Some observations on the Green function for the ball in the fractional Laplace framework

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.


Introduction
The Green function for the ball in a fractional Laplace framework naturally arises in the study of the representation formulas for the fractional Laplace equations. In particular, in analogy to the classical case of the Laplacian, given an equation with a known forcing term on the ball and vanishing Dirichlet data outside the ball, the representation formula for the solution is precisely the convolution of the Green function with the forcing term. As in the classical case, the Green function is introduced in terms of the Poisson kernel. For this, we will provide both the representation formulas for the problems # p´∆q s u " 0 in B r , u " g in R n zB r (1.1) and # p´∆q s u " g in P B r , u " 0 in P R n zB r (1.2) in terms of the fractional Poisson kernel and respectively the Green function. Moreover, we will prove an explicit formula for the Green function on the ball.
Here follow a few preliminary notions on the reference context and on the fractional Laplace operator, and the definition of the four kernels that play particular roles in our study: the s-mean kernel, the fundamental solution, the Poisson kernel and the Green function.
Let us briefly introduce the Schwartz space (refer to [12] for details) and the Fourier transform acting on this functional space (refer, for instance, to [7] for details). In other words, the Schwartz space consists of smooth functions whose derivatives (including the function itself) decay at infinity faster than any power of x; we say, for short, that Schwartz functions are rapidly decreasing. Endowed with the family of seminorms rf s α,N SpR n q " sup the Schwartz space is a locally convex topological space. We denote by S 1 pR n q the space of tempered distributions, the topological dual of SpR n q. We set the following notations for the Fourier and the inverse Fourier transform p f pξq :" Ff pξq :" ż R n f pxqe´2 πix¨ξ dx respectively q f pxq " F´1f pxq " ż R n f pξqe 2πix¨ξ dξ.
Here the original space variables are denoted as x P R n and the frequency variable as ξ P R n . We recall that the Fourier and the inverse transform are well defined for f P L 1 pR n q, whereas f pxq " F`F´1f qpxq " F´1`Ff qpxq almost everywhere if both f and p f P L 1 pR n q, and pointwise if f is also continuous. Also for all f, g P L 1 pR n q The pointwise product is taken into the convolution product and vice versa, namely for all f, g P L 1 pR n q Fpf˚gq " Fpf q Fpgq.
On the Schwartz space, the Fourier transform gives a continuous bijection between SpR n q and SpR n q. We say that p f is the Fourier transform of f in a distributional sense, for f that satisfies ż R n |f pxq| 1`|x| p dx ă 8 for some p P N if, for any ϕ P SpR n q we have that ż We remark that the integral notation is used in a formal manner whenever the arguments are not integrable.
Let s P p0, 1q be fixed. We introduce the fractional Laplacian for u belonging to the Schwartz space.
Definition 1.2. The fractional Laplacian of u P SpR n q in x P R n is defined as p´∆q s upxq :" Cpn, sqP.V. ż R n upxq´upyq |x´y| n`2s dy " Cpn, sq lim εÑ0 ż R n zBεpxq upxq´upyq |x´y| n`2s dy, (1.4) where Cpn, sq is a constant depending only on n and s.
Let ε be positive, sufficiently small. Then indeed, for u P L 1 s pR n q and C 2s`ε (or C 1,2s`ε´1 for s ě 1{2) in a neighborhood of x P R n , the fractional Laplacian is well defined in x as in (1.4). We refer for the proof to [13], where an approximation by Schwartz functions is performed. We often use this type of regularity to obtain pointwise solutions to the posed equations. We will always write C 2s`ε for both s ă 1{2 and s ě 1{2. Remark 1.4. Definition (1.4) is also well posed in x when u P L 8 pR n q and C 2 in a neighborhood of x.
In definition (1.4), the singular integral can be substituted with a weighted second order differential quotient by performing the changes of variablesỹ " x`y and y " x´y and summing up. In this way, the singularity at the origin can be removed, as we observe in the following lemma. Lemma 1.5 . Let p´∆q s be the fractional Laplace operator defined by (1.4). Then for any smooth u p´∆q s upxq " Cpn, sq 2 ż R n 2upxq´upx`yq´upx´yq |y| n`2s dy. (1.6) For u P SpR n q the fractional Laplace operator can also be expressed as a Fourier transform, according to the following lemma (see [3] for the proof). Lemma 1.6 . Let p´∆q s be the fractional Laplace operator defined by (1.4). Then for any u P SpR n q p´∆q s upxq " F´1´`2π|ξ|˘2 s p upξq¯. (1.7) In order obtain the solution to the problems (1.1) and (1.2) we will solve also the equations on the whole space R n p´∆q s u " 0 (1.8) and p´∆q s u " f, (1.9) using the s-mean kernel and the fundamental solution.
We refer usually to pointwise solutions, nevertheless distributional solution will also be employed. Following the approach in [13], we introduce a suitable functional space where distributional solutions can be defined. Let S s pR n q :" ! f P C 8 pR n q s. t. @α P N n 0 , sup xPR n`1`| x| n`2s˘| D α f pxq| ă`8 ) .
The linear space S s pR n q endowed with the family of seminorms rf s α SspR n q :" sup xPR n`1`| x| n`2s˘| D α f pxq| is a locally convex topological space. We denote with S 1 s pR n q the topological dual of S s pR n q.
We notice that if ϕ P SpR n q then p´∆q s ϕ P S s pR n q, which makes this framework appropriate for the distributional formulation. In order to prove this claim, we observe that for any x P R n zB 1 the bound |p´∆q s ϕpxq| ď c n,s |x|´n´2 s (1.10) follows from the upcoming computation and the fact that ϕ P SpR n q |p´∆q s ϕpxq| ď ϕpxq´ϕpx´yq´ϕpx`yqˇ| y| n`2s dỳ 2 ż R n zB |x| 2ˇϕ pxq´ϕpx`yqˇ| y| n`2s dy ď c n,s "´s up zPR n p1`|z|q n`2 |D 2 ϕpzq|¯|x|´n´2 s´s up zPR n p1`|z|q n |ϕpzq|¯|x|´n´2 s`} ϕ} L 1 pR n q |x|´n´2 s  .
Hence, by iterating the presented argument, one proves that p´∆q s ϕ P S s pR n q, which assures our claim. Then we have the following definition.
Definition 1.7. Let f P S 1 pR n q, we say that u P S 1 s pR n q is a distributional solution of p´∆q s u " f in R n if ă u, p´∆q s ϕ ą s " ż R n f pxqϕpxq dx for any ϕ P SpR n q, (1.11) where ă¨,¨ą s denotes the duality pairing of S 1 s pR n q and S s pR n q and the latter (formal) integral notation designates the pairing SpR n q and S 1 pR n q.
We also use the integral notation for the pairing ă¨,¨ą s in a purely formal manner whenever the arguments are not integrable ă u, ψ ą s " ż R n upxqψpxq dx.
We notice that L 1 s pR n q Ă S 1 s pR n q, in particular for any u P L 1 s pR n q and ψ P S s pR n q we have thaťˇˇă u, ψ ą sˇď ż R n |upxq| |ψpxq| dx ď ż R n |upxq| 1`|x| n`2s p1`|x| n`2s q|ψpxq| dx ď rψs 0 SspR n q }u} L 1 s pR n q . (1.12) Let us now introduce the four functions A r , Φ, P r and G, namely the s-mean kernel, the fundamental solution, the Poisson kernel and the Green function. Definition 1.8. Let r ą 0 be fixed. The function A r is defined by cpn, sq r 2s p|y| 2´r2 q s |y| n y P R n zB r , where cpn, sq is a constant depending only on n and s. Definition 1.9. For any x P R n zt0u the function Φ is defined by where apn, sq is a constant depending only on n and s. Definition 1.10. Let r ą 0 be fixed. For any x P B r and any y P R n zB r , the Poisson kernel P r is defined by P r py, xq :" cpn, sq˜r 2´| x| 2 |y| 2´r2¸s 1 |x´y| n . (1.15) The Poisson kernel is used to build a function which is known outside the ball and s-harmonic in B r . Indeed, it is used to give the representation formula for problem (1.1), as stated in the following theorem. Theorem 2.11. Let r ą 0, g P L 1 s pR n q X CpR n q and let u g pxq :" (1.16) Then u g is the unique pointwise continuous solution to the problem (1.1) Definition 1.11. Let r ą 0 be fixed. For any x, z P B r and x ‰ z, the function G is defined by Gpx, zq :" Φpx´zq´ż From this latter definition, a formula that is more suitable for applications can be deduced. Indeed, one of the main goals is to prove this simpler, explicit formula for the Green function on the ball, by means of elementary calculus techniques. At this purpose, Theorem 3.4 establishes a symmetrical expression for G. Theorem 3.4. Let G be the Green function on the ball B r for a positive r. Then if n ‰ 2s and κpn, sq is a constant depending only on n and s. For n " 2s, the following holds This result is not new (see [2]), however, the proof we provide uses only calculus techniques, therefore we hope it will be accessible to a wide audience. It makes elementary use of special functions like the Euler-Gamma function, the Beta and the hypergeometric function, that are introduced in the Appendix (we refer to [1] for details). Moreover, the point inversion transformations and some basic calculus facts, that are also outlined in the Appendix, are used in the course of this proof.
The main property of the Green function is stated in the upcoming Theorem 3.5. The function G is used to build the solution of an equation with a given forcing term in a ball and vanishing Dirichlet data outside the ball, by convolution with the forcing term. While this convolution property in itself may be easily guessed from the superposition effect induced by the linear character of the equation, the main property that we point out is that the convolution kernel is explicitly given by the function G. Theorem 3.5. Let r ą 0, h P C 0,ε pB r q, and let upxq :" Then u is the pointwise continuous unique solution to the problem (1.2) The proof is classical, and makes use of the properties and representation formulas involving the two functions Φ and P r .
We are also interested in the actual values of the normalization constants used in the definitions of the s-mean kernel (and the Poisson kernel) and of the fundamental solution. The only difficulty we encounter is that, having the fundamental solution different expressions with respect to n ě 1 and s P p0, 1q, we must worry about the values of the constants in three different cases: n ą 2s, n ă 2s (that is when n " 1 and s ą 1{2) and n " 2s (n " 1, s " 1{2) . Indeed: These constants are used for normalization purposes, and we explicitly clarify how their values arise. However, these values are only needed to compute the constant κpn, sq introduced in Theorem 3.4, and won't play much role for the rest of our discussion. The value of κpn, sq is given in terms of the Euler-Gamma function in the different cases n ‰ 2s and n " 2s. Theorem 3.7. The constant κpn, sq introduced in definition identity (1.18) is κpn, sq " Γp n 2 q 2 2s π n 2 Γ 2 psq for n ‰ 2s, κ´1, 1 2¯" 1 π for n " 2s.
One interesting thing that we want to point out here is related to the two constants Cpn, sq and cpn, sq. The first one, Cpn, sq is given by [3] in the definition of the fractional Laplacian, and we defined it here in (1.5). The constant cpn, sq is introduced in [10] in the definition of the s-mean kernel and the Poisson-kernel, and is here given in (1.23). These two constants often appear in the definition of the fractional Laplacian, however they are used for different normalization purposes. We observe that they have similar asymptotic properties, however they are not equal. In the following proposition we compute the value of Cpn, sq defined in (1.5), and we compare it with the constant cpn, sq from (1.23). Theorem 3.9. a) The constant Cpn, sq introduced in (1.5) is given by . (1.24) b) The quotient between the constants Cpn, sq introduced in (1.5) and cpn, sq defined in (1.23) is given by Cpn, sq cpn, sq " 2 2s Γps`1qΓp n 2`s q Γp n 2 q . This paper is structured as follows: in the Preliminaries (Section 2) we introduce some kernels related to the fractional Laplacian. In Subsection 2.1 we define the s-mean value property by means of the s-mean kernel and prove that if a function is s-harmonic, then it has the s-mean value property. Subsection 2.2 deals with the study of the function Φ as the fundamental solution of the fractional Laplacian. By means of the fundamental solution, we obtain the representation formula for the equation (1.9). The fractional Poisson kernel is introduced in Subsection 2.3, and the representation formula for equation (1.1) is obtained. Section 3 focuses on the Green function G, presenting two main theorems: Theorem 3.4 gives a more basic formula of the function G for the ball and is treated in Subsection 3.1; Theorem 3.5, that illustrates how the solution to equation (1.2) is built by means of the function G, is dealt with in Subsection 3.2. The computation of the normalization constants introduced along this notes is done in Subsection 3.3. The Appendix A introduces three special functions (Gamma, Beta and hypergeometric), the point inversion transformations and some calculus identities that we use throughout this paper.

Preliminaries
In this section, we investigate the properties of the s-mean kernel, the fundamental solution and the Poisson kernel. We introduce the representation formulas for equations (1.8), (1.9) and (1.1).
2.1. The s-mean value property. In this subsection, we define the s-mean value property of the function u, namely, an average property defined by convolution of u with the s-mean kernel.
Definition 2.1 (s-mean value property). We say that u P L 1 s pR n q and continuous in a neighborhood of x has the s-mean value property at a point x P R n if, for any r ą 0 arbitrarily small, upxq " A r˚u pxq. (2.1) We say that u has the s-mean value property in Ω Ď R n if for any r ą 0 arbitrarily small, identity (2.1) is satisfied at any point x P Ω.
The special function A r from definition (1.13) is used to state the s-mean value property (this makes it reasonable to say that A r plays the role of the s-mean kernel). We consider r ą 0 and since A r is supported in R n zB r , the convolution of A r and a smooth function u is to be taken only on the set R n zB r , thus A r˚u pxq " cpn, sqr 2s ż R n zBr upx´yq p|y| 2´r2 q s |y| n dy.
The main result that we state here is that if a function has the s-mean value property, then it is s-harmonic (where by s-harmonic we mean that u satisfies the classical relation p´∆q s u " 0). Theorem 2.2. Let u P L 1 s pR n q and C 2s`ε in a neighborhood of x P R n . If u has the s-mean value property at x, then u is s-harmonic at x.
Proof. The function u has the s-mean value property for any r ą 0 arbitrarily small, namely We use identity (A.20) from the Appendix, which states that ż R n zBr A r pyq dy " 1 to obtain 0 " upxq´ż R n zBr A r pyqupx´yq dy " cpn, sqr 2s ż R n zBr upxq´upx´yq p|y| 2´r2 q s |y| n dy.
This holds for any r sufficiently small. We divide by r 2s and by cpn, sq and obtain ż R n zBr upxq´upx´yq p|y| 2´r2 q s |y| n dy " 0. to obtain the desired conclusion that p´∆q s upxq " 0. We prove this by splitting the integral on R n zB R for R large and on the annulus B R zB r . Indeed, we consider R ą r ? 2 and we write the integral (2.2) as ż R n zBr upxq´upx´yq p|y| 2´r2 q s |y| n dy " ż R n zB R upxq´upx´yq p|y| 2´r2 q s |y| n dỳ ż B R zBr upxq´upx´yq p|y| 2´r2 q s |y| n dy " I 1 pr, Rq`I 2 pr, Rq.
Since u P L 1 s pR n q then |upxq´upx´yq| |y| n`2s P L 1 pR n zB R q, thus we can use the dominated convergence theorem, send r Ñ 0 and conclude that Now, for r ă |y| ă R and u P C 2s`ε (for s ă 1{2) in a neighborhood of x we have the boundˇˇu pxq´upx´yqˇˇď c|y| 2s`ε , while for s ě 1{2 and u P C 1,2s`ε´1 we use that Notice that y¨∇upxq p|y| 2´r2 q s |y| n and y¨∇upxq |y| 2s`n are even functions, hence they vanish when integrated on the symmetrical domain B R zB r . Therefore, by setting By summing up the two limits for r Ñ 0 of I 1 pr, Rq and I 2 pr, Rq, identity (2.3) follows and hence the conclusion that p´∆q s upxq " 0.

The fundamental solution.
We claim that the function Φ plays the role of the fundamental solution of the fractional Laplacian. The following theorem provides the motivation for this claim, namely the fractional Laplacian of Φ is equal in the distribution sense to the Dirac Delta function evaluated at zero.
The computation of the Fourier transform of the fundamental solution is required in order to prove Theorem 2.3. Proposition 2.4 . a) For n ą 2s, let f P L 1 pR n q X CpR n q with q f P S s pR n q, b) for n ď 2s, let f P L 1 pRq X CpRq X C 1`p´8 , 0q Y p0,`8q˘with q f P S s pRq such that |f pxq| ď c 1 |x| 2s for x P R |f pxq| ď c 2 |x| for |x| ą 1 Proof. We notice the conditions imposed to the functions f assure that the integrals are well defined. Indeed, since Φ P L 1 s pR n q Ă S 1 s pR n q the left hand side is finite thanks to (1.12). The right hand side is also finite since, for n ą 2s, |f pxq|`}f } L 1 pR n q and for n ď 2s we have that ż a) We prove at first the result in the case n ą 2s. We have to prove that apn, sq We use the Fourier transform of the Gaussian distribution as a starting point of the proof. For any δ ą 0 we have that In particular for any f P L 1 pR n q and q f P S s pR n q (which is a subspace of L 2 pR n q), we have that ż We multiply by δ n 2´s´1 , integrate in δ from 0 to 8, call I 1 the first integral and I 2 the latter. We have that and that We perform in I 1 the change of variable α " δ|x| 2 and obtain that Since n 2´s´1 ą´1, the integral in α is finite. We set c 1 :" and have that On the other hand, in I 2 , we change the variable α " |x| 2 δ and obtain Since s´1 ą´1, the integral in α is finite. We then set and we have that We take apn, sq " c 1 c 2 p2πq 2s and we obtain that hence the result. We use definition (A.1) of the Gamma function and perform the change of variable πα " t in (2.4). We obtain that We do the same in the definition (2.5) of c 2 . With the same change of variables πα " t, we obtain that Therefore c 1 c 2 " π 2s´n 2 Γp n 2´s q Γpsq .
Let R ą 0 be as large as we wish (we will make R go to 8 in sequel). Then We use the change of variablesx " 2πx (but still write x as a the variable of integration for simplicity) and letR " 2πR. Then We have that In the integral şR 0 x 2s´1 cospξxq dξ, we integrate by parts, change variables |ξ|x " t, notice that lim xÑ0 x 2s´1 " 0 and obtain that (2.6) We claim that and that (this holds also for n " 2s) (2.8) We integrate by parts and obtain thaťˇˇˇż Using the conditions imposed on f , we have that for ξ large For ξ small we have that |f pξq| ξ ď c 1 |ξ| 2s´1 and it follows that Furthermore, by changing variables t " ξR we have that .
and in the same way we obtaiňˇˇˇż and we have proved the claim (2.7). In order to prove the second claim we estimate the differencěˇˇˇż We then have thaťˇˇˇż Hence we obtain We consider the closed path Ω ρ " B´`r0, ρsˆr0, ρs˘X B ρ p0q¯. We take the contour integral ş Ωρ z 2s´2 e´z dz, and let γ ρ " BB ρ p0q X`r0, ρsˆr0, ρs˘(the boundary of the quarter of the circle). By Cauchy's Theorem, the contour integral is 0 (there are no poles inside Ω ρ ), therefore Integrating along γ ρ , by using polar coordinates z " ρe iθ and then the change of variables cos θ " t we have thaťˇˇˇż Hence lim ρÑ8 ż γρ z 2s´2 e´z dz " 0 and we are left only with the integrals along the real and the imaginary axis, namely Here the left hand side returns the Gamma function according to definition (A.1), we compute i 1´2s "`cospπ{2q`i sinpπ{2q˘1´2 s " sinpπsq`i cospπsq and in (2.9) we obtain that ż 8 0 t 2s´2 sin t dt "´Γp2s´1qIm´sinpπsq`i cospπsq¯"´cospπsqΓp2s´1q.
By sending R to infinity in (2.6) we finally obtain that Taking ap1, sq " 1 2 cospπsqΓp2sq , hence the result.
Remark 2.6. We make a remark on the constant ap1, sq as given in definition (1.21). This constant is introduced in order to normalize the Fourier transform of the fundamental solution. We point out that we can simplify the value ap1, sq using (A.6) and (A.4), as follows We continue proving the Proposition 2.4 for n " 2s. We have that We change variablesx " 2πx (but still write x as a the variable of integration for simplicity) and letR " 2πR. Then we have that We integrate by parts the integral şR 0 log x cospξxq dx and obtain that żR We thus have that We claim that Moreover integrating by parts we have thaťˇˇˇż We have that for ξ large On the other hand by using the change of variables t " ξR And finally and since the same bounds hold for Also, the proof of the claim (2.8) assures us that This concludes the proof of the Proposition.
By applying this latter Proposition, we prove Theorem 2.3.
Proof of Theorem 2.3. For any f P SpR n q we have that F´1´|ξ| 2s p f pξq¯P S s pR n q.
Notice that |ξ| 2s p f pξq P L 1 pR n q X CpR n q, since Moreover, for n ď 2s all the conditions in Proposition 2.4 are satisfied. Indeed, for |ξ| ą 1.
and for |ξ| ą 1ˇˇˇd which proves that all the assumptions on f are satisfied. Thanks to Proposition 2.4 we have that Therefore in the distribution sense Before continuing with the main result of this section, we introduce the following lemma, that will be the main ingredient in the proof of the upcoming Theorem 2.9. Lemma 2.8 . Let f P C 8 c pR n q, then f˚Φ P L 1 s pR n q. Moreover, let ϕ be an arbitrary function such that q ϕ P S s pR n q and a) for n ą 2s, ϕ P L 1 pR n q X CpR n q, b) for n ď 2s, ϕ P L 1 pRq X CpRq X C 1`p´8 , 0q Y p0, 8q˘q and Proof. To prove that f˚Φ P L 1 s pR n q, we suppose that supp f Ď B R and we compute We prove that the integral is finite in the three cases n ą 2s, n ă 2s and n " 2s, which leads to the bound ż R n |f˚Φpxqpxq| 1`|x| n`2s dx ď c n,s,R }f } 8 . (2.14) Notice that we are only using that f P C c pR n q. We take for simplicity R " 1. For n ą 2s we have that For x small, we have that 1`|x| n`2s ě 1 hence For x large, we use that |x´y| ě |x|´|y| and 1`|x| n`2s ą |x| n`2s , thus Thus for n ą 2s, the integral in (2.13) is bounded. Meanwhile, for n ă 2s for x small the same bound as in the n ą 2s case holds. For x large, we have that In the case n " 2s, we have that |x´y| ď |x|`|y| and ż B1ˆżB2 log |x´y| 1`|x| 2 dx˙dy ď c therefore we have that also for n " 2s, the integral (2.13) is bounded. Hence, f˚Φ P L 1 s pR n q, as stated. In order to prove identity (2.12) we notice that by the Fubini-Tonelli theorem we have that ż We denote The operation¯is well defined for f P C 8 c pR n q and q ϕ P S s pR n q, furthermore it is easy to see that Fpf¯q ϕqpxq " q f pxqϕpxq.
We notice at first that since ϕ and q ϕ are continuous, Fp q ϕq " ϕ on R n . We define and we write (2.15) as To be able to apply Proposition 2.4, we have to verify the conditions on ψ. Since we have that ψ P L 1 pR n q. Also, ψ P CpR n q as a product of continuous functions. We claim that f¯q ϕ P S s pR n q . Indeed, suppose suppf Ď B R for R ą 0. Then for |x| ď 2R we have that For |x| ą 2R we have that |x| n`2s |f¯q ϕpxq| ď }f } 8 r q ϕs 0 SspR n q |x| n`2s ż B R |x`y|´n´2 s dy and we remark that |y| ď |x|{2 (otherwise y R suppf ). Then we use the bound |x`y| ě |x|´|y| ě |x|´| x| 2 " |x| 2 and we have that We can iterate the same method to prove that p1`|x| n`2s q|D α f¯q ϕpxq| is bounded since D α f¯q ϕpxq " f¯D α q ϕf q ϕpxq and D α q ϕf P SpR n q. For n ď 2s we have that and finally for |x| ą 1 and for |x| ď 1, since f P C 8 pRq Hence all the assumptions on ψ are satisfied. Taking into account (2.17) and applying Proposition 2.4 we have that ż and from (2.16) we conclude that ż The function Φ gives the representation formula for equation (1.9) both in the distributional sense and pointwise. Theorem 2.9. Let f P C 0,ε c pR n q and let u be defined as upxq :" Φ˚f pxq.
Then u P L 1 s pR n q X C 2s`ε pR n q and in the distributional sense p´∆q s upxq " f pxq.
Moreover, p´∆q s upxq " f pxq pointwise in R n .
Proof. We prove at first the statement for f P C 8 c pR n q. By approximation, we prove that the results holds when the known term has C 0,ε c pR n q regularity. We have proved in Lemma 2.8 that u P L 1 s pR n q. In order to prove that u satisfies equation (1.9), we use identity (2.12) (we are entitled to use this result since, as proved in Theorem 2.3, all the hypothesis are satisfied). Hence The last equality follows since q f P L 1 pR n q, which is assured by the infinite differentiability of f . Indeed, | q f |pxq ď p1`|x|q´n´1}F´1pf pn`1q q} L 8 pR n q ď p1| x|q´n´1}f pn`1q } L 1 pR n q . We conclude that u is the distributional solution of p´∆q s upxq " f pxq.
We consider now f P C 0,ε c pR n q. The fractional Laplacian is well defined, since u P L 1 s pR n q (notice that when we proved in Lemma 2.8 that u P L 1 s pR n q, we have used only that f P C c pR n q). Let R ą 0 such that suppf Ď B R . Surely u is continuous since it is the convolution of a locally integrable function with a continuous, compactly supported function. Furthermore, u is C 2s`ε pR n q. We refer for the proof of this statement to Proposition 2.1.10 from [13] and underline here the main elements for the case n ą 2. Let Ψ be the fundamental solution of the Laplace operator, defined as For any f P C 0,ε c pR n q both in a distributional sense and, by using a density argument (approximating with C 8 c pR n q functions), pointwise we have that f˚Φpxq " f˚p´∆q 1´s Ψpxq " p´∆q 1´s`f˚Ψ˘p xq, in view of the next identity that follows from Proposition 2.4 and the fact that We know that (refer to Lemma 4.4 from [8]) for f P C ε pB R q, g :" f˚Ψ P C 2,ε pB R{2 q and the norm of g depends only on }f } C 0,ε pR n q . Moreover, g P L 8 pR n q, as are its first and second derivatives, and it follows that g P C 2,ε pR n q. Thanks to Proposition 2.1.9 from [13] we can conclude that f˚Φ P C 2s`ε pR n q and this proves the Holder continuity of u.
In order to prove equation (1.9), we take a sequence of functions pf k q k P C 8 c pR n q such that }f k´f } L 8 pR n q kÑ8 ÝÑ 0 and we consider u k " Φ˚f k . Then by the result achieved for C 8 c pR n q, we have that for any ϕ P SpR n q ă u k , p´∆q s ϕ ą s " moreover, using (1.12) and (2.14) we have that ă u k´u , p´∆q s ϕ ą s ď rp´∆q s ϕs 0 SspR n q }u k´u } L 1 s pR n q ď c n,s,R rp´∆q s ϕs 0 SspR n q }f k´f } L 8 pR n q kÑ8 ÝÑ 0.
We thus obtain that for any ϕ P SpR n q ă u, p´∆q s ϕ ą s " Hence in the distributional sense p´∆q s upxq " f pxq on R n . To obtain the pointwise solution, we notice that in particular for any ϕ P C 8 c pR n q we have that ż R n upxqp´∆q s ϕpxq dx " ż R n f pxqϕpxq dx and that ż R n p´∆q s upxqϕpxq dx is well defined. This follows from the continuity of the mapping R n Q x Þ Ñ p´∆q s upxq, according to Proposition 2.1.7 from [13], as u P C 2s`ε pR n q. We use Fubini-Tonelli's Theorem and by a change of variables, we observe that we can pass to operator from ϕ to u We conclude that pointwise in R n p´∆q s upxq " f pxq.
2.3. The Poisson kernel. We claim that P r plays the role of the fractional Poisson kernel. Indeed, the function P r arises in the construction of the solution to Dirichlet problem with vanishing Laplacian inside the ball and a known forcing term outside the ball, as stated in Theorem 2.11. In order to prove this Theorem, we rely on a representation formula for a C 8 c pR n q function, namely, any function C 8 c pR n q can be represented as a convolution of the function Φ with a second function belonging to C 8 pR n q and vanishing at infinity like |x|´n´2 s . More precisely we have the following lemma. Lemma 2.10 . For any f P C 8 c pR n q there exists a function ϕ P C 8 pR n q such that f pxq " ϕ˚Φpxq, and as |x| Ñ 8, ϕpxq " Op|x|´n´2 s q.
Proof. For f P C 8 c pR n q, we define ϕ as ϕpxq :" p´∆q s f pxq.
The bound established in (1.10) assures the asymptotic behavior of ϕ, while it is not hard to see that ϕ P C 8 pR n q. Then by using Theorem 2.9 we have that pointwise on the support of f ϕ˚Φpxq " p´∆q s f˚Φpxq " p´∆q s pΦ˚f qpxq " f pxq.
Having Lemma 2.10 and making use of some results from the Appendix, we proceed with the proof of Theorem 2.11. Theorem 2.11. Let r ą 0, g P L 1 s pR n q X CpR n q and let u g pxq :" gpxq if x P R n zB r .

(2.18)
Then u g is the pointwise continuous unique solution to the problem (1.1) Proof of Theorem 2.11. Notice that u g P L 1 s pR n q. By definition, u g P L 1 s pR n zB r q. On the other hand, u g P L 8 pB r q. Indeed, take R ą 2r, then by using (A.23), the inequality |x´y| ě |y|´|x| ą |y|´r and for |y| ą R the bound |y| n`2s p|y| 2´r2 q s |x´y| n ď |y| n`2s p|y| 2´r2 q s p|y|´rq n ă 2 n`s |gpyq|`2 n`s cpn, sqr 2s ż |y|ąR |gpyq| |y| n`2s dy.
Since g P L 1 s pR n q, the last integral is bounded, and it follows that u g is bounded in B r . Concluding u g P L 1 s pR n q, as stated. Let us fix x P B r and prove that u g has the s-mean value property in x. If this holds, indeed, Theorem 2.2 implies that p´∆q s upxq " 0, and given the arbitrary choice of x, the same is true in the whole B r . We prove the continuity and the uniqueness of u g as the solution to the Dirichlet problem in the final part of the proof.
We prove that u g has the s-mean value property in the following manner: at first, we prove the result in the particular case in which the forcing term g is defined as Φpz´yq dz, for a fixed R ą r. Then, in a second step, we prove the result for any function g P C 8 c pR n q by means of the representation formula provided in Lemma 2.10. At last, we prove the desired result for g in L 1 s pR n q X CpR n q, employing approximations by C 8 c pR n q functions. Let x P B r be fixed. We claim that for any ρ such that 0 ă ρ ă r´|x| A ρ˚ug pxq " u g pxq. (2.20) We start proving the claim (2.20) for the case in which the forcing term g is defined as gpyq :" ż Rą|z|ąr Φpz´yq dz for any y P R n , (2.21) for a fixed R. Notice that g P L 1 s pR n q: the computations are similar to those performed in the proof of the claim (2.13) and will be omitted. Therefore, u g P L 1 s pR n q, and this entitles us to compute the s-mean value of u g . By definition (2.18) of u g and definition (2.21) of g, we have that for x P B r u g pxq " ż |y|ąr P r py, xqgpyq dy " ż |y|ąr P r py, xqˆż Rą|z|ąr Φpz´yq dz˙dy " ż Rą|z|ąrˆż|y|ąr P r py, xqΦpz´yq dy˙dz.
Using identity (A.36) of the Appendix, for the fixed x P B r , we obtain that u g pxq " We use formula (2.22) to obtain that In this last line we use identity (A.34), noticing that for any |t| ą r we have chosen ρ such that |t´x| ě |t|´|x| ą r´r`ρ " ρ. We then observe that A ρ˚ug pxq " u g pxq, thus the claim that u g has the s-mean value property in x, in the particular case in which the forcing term g is given by (2.21).
Now we prove the claim (2.20) in the case in which the forcing term g is in C 8 c pR n q. By Lemma 2.10, there exists a function ϕ P C 8 pR n q such that for any y belonging to the support of g gpyq " ż R n Φpz´yqϕpzq dz.
Notice that at infinity ϕpzq " Op|z|´n´2 s q, and it follows that g P L 8 pR n q. Indeed, g has compact support, say suppg Ď B R and Φpz´yq|ϕpzq| dz. Now, for |y| ă R and |z| ą R`1, for n ą 2s |z´y| ě |z|´|y| ě 1, For n ă 2s |z´y| ď |z|`|y| ď 2|z| therefore ż For n " 2s, log |z´y| ď log`|z|`|y|˘ď log`2|z|˘ and ż We recall now that at infinity ϕpzq " Op|z|´n´2 s q and it follows that the integral ż Φpz´yq|ϕpzq| dz is bounded in all three cases. Moreover, Φ is locally integrable and we conclude that g P L 8 pR n q. For r ą 0 fixed, we write g as Consequently A ρ˚ug pxq " u g pxq, thus u g has the s-mean value property in x. Therefore, for g P C 8 c pR n q, the claim (2.20) is proved.
We now prove the claim (2.20) for any forcing term g P L 1 s pR n q X CpR n q. In particular, let η k P C 8 c pR n q such that η k pxq P r0, 1s, η k " 1 in B k and η k " 0 in B k`1 . Then g k :" η k g P C 8 c pR n q and we have that g k ÝÑ kÑ8 g pointwise in R n , in norm L 1 s pR n q and uniformly on compact sets. We have proved the claim (2.20) for any g P C 8 c pR n q, therefore for any k ě 0 u g k pxq has the s-mean value property in x. Precisely, for any ρ ą 0 small independent of k, A ρ˚ug k˘p xq " u g k pxq. (2.25) We claim that lim kÑ8 u g k pxq " u g pxq (2.26) and that for any ρ ą 0 small lim kÑ8`A ρ˚ug k˘p xq " A ρ˚ug pxq. (2.27) In order to prove claim (2.26) we notice that, by definition (2.18) of u g , we have that u g k pxq " ż R n zBr g k pyqP r py, xq dy and that u g pxq " ż R n zBr gpyqP r py, xq dy.
Let R ą 2r and we recall that for x P B r and y P R n zB R we have the bound (2.19) |y| n`2s p|y| 2´r2 q s |x´y| n ă 2 n`s .
Let ε be any arbitrarily small quantity, for k large we have that |u g k pxq´u g pxq| ď ż R n zBr |g k pyq´gpyq|P r py, xq dy " cpn, sq ż R n zB R |g k pyq´gpyq| pr 2´| x| 2 q s p|y| 2´r2 q s |x´y| n dỳ ż B R zBr |g k pyq´gpyq|P r py, xq dy ď 2 n`s cpn, sqpr 2´| x| 2 q s ż R n zB R |g k pyq´gpyq| |y| n`2s dỳ sup yPB R zBr |g k pyq´gpyq| ż B R zBr P r py, xq dy ď cpn, s, rq ż R n zB R |g k pyq´gpyq| |y| n`2s dy`sup yPB R zBr |g k pyq´gpyq| ď ε by the convergence in L 1 s pR n q norm, the uniform convergence on compact sets of g k to g and integrability in R n zB r of the Poisson Kernel (by identity (A.23)). Hence, claim (2.26) is proved. In order to prove claim (2.27), we notice that, for any ρ ą 0 small we have that |A ρ˚ug k pxq´A ρ˚ug pxq| " ż |y|ąρ A ρ pyq|u g k px´yq´u g px´yq| dy Let R ą 2ρ, and thanks to the bound (2.19) for |y| ą R, the convergence in L 1 s pR n q norm, the uniform convergence on compact sets of g k to g and the integrability in R n zB ρ of the Mean Kernel (by identity (A.20)) we have that for k large I 1 " cpn, sqr 2s ż |y|ąρ |x´y|ěr |g k px´yq´gpx´yq| p|y| 2´ρ2 q s |y| n dy ď 2 n`s cpn, s, rq Once more, for R ą 2r and |z| ą R we use the bound (2.19) and we have that Therefore again by the convergence in L 1 s pR n q norm, the uniform convergence on compact sets of g k to g we have that I 2 ď ε 2 and it follows that thus u g has the s-mean value property at x. This concludes the proof of the claim (2.20).
We now prove the continuity of u g . Of course, u g is continuous in B r and in R n zB r . We need to check the continuity at the boundary of B r .
We send at first µ Ñ 0 and afterwards ε Ñ 0 and obtain that This solution is unique. We argue by contradiction and assume there exist u 1 and u 2 two different, continuous solutions of the Dirichlet problem. Then u " u 1´u2 would be a continuous, non constant solution to the problem Since u is continuous, suppose it has a positive maximum in B r . Therefore, there exists x 0 P B r such that upx 0 q " max xPBr upxq and upx 0 q ą 0. But, since u is zero outside B r , upx 0 q " max xPR n upxq. Thus, for x 0 P B r and u non constant upx 0 q´upx 0´y q´upx 0`y q |y| n`2s dy ą 0, which is a contradiction. Therefore u 1 " u 2 and the solution of the Dirichlet problem for the fractional Laplace equation is unique.

The Green function for the ball
The purpose of this section is to prove Theorems 3.4 and 3.5. Theorem 3.4 introduces a formula for the Green function on the ball, that is more suitable for applications, while in Theorem 3.5 the solution to the Dirichlet problem with vanishing data outside the ball and a given term inside the ball is built in terms of the Green function. We also compute the normalization constants needed in the formula of the Green function on the ball.

3.1.
Formula of the Green function for the ball. This subsection focuses on the proof of Theorem 3.4, which is divided in three cases n ą 2s, n ă 2s and n " 2s.
We introduce some calculus observations needed in the case n ą 2s.
We notice that for k ą 0, all the terms of the sum vanish. We are left with only with the term k " 0, for which we obtain that Furthermore, Γp1´sqΓpsq " π sinpπsq (as in identity (A.5)) and it follows that We recall here the statement of the Theorem 3.4 and continue with its proof. where r 0 px, zq " pr 2´| x| 2 qpr 2´| z| 2 q r 2 |x´z| 2 (3.3) and κpn, sq is a constant depending only on n and s. For n " 2s, the following holds Proof of Theorem 3.4 for n ą 2s. Let x, z P B r be fixed. We insert the explicit formula (1.14) of Φ into Definition 1.17 of the Green function and obtain that Gpx, zq " apn, sq`|z´x| 2s´n´A px, zq˘, (3.5) where Apx, zq :" ż |y|ąr P r py, xq |y´z| n´2s dy. Then we have that Apx, zq " cpn, sq ż |y|ąr pr 2´| x| 2 q s |y´z| n´2s p|y| 2´r2 q s |y´x| n dy " cpn, sq ż |y|ąr pr 2´| x| 2 q s |y´x|´n |y´z| 2s p|y| 2´r2 q s dy |y´z| n .
We proceed as follows: we use the point inversion transformation (refer to the Appendix) and after that an hyperspherical change of coordinates and the change of variables (A.26), to arrive at a simpler result. We then make use of identities (3.1) and (3.2), we introduce yet another constant with an integral definition, finally use Proposition 3.3 and we are able to finish the computation. We delegate the explicit calculus of the constants to the next subsection of the paper.
We perform a point inversion of y and of x with center at z. The inversion is defined by The transformed points are x˚P R n zB r and y˚P B r . We use formulas (A.18c), (A.18d) and (A.18e) to obtain r 2´| x| 2 " pr 2´| z| 2 qp|x˚| 2´r2 q |x˚´z| 2 , |y´x| " pr 2´| z| 2 q |y˚´x˚| |y˚´z||x˚´z| , dy |y´z| n " dy| y˚´z| n .
Notice that for n " 3 the usual spherical coordinates can be used y 1 " ρ sin θ sin θ 1 , y 2 " ρ sin θ cos θ 1 and y 3 " ρ cos θ and the computations follow as in the more general case. Similar computations are employed in the case n " 2 and n " 1.
Proof of Theorem 3.4 for n ă 2s. We consider without loss of generality r " 1 (by rescaling, the statement of the theorem is verified in the more general case). We insert the explicit formula (1.14) of Φ into definition (1.17) of the Green function to obtain Gpx, zq " ap1, sq`|z´x| 2s´1´A px, zq˘, (3.10) where Apx, zq :" ż Rzp´1,1q P 1 py, xq |z´y| 1´2s dy.
Proof of Theorem 3.4 for n " 2s. Without loss of generality, we assume r " 1. In the definition (1.17) of the Green function we insert the explicit formulas (1.14) and (1.15) of the function Φ, respectively P 1 . Moreover, we use the explicit values of the constant a´1, 1 2¯f rom (1.22) and c´1, 1 2¯f rom (1.23). We obtain that We perform the change of variables v " yx´1 y´x . Since 1´v 2 ě 0, we have that |v| ď 1. We set w :" xz´1 z´x and observe that |w| ě 1. We have that x´y " 1´x 2 v´x and it follows that We use identity (A.35) and since |w| ě 1 and |x| ď 1 we obtain that Apx, zq " π log´|w|`pw 2´1 q 1{2¯´π log 2`π log 2`π log |x´z| " π logp1´zx`ap1´x 2 qp1´z 2 qq.

Inserting this into (3.15) we obtain that
Gpx, zq " 1 π logˆ1´z x`ap1´x 2 qp1´z 2 q |x´z|˙, which the desired result. This completes the proof of Theorem 3.4 for n " 2s.

3.2.
Representation formula for the fractional Poisson equation. This subsection is dedicated to the proof of Theorem 3.5, which we recall here.
Theorem 3.5. Let r ą 0, h P C 0,ε pB r q, and let upxq :" Then u is the pointwise continuous unique solution to the problem (1.2) Proof of Theorem 3.5. We identify h with its C 0,ε c pR n q extension, namely we considerh P C 0,ε c pR n q with B r Ă supph such thath " h on B r . Then, by definition (1.17) of the function G, we have that in B r upxq "  gpxq :" h˚Φpxq for any x P R n .
As we have seen in Theorem 2.9, g P L 1 s pR n q X CpR n q. Consider now the two functions v 0 pxq " gpxq in R n and v 1 pxq " gpxq if x P R n zB r then, for any x P R n upxq " v 0 pxq´v 1 pxq.
Furthermore, inside B r , by Theorem 2.9 and Theorem 2.11 p´∆q s upxq " hpxq´0 " hpxq, hence u is solution (1.2). Also, Theorem 2.9 and Theorem 2.11 assure the continuity of u in R n . The solution is unique. We refer to the proof of the uniqueness for the solution of problem (1.1) in Theorem 2.11.

Computation of the normalization constants.
This subsection is dedicated to the computation of the constants encountered in the course of the proof of Theorem 3.4.
Proof. We consider the change of variable t 2 " τ and obtain 2 ż 1 0 t n´2s´1 p1´t 2 q s´1 dt " We use identities (A.9) and (A.10) referred to the Beta function to obtain that which is exactly the result.
By inserting the value of apn, sq from definition (1.21) and the value of kpn, sq from identity (3.16) we obtain that κpn, sq " apn, sqkpn, sq For n ă 2s, we recall definitions (3.13), (3.14) and (1.21), we use identities (A.5), (A.7) and (A.3) relative to the Gamma function and obtain that κp1, sq "´ap1, sqkp1, sq It is easily verified that this is the general formula given in Theorem 3.7, evaluated at n " 1. On the other hand, we recall that κ´1, 1 2¯" 1 π , as we have seen in the proof of Theorem 3.4 for n " 2s. This concludes the proof of Theorem 3.7.
We make now a remark on the constants Cpn, sq defined by (1.5) and cpn, sq defined in (1.23). These two constants are both used in various works in the definition of the fractional Laplacian, but as we have seen in the course of this paper, they arise for different normalization purposes. The constant Cpn, sq as defined by [3] is consistent with the Fourier expression of the fractional Laplacian, meanwhile cpn, sq as introduced in [10] is used to normalize the Poisson kernel (and the s-mean kernel), and is consistent with the constants used for the fundamental solution and the Green function. The point that we want to make here is that the two constants are not equal. We compute the value of the constant Cpn, sq and clarify the relation between cpn, sq and Cpn, sq in the following Proposition 3.9. In order to prove this result, we need the direct computation of the fractional Laplacian of a particular function. Namely: Lemma 3.8 . Let upxq " p1´|x| 2 q s . Then in B 1 p´∆q s upxq " Cpn, sq ω n 2 Bps, 1´sq, where B is the Beta function defined in (A.8).
The more general case of this result (more precisely, for the function upxq " p1´|x| 2 q p for any p ą´1) was proved in [4] and [5]. For the sake of completeness, we perform here the computation in this particular case, with few modifications respect to the more general proof.
Proof of Lemma 3.8. The proof follows from four main identities. We first work in the one dimensional case, and then see how the n dimensional case relates to the one dimensional case. Consider the operator Lupxq as the regional fractional Laplacian in dimension 1 restricted to p´1, 1q and with no normalization constant Lupxq :" P.V.
The computation for x " 0 of the restricted fractional Laplacian gives Lup0q " Bp1´s, sq´1 s . (3.17) Then, for any |x| ă 1 we have that We then compute the fractional Laplacian in dimension one in B 1 to be p´∆q s upxq " Cp1, sqBp1´s, sq. Finally, in the n dimensional case for |x| ă 1 we prove that where h n P R is the n th component of a vector h. Notice that the argument of the fractional Laplacian inside the integral is one dimensional. The conclusion follows immediately. We now prove identity (3.17). Since the integrand function is even we have that Lup0q " P.V. ż 1 1 1´p1´|y| 2 q s |y| 1`2s dy " 2 lim εÑ0 ż 1 ε 1´p1´y 2 q s y 1`2s dy.
We then use the change of variables ω " y 2 to obtain that We use the integral formulation (A.9) of the Beta function we compute the limit of 1´p1´ε 2 q s ε 2s to be 0 and it immediately follows that Lup0q " Bp1´s, sq´1 s , that is identity (3.17). We now prove identity (3.18) for any |x| ă 1. We use the change of variables ω " x´y 1´xy , that is y " ω´x ωx´1 . We notice that |ω| ď 1, 1´y 2 " p1´x 2 qp1´ω 2 q p1´ωxq 2 and x´y " ωp1´x 2 q 1´ωx . We also remark that 1´ωx ě 0, fact that will allow us to remove the absolute value in the computations. We thus have Lupxq " P.V.
We now compute Ipxq. We do a Taylor expansion of the function p1´ωxq´1 in 0, that is p1´ωxq´1 " ř 8 k"0 pxωq k and insert this into the definition of Ipxq. We have that We recognize that the odd part of the sum vanishes, and we are left with the sum of the even powers where in the last line we have used the change of variables t " ω 2 . Integrating by parts, we have that Ipxq "´lim Notice that for k ě 1 lim εÑ0 ε 2k´2s p1´ε 2 q s k´s " 0, we then use the integral representation of the Beta function from identity (A.9) and it follows that
We continue with the n dimensional case and prove identity (3.20) in B 1 . To do this, without loss of generality and up to rotations, we consider x " p0, 0, . . . , x n q with x n ě 0, we use polar coordinates and write x´y " th, with h P BB 1 and t ě 0. Then p´∆q s upxq Cpn, sq " P.V. ż R n p1´|x| 2 q s´p 1´|y| 2 q s |x´y| n`2s dy Now we change the variable t "´|x|h n`τ a |h n x| 2´| x| 2`1 . We notice that 1´|x`ht| 2 " p1´τ 2 qp1´|x| 2`| h n x| 2 q and obtain that ant thus the identity (3.20). Finally, since the one dimensional fractional Laplacian is constant, we obtain p´∆q s upxq " Cpn, sqBp1´s, sq ω n 2 and this concludes the proof of the Lemma 3.8. b) The quotient between the constants Cpn, sq introduced in (1.5) and cpn, sq defined in (1.23) is given by Cpn, sq cpn, sq " 2 2s Γps`1qΓp n 2`s q Γp n 2 q . Proof of Theorem 3.9. By Lemma 3.8 we have that in B 1 p´∆q s upxq " Cpn, sq ω n 2 Bp1´s, sq.
For n ‰ 2s, we use the representation formula for the problem with a given forcing term inside the ball and vanishing data outside the ball (we refer to Theorem 3.5) and obtain that dt˙dy " ω n Bˆs, n 2˙.

ΓpsqΓp1´sq
Γp n 2 q 2 2s π n 2 Γ 2 psq π n 2 sΓp n 2 q ΓpsqΓp n 2 q Γps`n 2 q , therefore Cpn, sq " 2 2s sΓp n 2`s q π n 2 Γp1´sq . The picture is a little different for n " 2s. We use equation (3.19)  In oder to compute the quotient Cpn, sq cpn, sq , we use the value of cpn, sq from definition (1.23) and the identity (A.5). We obtain that Cpn, sq cpn, sq " 2 2s Γps`1qΓpn{2`sq Γpn{2q , thus the desired result. We remark also that the term π 2s depends on the normalization factor used in the Fourier transform.
Appendix A. Appendix A.1. The Gamma, Beta and hypergeometric functions. We recall here a few notions on the special functions Gamma, Beta and hypergeometric (refer to [1] §6 and §15 for details).

Beta function
The Beta function has an integral representation (see [1], §6.2.1): for x, y ą 0 Bpx, yq :" An equivalent representation is Finally, we have the identity

Hypergeometric functions
The hypergeometric function can be expressed in different manners, and under different conditions. We recall the ones useful for our own purposes.
(1) Gauss series (refer to §15.1.1) where pqq k is the Pochhammer symbol defined by: The interval of convergence of the series is |w| ď 1. The Gauss series, on its interval of convergence, diverges when c´a´b ď´1, is absolutely convergent when c´a´b ą 0 and is conditionally convergent when |w| ă 1 and´1 ă c´a´b ď 0. Also, the series is not defined when c is a negative integer´m, provided a or b is a positive integer n and n ă m.
(3) Linear transformation formulas(refer to §15.3.3 - §15.3.6) From the integral representation (A.14), the following transformations can be deduced. Let r ą 0 to be fixed. Unless otherwise specified, the results in the Appendix hold for n ě 1 and any value of s P p0, 1q. We prove the results in the case n ą 2s, but they are easily adaptable to the case n " 1.
Definition A.1. Let x 0 P B r be a fixed point. The inversion with center x 0 is a point transformation that maps an arbitrary point y P R n ztx 0 u to the point K x0 pyq such that the points y, x 0 , K x0 pyq lie on one line, x 0 separates y and K x0 pyq and This is a bijective map from R n ztx 0 u to itself. Of course, K x0`Kx0 pxq˘" x. When this does not generate any confusion, we will use the notation y˚:" K x0 pyq and x˚:" K x0 pxq to denote the inversion of y and x respectively, with center at x 0 .
Remark A.2. It is not hard to see, from definition (A.16), that (A.17) Proposition A. 3 . Let x 0 P B r be a fixed point, and x˚and y˚be the inversion of x P R n ztx 0 u respectively y P R n ztx 0 u with center at x 0 . Then: a) Points on the sphere BB r are mapped into points on the same sphere, Points outside the sphere BB r are mapped into points inside the sphere, Proof. Elementary geometrical considerations will be used to prove most of the claims of this lemma. We refer to Figure 1 for the proof of claims a) to c) and to Figure 2 for the claim e).
d) Without loss of generality, we consider the point inversion of radius one with center at zero y˚"´1 |y| 2 y.
The solution to the system b˚2´β 2`a˚2 2at 1 t 2 " b bå a˚" bb˚" ρ is given by β " ρα ab .
Notice that for n " 3 the usual spherical coordinates can be used y 1 " ρ sin θ sin θ 1 , y 2 " ρ sin θ cos θ 1 and y 3 " ρ cos θ and the computation follow as in the more general case. Similar computations are employed in the case n " 2 and n " 1.
Lemma A.6 . For any r ą 0 and any x P B r cpn, sq ż Br dy pr 2´| y| 2 q s |x´y| n´2s " 1. (A.33) Proof. Let Let y˚be the inversion of y with center at x (notice that |y˚| ą r). Then by (A.18c) we have that |x´y| 2 r 2´| y| 2 " r 2´| x| 2 |y˚| 2´r2 , and by (A.18d) dy |x´y| n " dy| x´y˚| n .