)& ={\gamma }_1(n,s){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{A}}_{\tau }f\left(x\right)}{{\tau }^{1+s}}}d\tau ,\hfill \end{array}$$

(9)

where the normalization constant ${\gamma }_1(n,s)$ is given by:

$${\gamma }_1(n,s)={\displaystyle \frac{2^{1+s}\mathrm{\Gamma }\left(1+\frac{s+n}{2}\right)}{\mathrm{\Gamma }\left(1+\frac{n}{2}\right)\left|\mathrm{\Gamma }\left(-\frac{s}{2}\right)\right|}}.$$

(10)

In the next theorem, we will use generalized spherical means to derive formulas that reduce any n-dimensional fractional Laplacian to a one-dimensional operator, either on the entire real line $\mathbb{R}$ or on the half-line $(0,\infty )$, making the computation of the fractional Laplacian more straightforward and efficient.

We begin by recalling the definition of the fractional power of the second derivative of order $s\in (0,2)$, which is given by:

$${\left(-{\partial }_r^2\right)}^{{\displaystyle \frac{s}{2}}}\phi \left(r\right)={\displaystyle \frac{\mathrm{\Gamma }(1+s)}{\pi }}sin\left({\displaystyle \frac{\pi s}{2}}\right){\int }_0^{\infty }{\displaystyle \frac{2\phi \left(r\right)-\phi (r+\tau )-\phi (r-\tau )}{{\tau }^{1+s}}}d\tau .$$

In particular, for $r=0$, we obtain:

$${\left(-{\partial }_r^2\right)}^{{\displaystyle \frac{s}{2}}}\phi \left(r\right){|}_{r=0}={\displaystyle \frac{\mathrm{\Gamma }(1+s)}{\pi }}sin\left({\displaystyle \frac{\pi s}{2}}\right){\int }_0^{\infty }{\displaystyle \frac{2\phi \left(0\right)-\phi \left(\tau \right)-\phi (-\tau )}{{\tau }^{1+s}}}d\tau .$$

Based on the Theorem 1, we obtain the following Corollary 2.

Corollary 2.

Let $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$. Then,

$${(-\mathrm{\Delta })}^{{\displaystyle \frac{s}{2}}}f\left(x\right)=c_{\alpha }(n,s){\left(-{\partial }_t^2\right)}^{{\displaystyle \frac{s}{2}}}{\mathcal{M}}_t^{\alpha }f\left(x\right)){|}_{t=0},$$

(11)

where

$$c_{\alpha }(n,s)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\frac{1}{2})\mathrm{\Gamma }(\alpha +(n+s)/2)}{\mathrm{\Gamma }(\alpha +n/2)\mathrm{\Gamma }(\alpha +\frac{s+1}{2})}}.$$

Proof. 

We begin by considering the operator ${\mathcal{M}}^{\alpha }u(x,t)$ at $t=0$:

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\alpha }f(x,0)& ={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}f\left(x\right)dy\hfill \\ \hfill & =f\left(x\right){\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}dy.\hfill \end{array}$$

Passing to polar coordinates and applying the Beta integral formula, we find:

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\alpha }f(x,0)& =f\left(x\right){\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\sigma }_{n-1}{\int }_0^1{(1-r^2)}^{\alpha -1}{r}^{n-1}dr\hfill \\ \hfill & =f\left(x\right)·{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}·{\sigma }_{n-1}·{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(n/2)}{2\mathrm{\Gamma }(\alpha +n/2)}}\hfill \\ \hfill & =f\left(x\right).\hfill \end{array}$$

Since ${\mathcal{M}}^{\alpha }f(x,\tau )$ is even in the argument $\tau$ and ${\mathcal{M}}^{\alpha }f(x,0)=f\left(x\right)$, then

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,\tau )}{{\tau }^{1+s}}}d\tau & ={\displaystyle \frac{1}{2}}{\int }_0^{\infty }{\displaystyle \frac{2{\mathcal{M}}^{\alpha }f(x,0)-{\mathcal{M}}^{\alpha }f(x,\tau )-{\mathcal{M}}^{\alpha }f(x,-\tau )}{{\tau }^{1+s}}}d\tau \hfill \\ & =c_{\alpha }(n,s){\left(-{\partial }_t^2\right)}^{{\displaystyle \frac{s}{2}}}{\mathcal{M}}_r^{\alpha }f\left(x\right)){|}_{r=0}\hfill \end{array}$$

where

$$c_{\alpha }(n,s)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\frac{1}{2})\mathrm{\Gamma }(\alpha +(n+s)/2)}{\mathrm{\Gamma }(\alpha +n/2)\mathrm{\Gamma }(\alpha +\frac{s+1}{2})}}.$$

□

4. Proof of the Main Results

Lemma 1.

Let $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$. Then, we have

$$-\mathrm{\Delta }fu\left(x\right)=2(n+2\alpha )\underset{t\to 0}{lim}{\displaystyle \frac{u\left(x\right)-{\mathcal{M}}_t^{\alpha }f\left(x\right)}{t^2}},x\in {\mathbb{R}}^n.$$

(12)

Proof. 

Since ${\mathcal{M}}^{\alpha }f(x,0)=f\left(x\right)$, we have the expression:

$$f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}\left[f\left(x\right)-f(x-ty)\right]dy.$$

Next, we consider the limit as $t\to 0$:

$$\begin{array}{cc}\hfill \underset{t\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^2}}& ={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}\underset{t\to 0}{lim}{\displaystyle \frac{1}{t^2}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}\left[f\left(x\right)-f(x-ty)\right]dy\hfill \\ \hfill & ={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}\underset{t\to 0}{lim}{\displaystyle \frac{1}{t^2}}{\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}\{t\nabla f\left(x\right)·y\hfill \\ & -{\displaystyle \frac{t^2}{2}}{D}^{2}f\left(x\right)y·y+{O\left(\right|ty|}^3\left)\right\}dy.\hfill \end{array}$$

The odd contributions cancel out, leaving us with:

$$\begin{array}{cc}\hfill \underset{t\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^2}}& =-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{2{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}\sum _{i,j=1}^n{\partial }_{i,j}^{2}f\left(x\right){\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}{y}_i{y}_{j}dy\hfill \\ \hfill & =-{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{2{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}\sum _{i=1}^n{\partial }_i^{2}f\left(x\right){\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}{y}_i^{2}dy.\hfill \end{array}$$

Since the integrals ${\int }_{\left|y\right|<1}{(1-|y|}^2{)}^{\alpha -1}{y}_i^{2}dy$ are identical for all i, we can simplify further:

$$\begin{array}{cc}\hfill \underset{t\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^2}}& ={\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{2{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}\mathrm{\Delta }f\left(x\right){\int }_{\left\{y\right|\left|y\right|\le 1\}}{(1-|y|}^2{)}^{\alpha -1}{\left|y\right|}^{2}dy\hfill \\ \hfill & =-\mathrm{\Delta }f\left(x\right){\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{2{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\displaystyle \frac{{\sigma }_{n-1}}{n}}{\int }_0^1{(1-r^2)}^{\alpha -1}{r}^{n+1}dr\hfill \\ \hfill & =-\mathrm{\Delta }f\left(x\right){\displaystyle \frac{\mathrm{\Gamma }(\alpha +n/2)}{2{\pi }^{n/2}\mathrm{\Gamma }\left(\alpha \right)}}{\displaystyle \frac{{\sigma }_{n-1}}{2n}}{\displaystyle \frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(1+n/2)}{\mathrm{\Gamma }(\alpha +1+n/2)}}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2(n+2\alpha )}}\mathrm{\Delta }f\left(x\right).\hfill \end{array}$$

□

Lemma 2

([27]). Let $x\in \mathbb{R},$$\nu \ge -1/2$ and $0<s<2$, we have

$${\left|x\right|}^s={\displaystyle \frac{2^{s+1}\mathrm{\Gamma }(\nu +\frac{s}{2}+1)}{\mathrm{\Gamma }(\nu +1)|\mathrm{\Gamma }(-\frac{s}{2})|}}{\int }_0^{\infty }\left(1-{\mathcal{J}}_{\nu }\left(tx\right)\right){\displaystyle \frac{dt}{t^{s+1}}}.$$

Proof of Theorem 2.

Let $f\in \mathcal{S}\left({\mathbb{R}}^n\right)$, the Schwartz space of rapidly decreasing smooth functions. From Lemma 1, we have:

$$\mathrm{\Delta }f\left(x\right)=-2(n+2\alpha )\underset{t\to 0}{lim}{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^2}}.$$

Consequently, the integrand on the right-hand side of (7) behaves as:

$${\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^{1+s}}}=O\left(t^{1-s}\right)\mathrm{as}t\to 0.$$

Since $s\in (0,2)$, this ensures that the integral converges near $t=0$.

Next, referring to [16], the generalized spherical mean-value operator ${\mathcal{M}}^{\alpha }f(x,t)$ can be expressed in terms of the Fourier transform as:

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\alpha }f(x,t)& =\left({\mathcal{F}}^{-1}\left[{\mathcal{J}}_{\alpha +n/2-1}\left(t\right|·\left|\right)\mathcal{F}f\right]\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}{\mathcal{J}}_{\alpha +n/2-1}\left(t\right|\xi \left|\right)\mathcal{F}f\left(\xi \right)e^{ix·\xi }d\xi ,\hfill \end{array}$$

(13)

where ${\mathcal{J}}_{\nu }$ denotes the normalized Bessel function of order $\nu$.

Furthermore, by the Fourier inversion formula, we have:

$$f\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}\mathcal{F}f\left(\xi \right)e^{ix·\xi }d\xi .$$

(14)

Subtracting (13) from (14), we obtain:

$$\begin{array}{cc}\hfill f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)& ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}\left(1-{\mathcal{J}}_{\alpha +n/2-1}\left(t\right|\xi \left|\right)\right)\mathcal{F}f\left(\xi \right)e^{ix·\xi }d\xi .\hfill \end{array}$$

Integrating both sides with respect to t, we obtain:

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^{1+s}}}dt& ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}\mathcal{F}f\left(\xi \right)e^{ix·\xi }\left({\int }_0^{\infty }{\displaystyle \frac{1-{\mathcal{J}}_{\alpha +n/2-1}\left(t\right|\xi \left|\right)}{t^{1+s}}}dt\right)d\xi .\hfill \end{array}$$

(15)

From Lemma 2, we have the integral representation:

$${\left|\xi \right|}^s={\gamma }_n(s,\alpha ){\int }_0^{\infty }\left(1-{\mathcal{J}}_{\alpha +n/2-1}\left(t\right|\xi \left|\right)\right){\displaystyle \frac{dt}{t^{1+s}}},0<s<2,$$

(16)

where

$${\gamma }_n(s,\alpha )={\displaystyle \frac{2^{1+s}\mathrm{\Gamma }\left(\alpha +\frac{n+s}{2}\right)}{\mathrm{\Gamma }\left(\alpha +\frac{n}{2}\right)\left|\mathrm{\Gamma }\left(-\frac{s}{2}\right)\right|}}.$$

Substituting (16) into (15), we obtain:

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^{1+s}}}dt& ={\displaystyle \frac{1}{{\gamma }_n(s,\alpha ){\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}{\left|\xi \right|}^s\mathcal{F}f\left(\xi \right)e^{ix·\xi }d\xi .\hfill \end{array}$$

Multiplying both sides by ${\gamma }_n(s,\alpha )$, we obtain:

$$\begin{array}{cc}\hfill {\gamma }_n(s,\alpha ){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}^{\alpha }f(x,t)}{t^{1+s}}}dt& ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}{\int }_{{\mathbb{R}}^n}{\left|\xi \right|}^s\mathcal{F}f\left(\xi \right)e^{ix·\xi }d\xi \hfill \\ \hfill & ={\left(-\mathrm{\Delta }\right)}^{s/2}f\left(x\right),\hfill \end{array}$$

since the fractional Laplacian in Fourier space is given by $\mathcal{F}\left[{(-\mathrm{\Delta })}^{s/2}f\right]\left(\xi \right)={\left|\xi \right|}^s\mathcal{F}f\left(\xi \right)$. This completes the proof. □

5. Decomposition of the Generalized Spherical Mean-Value Operator into Wave Solutions

The spherical mean-value operator plays a crucial role in the study of partial differential equations (PDEs) and it is fundamental to our work. In this section, we first examine its essential properties and then explore various methods for generating generalized spherical means for arbitrary functions.

The generalized spherical mean-value operator is of particular importance in solving the Euler–Poisson–Darboux (EPD) equation. Specifically, for $\alpha \ge {\displaystyle \frac{1-n}{2}}$, the function $v(x,t)=\left({\mathcal{M}}_t^{\alpha }\phi \right)\left(x\right)$ provides a unique solution to the singular Cauchy problem for the EPD equation [17]

$$\left\{\begin{array}{cc}{\partial }_t^{2}v(x,t)+{\displaystyle \frac{\alpha +\frac{n}{2}-1}{t}}{\partial }_{t}v(x,t)={\mathrm{\Delta }}_{x}v(x,t),\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ v(x,0)=\phi \left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \\ {\partial }_{t}v(x,0)=0,\hfill & x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(17)

Notably, when $\alpha ={\displaystyle \frac{1-n}{2}}$, Equation (17) reduces to the classical wave equation:

$$\left\{\begin{array}{cc}{\partial }_t^{2}v(x,t)={\mathrm{\Delta }}_{x}v(x,t),\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ v(x,0)=\phi \left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \\ {\partial }_{t}v(x,0)=0,\hfill & x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(18)

A direct approach to generating the spherical mean ${\mathcal{M}}_t^{\alpha }\phi$ for a function f is by solving Darboux’s Equation (17). The following proposition establishes a one-to-one correspondence between the solutions of the standard wave Equation (18) and Darboux’s differential Equation (17). This demonstrates that the generalized spherical mean-value operator can be derived from the solution of the wave equation.

Proposition 1.

Let $w(x,t)$ be a solution to the wave Equation (18) with initial data $\phi \left(x\right)$. The spherical mean ${\mathcal{M}}^{\alpha }\phi (x,t)$ can be expressed as the following integral representation of wave solutions:

$${\mathcal{M}}^{\alpha }\phi (x,t)={\displaystyle \frac{2\mathrm{\Gamma }\left(\alpha +\frac{n}{2}\right)}{\sqrt{\pi }\mathrm{\Gamma }\left(\alpha +\frac{n-1}{2}\right)}}{\int }_0^{1}w(x,\tau t){(1-{\tau }^2)}^{\alpha +{\displaystyle \frac{n-3}{2}}}d\tau .$$

(19)

Proof. 

We begin by applying the Riemann–Liouville integral transform ${\mathcal{R}}_{\alpha +n/2-1}$ to the wave Equation (18) with respect to the variable t. This transforms the equation into the following system:

$$\left\{\begin{array}{cc}{\mathcal{R}}_{\alpha +n/2-1}\left({\partial }_t^{2}w(x,·)\right)\left(t\right)={\mathrm{\Delta }}_x{\mathcal{R}}_{\alpha +n/2-1}w(x,·)\left(t\right),\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ v(x,0)=\phi \left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \\ {\partial }_{t}v(x,0)=0,\hfill & x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(20)

Using the properties of the Riemann–Liouville integral transform, we have:

$${\mathcal{R}}_{\alpha +n/2-1}\left({\partial }_t^{2}w(x,·)\right)\left(t\right)={\mathcal{L}}_{\alpha +n/2-1}\left({\mathcal{R}}_{\alpha +n/2-1}w(x,·)\right)\left(t\right),$$

(21)

where ${\mathcal{L}}_{\alpha +n/2-1}$ denotes the Bessel differential operator as defined in (6).

On the other hand, the solution $w(x,t)$ to the standard wave equation can be expressed in terms of the spherical mean ${\mathcal{M}}_t^{\alpha }\phi \left(x\right)$ of the initial data $\phi$. Thus, the function ${\mathcal{R}}_{\alpha +n/2-1}w(x,·)\left(t\right)$ satisfies the system (17). By the uniqueness of the solution to this system, we conclude that:

$${\mathcal{R}}_{\alpha +n/2-1}w(x,·)\left(t\right)=\left({\mathcal{M}}_t^{\alpha }\phi \right)\left(x\right).$$

This completes the proof. □