Mean Value Properties of Fractional Second Order Operators

COMMUNICATIONS ON doi:10.3934/cpaa.2015.14.83 PURE AND APPLIED ANALYSIS Volume 14, Number 1, January 2015 pp. 83–106

MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS

Fausto Ferrari Dipartimento di Matematica dell’ Universit`adi Bologna Piazza di Porta S. Donato, 5, 40126, Bologna, Italy

Abstract. In this paper we introduce a method to define fractional operators using mean value operators. In particular we discuss a geometric approach in order to construct fractional operators. As a byproduct we define fractional linear operators in Carnot groups, moreover we adapt our technique to define some nonlinear fractional operators associated with the p−Laplace operators in Carnot groups.

1. Introduction. In this paper we introduce and discuss a geometric method to define nonlocal operators arising from (local) mean value formulas. The idea we are going to discuss may be considered as indirectly and implicitly suggested in the book [25] by Landkof. We have to add, for completeness, that the afore mentioned book is largely based on the fundamental work by Marcel Riesz, see [34]. The idea that we want to discuss is quite natural and it is worth describing it in the simplest case. Let us consider the mean value operator of a continuous function φ defined for simplicity in n, R Z M[φ](x, r) = φ(σ)dHn−1(σ), ∂B(x,r) where, as usual, B(x, r) is the open Euclidean ball centered in x with radius r, R n−1 n−1 ∂B(x, r) is its boundary, ∂B(x,r) φ(σ)dH (σ) denotes the spherical mean, H is the n − 1-Hausdorff measure and Z Z n−1 1 n−1 φ(σ)dH (σ) := n−1 φ(σ)dH (σ). ∂B(x,r) H (∂B(x, r)) ∂B(x,r) Let us fix x ∈ Rn and for every r > 0 consider the function r → M[φ](x, r) − φ(x). Of course this function gives information about how far the mean value of φ at x is from the value of φ evaluated in x. It is also clear that, if M[φ](x, r) − φ(x) ≥ 0 for every r, then we are describing a function whose values are, averagely, larger than the value of φ(x). For example the function φ(x) = |x|2 in any ball centered at 0 satisfies this condition. It is well

2000 Mathematics Subject Classiﬁcation. Primary: 35H20, 35J60; Secondary: 35E05, 35J92. Key words and phrases. Mean operators, Carnot groups, fundamental solutions, nonlinear operators. The author is supported by the ERC starting grant project 2011 EPSILON (Elliptic PDEs and Symmetry of Interfaces and Layers for Odd Nonlinearities) n. 277749 and by RFO of the University of Bologna, Italy.

83 84 FAUSTO FERRARI known that the point wise behaviour for small r of r → M[φ](x, r) − φ(x), for each x in a fixed open set Ω, determines necessary and sufficient conditions to say that φ is harmonic in Ω, see e.g. [32] for a long list of authors who studied the relationships among mean properties and harmonic functions. However, in general, we would have to expect positive and negative values of M[φ](x, r) − φ(x) associated with different value of r, especially when r becomes larger and larger. So, it is, in a sense, natural to study the global behaviour of M[φ](x, r) − φ(x) by considering the integral of our function possibly with respect to a fixed measure µ. Namely we are interested in: Z ∞ (M[φ](x, r) − φ(x)) dµ(x, r). (1) 0 In this way we are able to evaluate the weight of the positive values of M[φ](x, r) − φ(x) with respect to its negative value and vice versa. On the other hand, we recall that whenever M[φ](x, r) − φ(x) vanishes as r → 0, we may also suppose that m X k m M[φ](x, r) − φ(x) = ak(x)r + o(r ), (2) k=1 as r → 0, for some given exponent m. For example, see one more time [32] for the historical details about the authorship of the following result, we know that continuous functions φ such that for every x ∈ Ω satisfy M[φ](x, r) − φ(x) = o(r2), (3) as r → 0, are harmonic functions indeed. Thus, in order to keep in account to the contribution of M[φ](x, r) − φ(x) for large r in a global framework, it is natural to multiply M[φ](x, r) − φ(x) by a singular weight as r → 0, and then considering the integral of that product in R+. For example we multiply M[φ](x, r)−φ(x) by r−s, where s > 0 is a positive number. As a consequence, we need to ask that previous integral (1) converges. Indeed, we would like that Z +∞ (M[φ](x, r) − φ(x))r−sdr < ∞. (4) 0 The approximation given in (2) depends on the definition of M, however, we can suppose to normalise this definition assuming that the size r is defined in such a way that, possibly changing the definition of the operator M, we get M[φ](x, r) − φ(x) lim = a2(x) ∈ R. r→0+ r2 More precisely, if u is sufficiently smooth, then R n−1 ∂B(x,r) φ(σ)dH (σ) − φ(x) lim = Cn∆φ(x), r→0+ r2 where Cn is a positive constant independent with x. Hence, φ satisfies Cn∆φ(x) = f(x) if and only if M[φ](x, r) − φ(x) = f(x)r2 + o(r2), as r → 0. As a consequence, MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 85 we restrict ourselves to consider mean value operators such that 2 2 M[φ](x, r) − φ(x) = a2(x)r + o(r ). (5) Indeed, if M[φ](x, r) − φ(x) = arβ + o(rβ), for some β > 0, then

2 2 2 M[φ](x, r β ) − φ(x) = ar + o(r ),

2 and defining M˜ [φ](x, r) := M[φ](x, r β ) our approach applies to M˜ on balls of different sizes. R 1 2−s Thus, recalling (4) and (5), on one side we need that 0 r dr converges. That is, we have to ask that s < 3. Moreover, requiring that the integral (4) converges, even for s → 3−, we may choose a normalising constant c(s, n) such that Z 1 c(s) c(s) c(s) lim c(s) r2−sdr = lim ( − lim r3−s) = lim = 1. s→3− 0 s→3− 3 − s r→0+ 3 − s s→3− 3 − s In particular this means that c(s) = 3 − s. On the other hand, assuming that the behaviour of M[φ](x, r) − φ(x), for large r, is uniformly bounded by a constant, we need to impose that s > 1 in order to have the convergence of the integral Z +∞ −s Mr[φ(y) − φ(x)]r dr. 1 Hence, it is natural to put in evidence this remark by writing s = 1 + 2α, where here α ∈]0, 1[ and to define, up to a multiplicative constant, the operator Z +∞ Lαφ(x) = (M[φ](x, r) − φ(x)) r−1−2αdr (6) 0 that is well defined, for example, in S(Rn). Applying the coarea formula we get immediately Z α 1 φ(y) − φ(x) L φ(x) = n+2α dy, nω n |x − y| n R n where ωn denotes the volume of the unit ball in R . In this approach we just used a mean value operator defined on the boundaries of a family of sets. Indeed, in our example, we considered the Euclidean balls. Nevertheless, a similar construction can be made, in principle, using other sets. The operator that we obtained is, up to some possible different normalisation constants, related to the Euclidean fractional Laplace operator. Indeed multiplying the integral in (6) with the right constant c = cα,n, see e.g. [9], we get that Lα := −(−∆)α, where (−∆)α is the Euclidean fractional Laplace operator for α ∈]0, 1[. Moreover it α is also true that, see one more time [9], limα→1− L φ(x) = ∆φ(x). After this remark it is quite natural to generalise such idea to a metric context in order to construct fractional operators naturally arising from assigned mean value formulas. Hence, one of the aims of this paper consists in revisiting the classical methods based on the mean value properties that, in the case of the nonlocal operators have been apparently considered less flexible with respect to the extension method proposed in [8]. An application of this approach is given by the proof of the Harnack inequality that we show in Section2. Indeed, by using the Caffarelli-Silvestre approach based on the extension method, see [8], the Harnack inequality boils down thanks to a sophisticated theory based 86 FAUSTO FERRARI on Muckenhoupt weights, see [10]. With our approach, the proof of the Harnack inequality follows from a sort of global mean equation, by mimiking the classical idea, see e.g. in [18], based on the application of the mean value property to positive harmonic functions on concentric balls. Since the definition of these nonlocal operators depends on the mean value operators kept in account, the proof of Harnack inequality may be, in principle, straightforwardly extended to all those situations where the mean value operators are defined on balls that satisfy the doubling property with respect to the assigned measure. For example to Carnot groups. It is worth saying that the proof of the Harnack inequality for fractional Laplace operators in Carnot groups has been recently obtained in [11] by generalising the extension approach, due to Caffarelli and Silvestre, to the stratified groups. Nevertheless, the computation of the kernels involved in that project might be long. On the contrary, moving from the metric approach showed in this paper, we can describe the kernels of the fractional operators in the case of Carnot groups, in particular in the Heisenberg group case, see Remark1 below, because their structures depend on the gauge considered in the examined Carnot group. From this point of view this paper may be a considered as counterpart approach of [11]. Moreover, our technique, theoretically, may be useful in order to describe even the evolutionary case. Indeed this is currently object of a work in progress, where the proof of the Harnack inequality requires some extra cares. Summarising, we think that the approach dealt with in this paper can be useful to further applications in a metric framework. In particular we list below the main definition and the new results contained in this paper, see Section4. Definition 1.1. Let (X, d) be a metric space and µ a measure on X. Let M : C(X, R) → R be a mean value operator defined as 1 Z M[φ](x, r) = φ(σ)dµ(σ) µ(∂B(x, r)) ∂B(x,r) such that for every α ∈]0, 1[ Z 1 (M[φ](x, r) − φ(x)) r−1−2αdr < ∞. 0 For every every α ∈]0, 1[ and for every bounded function φ ∈ C(X, R) we define the linear operator Z +∞ Lαφ(x) = (M[φ](x, r) − φ(x)) r−1−2αdr. 0

Let dG be the distance induced by a gauge norm in the Carnot group G, such that B(x, r) = {y ∈ G : dG(x, y) < r}. Let Q and ∇H be respectively the homogeneous dimension and the horizontal gradient in G, see Section4 and [5] for the definitions. Theorem 1.2. Let G be a stratified Carnot group with homogeneous dimension Q and dG a gauge on it. Then for every s ∈]0, 1[ and for every bounded smooth function φ, Z Lαφ(x) = (φ(y) − φ(x))N (x, y)dy, G where 2 |∇H dG(x, y)| N (x, y) = c(Q) Q+2α dG(x, y) MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 87 and c(Q) is a positive constant depending only on Q. Remark 1. The operator Lα can be always normalized with a constant C = C(Q, α) in such a way that Z |∇ d (x, y)|2 lim c(α, Q) (φ(y) − φ(x)) H G = ∆ φ(x). α→1− d (x, y)Q+2α G G G Remark 2. In the particular case of the Heisenberg group Hn the α-subLaplacian kernel is up to a multiplicative constant: | κ − ξ |2 + | h − η |2 N n = , H n Q+2+2α dH (x, y) n n n where x = (κ, h, τ) ∈ H , y = (ξ, η, t) ∈ H , and dH is the distance induced by the gauge norm in the Heisenberg group, see (27) in Corollary1. A second target of this paper originates from the possibility of applying the geometric approach just described in order to define new fractional operators associated with nonlinear local operators. In particular the p−Laplace operators. Indeed, in the Section5, as example of this method, we define nonlocal nonlinear operators originated from a particular form of local mean value property. In this part we have been inspired by the characterisation of p−harmonic functions in the viscosity sense given in [28]. More precisely, for every smooth function (for simplicity we can assume that u ∈ S(Rn)), and for every α ∈]0, 1[, p > 1, we define the α−p−spherical harmonious operator as

Z +∞ max∂B(x,r) u+min∂B(x,r) u − u(x) Z α 2 ˜ u(y) − u(x) Lp u(x) =γ ˜ 1+2α dr + β n+2α dy r n | x − y | 0 R p−2 ˜ n where α ∈]0, 1[ andγ ˜ = p+n−2 and β = p+n−2 . Concerning this last nonlinear example see also [1], where the fractional infinity Laplacians and the dynamic pro- gramming principle were studied.1 The paper is organised as follows. In Section2 we give a geometric prove of the Harnack inequality of the fractional Laplace, in Section3 we discuss some conse- quences of this metric approach starting from the fundamental solution associated with some linear operators, in Section4 we trait the subject in the framework of the Carnot groups. In the last Section5 we remark that for every α ∈]0, 1[, the fractional operator −(−∆)α is related with the nonlinear operator div(|∇ · |p(α)−2∇), where p(α) = n−α 2 n−α+(1−α) , since they have the same fundamental solution opening, hopefully, a perspective of research in this nonlinear field even in Carnot groups. For the reader’s convenience, we decided to put in Appendix some results more o less well known, which however are not easy to find in literature with complete proofs. In particular we recall some classical arguments that could be useful to put in evidence the geometric character of our definition. More precisely we give a direct proof of the fact that |x|−n+2α is a solution of (−∆)αu = 0 in Rn \{0} for every α ∈]0, 1[ and for every n ∈ N. Said otherwise, |x|−n+2α is the fundamental solution

1The author wishes to thank the anonymous referee for pointing out [1]. 88 FAUSTO FERRARI of the fractional laplacian −(−∆)α. Eventually, we use this result to prove that the so called α Kelvin transformation x K u(x) =| x |α−n u( ), α | x |2 satisﬁes Proposition4, that is, for every ∈ Rn \{0}and α ∈]0, 1[, x (−∆)α(K u)(x) =| x |−2α−n (−∆)αu( ), 2α |x|2 see [3] and [31] for a probabilistic approach.

2. Geometric proof of the Harnack inequality of positive α−harmonic functions in the Euclidean case. In this section we give a geometric proof of the Harnack inequality for the α−harmonic functions. More precisely, we say that a continuous function C(R) is an α− subharmonic (superharmonic) function in Ω ⊆ Rn if for every x ∈ Ω, Lαφ(x) ≥ (≤)0.

In particular if Lαφ(x) = 0 for every x ∈ Ω ⊆ Rn, we say that φ is α−harmonic in Ω. The α-potential theory has been developed in [34], see also [25], apparently with- out to put in evidence the role of the fractional Laplace operator −(−∆)α. There exists a proof of Harnack inequality based on this approach, but it is not easy to understand. Recently, in [8], the proof of the Harnack inequality for positive α−harmonic function has been reduced to the proof of the Harnack inequality of a positive function v deﬁned on Rn × R+ such that v(x, 0) = φ(x) that is solution of a local operator in divergence form with a weight. We give a a geometric proof of the Harnack inequality for positive α−harmonic function in an open set Ω.

Proposition 1. Let φ be a positive continuous function in Rn such that Lαφ(x) = 0 in Ω, α ∈]0, 1[. Then for every ball Br ⊂ B3r ⊂ Ω

sup φ ≤ 3n+2α inf φ B Br r Proof. We denote 1 Z Z M[φ](x, r) = φ(σ)dHn−1(σ) = φ(σ)dHn−1(σ), | ∂B(x, r) | ∂B(x,r) ∂B(x,r) where Hn−1 is the (n − 1)−Hausdorﬀ measure, B(x, r) is the Euclidean ball with n−1 n centre x and radius r, so that | ∂B(x, r) |= nωnr , and | B(x, r) |= ωnr . In particular ωn denotes the measure of the euclidean ball of radius 1. Moreover, by the co-area formula Z Z r Z φ(ξ)dξ = ( φ(σ)dHn−1(σ))dt. B(x,t) 0 ∂B(x,t) MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 89

Hence Z r Z Z r Z n−1 −1−2α −1 n−1 −n−2α ( φ(σ)dH (σ))t dt = (nωn) ( φ(σ)dH (σ))t dt 0 ∂B(x,t) 0 ∂B(x,t) Z Z r Z −1 −n−2α t=r −1 −n−2α−1 = (nωn) [t φ(ξ)dξ]t=0 + (n + 2α)(nωn) ( φ(ξ)dξ)t dt B(x,t) 0 B(x,t) Z Z r Z = n−1r−2α φ(ξ)dξ + (n + 2α)n−1 ( φ(ξ)dξ)t−1−2αdt. B(x,r) 0 B(x,t) (7)

Analogously Z +∞ Z ( φ(σ)dHn−1(σ))t−1−2αdt ∂B(x,t) Z +∞ Z −1 n−1 −n−2α =(nωn) ( φ(σ)dH (σ))t dt ∂B(x,t) Z −1 −n−2α t=+∞ =(nωn) [t φ(ξ)dξ]t= (8) B(x,t) Z +∞ Z −1 −n−2α−1 + (n + 2α)(nωn) ( φ(ξ)dξ)t dt B(x,t) Z Z +∞ Z =−n−1−2α φ(ξ)dξ + (n + 2α)n−1 ( φ(ξ)dξ)t−1−2αdt. B(x,) B(x,t) Since Z +∞ α α −1−2α −(−∆) φ(x) = L φ(x) = cα,n (M[φ](x, r) − φ(x)) r dr, 0 if φ is a positive function for every x ∈ Rn, such that −(−∆)αφ(x) = 0, in some subset Ω ⊆ Rn, then Z Z +∞ −1−2α −1−2α 0 = cα,n (M[φ](x, r)−φ(x)) r dr + cα,n (M[φ](x, r)−φ(x)) r dr, 0 for every x ∈ Ω ⊆ Rn. Let us denote for every positive number , Z −1−2α H(x, ) = cα,n (M[φ](x, r) − φ(x)) r dr. 0

Then, for every x ∈ Ω ⊆ Rn, Z +∞ Z +∞ −1−2α −1−2α 0 = H(x, ) + cα,n M[φ](x, r)r dr − cα,nφ(x) r dr Z +∞ −1−2α cα,n −2α = H(x, ) + cα,n M[φ](x, r)r dr − φ(x). 2α Hence Z +∞ 2α 2α −1−2α cα,n 0 = H(x, ) + cα,n M[φ](x, r)r dr − φ(x) 2α 90 FAUSTO FERRARI and recalling (8) we obtain Z +∞ cα,n 2α 2α −1−2α 2α φ(x) = H(x, )+cα,n M[φ](x, r)r dr = H(x, ) 2α (9) Z Z +∞ Z ! 2α −1 −2α −1 −1−2α +cα,n −n φ(ξ)dξ+(n+2α)n ( φ(ξ)dξ)t dt B(x,) B(x,t) and keeping in mind that φ ≥ 0 in all of Rn we get that for every x, y ∈ B(¯x, R) such that B(¯x, 3R) ⊂ Ω, and −(−∆)αφ = 0 in Ω. Then for every t ≥ R,B(x, t) ⊂ B(y, 3t), Z cα,n 2α −1 φ(x) ≤ H(x, ) − cα,nn φ(ξ)dξ 2α B(x,) Z R Z −1 2α −1−2α + cα,n(n + 2α)n ( φ(ξ)dξ)t dt B(x,t) Z +∞ Z −1 n 2α −1−2α + cα,n(n + 2α)n 3 ( φ(ξ)dξ)t dt R B(y,3t) moreover recalling the representation formula (9) we get: Z 2α −1 = H(x, ) − cα,nn φ(ξ)dξ B(x,) Z R Z −1 2α −1−2α + cα,n(n + 2α)n ( φ(ξ)dξ)t dt B(x,t) Z +∞ Z −1 n+2α 2α −1−2α + cα,n(n + 2α)n 3 ( φ(ξ)dξ)τ dt 3R B(y,τ) =2αH(x, ) − 3n+2α2αH(y, ) Z R Z −1 2α −1−2α + cα,n(n + 2α)n ( φ(ξ)dξ)t dt B(x,t) Z 3R Z ! −3n+2α ( φ(ξ)dξ)τ −1−2αdτ B(y,τ) Z Z ! n+2α −1 n+2α 3 cn,α + cα,nn 3 φ(ξ)dξ − φ(ξ)dξ + φ(y) B(y,) B(x,) 2α

cn,α Dividing the both sides by 2α , we get: 2α φ(x) ≤ 2α(H(x, ) − 3n+2αH(y, )) cn,α Z R Z + 2α(n + 2α)n−12α ( φ(ξ)dξ)t−1−2αdt B(x,t) Z 3R Z ! −3n+2α ( φ(ξ)dξ)t−1−2αdt B(y,t) Z Z ! + 2αn−1 3n+2α φ(ξ)dξ − φ(ξ)dξ + 3n+2αφ(y). B(y,) B(x,) MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 91

Letting → 0, the second couple of integrals may be handled with the De l’Hˆopitalrule, we get φ(x) ≤ − 2α(n + 2α)n−1φ(x) + 2α(n + 2α)n−13n+2αφ(y) + 2αn−1 3n+2αφ(y) − φ(x) + 3n+2αφ(y) that is

(1 + 2αn−1 + 2α(n + 2α)n−1)φ(x) ≤ 3n+2α(1 + 2αn−1 + 2α(n + 2α)n−1)φ(y) or equivalently φ(x) ≤ 3n+2αφ(y).

3. Towards a metric definition of the fractional laplacian. In the case of the fractional Laplace operator we know that Z +∞ α −1−2α −(−∆) φ(x) = cα,n (M[φ](x, r) − φ(x)) r dr. 0 From this point of view it is natural to put in evidence that the definition is geometric. It only appears the mean value operator. It is worth to say that in this way the definition of our operator, in principle, needs only a metric structure and a measure. We remark that if Z M[φ](x, r) = φ(y)dy, g(x,y)=r we are interested to find s such that ! Z +∞ Z φ(y) − φ(x) dy r−sdr 0 g(x,y)=r | ∇g(x, y) | converges. In particular, denoting h(x, r) := Hn−1({y : g(x, y) = r)}) and assuming that h is well defined in this case, by Federer coarea formula we get

Z +∞ Z ! Z φ(y) − φ(x) −s φ(y) − φ(x) dy dy r dr = s . | ∇g(x, y) | n g(x, y) h(x, g(x, y)) 0 g(x,y)=r R So for example, if g(x, y) = |x − y| Pizzetti’s formula holds and φ is bounded, 1 then a1 = 0 and a2(x) = 2n ∆φ(x). In this case the integral will converge in a neighbourhood of 0 whenever −s + 2 > −1, that is if s < 3. On the other hand assuming that the behaviour of M[φ](x, r) − φ(x), for large r, is bounded by a constant, we need to ask that s > 1. It is natural to put in evidence this remark by writing s = 1 + 2α, where here α ∈]0, 1[ and to deﬁne the operator Z +∞ Lαφ = (M[φ](x, r) − φ(x)) r−1−2αdr, 0 because |∇|x − y|| = 1. We can distinguish two types of results depending on the form of the operator. 92 FAUSTO FERRARI

3.1. The divergence case. Let L = div(A(x)∇), where A = AT is a matrix. Suppose that there exists λ ≥ 0 such that hA(x)ξ, ξi ≥ λ | ξ |2, for every ξ ∈ Rn. Assume that L is endowed with a fundamental solution Γ. Deﬁne 1 Ω (x) = {y ∈ n : Γ(x, y) > }, r R r see [7].

Theorem 3.1. Let L = div(A(x)∇) be an elliptic operator, possibly degenerate endowed with a fundamental solution sufficiently smooth. Assume that there exists positive constants γ and C such that Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i u(σ) dHn−1(σ) − u(x) ∼ rγ , (10) ∂Ωr | ∇Γ(x, σ) | as r → 0+ and Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i n−1 u(σ) dH (σ) − u(x) ≤ C, (11) ∂Ωr | ∇Γ(x, σ) | for every r. Then for every s > 1 such that γ − 1 < s the following operator is well defined in S(Rn): γ Z hA(y)∇Γ(x, y), ∇Γ(x, y)i Lsu(x) = (u(y) − u(x)) dy. (1−s) γ +1 2 n 2 R Γ(x, y) Proof. In general, if L = div(A(x)∇),A = AT , and L is endowed with a fundamental solution Γ, then the following mean value formula holds for solution of Lu = 0 Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i u(x) = u(σ) dHn−1(σ), ∂Ωr (x) | ∇Γ(x, σ) | at least for almost all r. Hence, if (10) and (11) hold we fix our normalisation in order to obtain M[φ](x, r) − φ(x) lim = cn(x). r→0+ r2

In this case let B(x, r) = Ω 2 (x). We are interested in the convergence of the r γ integral ! Z +∞ Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i u(σ) dHn−1(σ) − u(x) r−sdr. (12) 0 ∂Br (x) | ∇Γ(x, σ) |

In general if B(x, r) = Ω 2 (x), then r γ Z n−1 2 2 Mr[φ](x, r) − φ(x) = (φ(σ) − φ(x))Kγ (x, σ)dH (σ) = cnr + o(r ), ∂B(x,r) (13) where hA(x)∇Γ(x, σ), ∇Γ(x, σ)i K (x, σ) = . γ | ∇Γ(x, σ) | MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 93

− γ As a consequence, with a change of variable r = t 2 we get, Z +∞ Z ! n−1 −s (u(σ) − u(x))Kγ dH (σ) r dr 0 ∂B(x,r)   Z +∞ Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i n−1 −s =  (u(σ) − u(x)) dH (σ) r dr 0 ∂Ω 2 (x) | ∇Γ(x, σ) | r γ Z +∞ Z ! γ hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i n−1 sγ−γ −1 = (u(σ)−u(x)) dH (σ) t 2 2 dt 2 | ∇Γ(x, σ) | 0 ∂Ωt−1 (x) γ Z hA(y)∇Γ(x, y), ∇Γ(x, y)i = (u(y) − u(x)) dy. (1−s) γ +1 2 n 2 R Γ(x, y) (14)

Actually, if A = I and Γ(r) = cr2−n, it results in that

1−n Kγ (x, y) =| ∇Γ(x, σ) |= c|x − y| and on ∂B(x, r) = Ωrn−2 (x) we get 1−n Kγ (x, y) = cr .

Notice that y ∈ ∂B(x, r) if and only if |x − y| = cnr and B(x, r) = Ωrn−2 (x). Hence in this particular case we have Z hA(σ)∇Γ(x, σ), ∇Γ(x, σ)i n−1 Mr[φ](x, r) − φ(x) = φ(σ) dH (σ) − φ(x) ∂B(x,r) | ∇Γ(x, σ) | Z 1−n n−1 2 2 = c|x − σ| φ(σ)dH − φ(x) = cnr + o(r ). ∂B(x,r) (15)

3.2. Variable coeﬃcients case, the nondivergence case. In this case, we ex- amine operators that are not in divergence form. Let L = Tr(A(x)D2u(x)) deﬁne in Rn,A = AT . Suppose that hA(x)ξ, ξi ≥ λ | ξ |2, for every ξ ∈ Rn, λ ≥ 0. Let for every r > 0

n −1 2 E(x, r) = {y ∈ R : hA(x) (x − y), (x − y)i = r }. and (phA−1(x)(x − y), (x − y)i)−2α−1|A−1(x) · (x − y)| K (x − y) = , α Hn−1(∂E(x, η(x − y))) where η(x − y) = phA(x)−1(x − y), (x − y)i and A−1 denote the inverse matrix of A.

Theorem 3.2. Let L be a non divergence operator defined in Rn with a matrix of coefficients sufficiently smooth. If for α > 0 and for every u ∈ S(Rn) Z +∞ Z ! 1 n−1 −1−2α n−1 u(y)dH (y) − u(x) r dr (16) 0 H (∂E(x, r)) ∂E(x,r) 94 FAUSTO FERRARI is convergent, then in S(Rn) is well defined the following operator for every u ∈ S(Rn): Z α L u(x) = (u(y) − u(x))Kα(x − y)dy. n R Proof. For continuous coefficients of A, can be characterized using the approximated mean value formulas. Recalling the result by Fulks, see [16], [33], we consider Z 1 1 n−1 u(x + rA(x) 2 y)dH (y) − u(x) ωn {|y|=1} Z (17) 1 n−1 = n−1 u(y)dH (y) − u(x), H (∂E(x, r)) ∂E(x,r) where n −1 2 E(x, r) = {y ∈ R : hA(x) (x − y), (x − y)i = r }. Since for sufficiently smooth function we have Z 1 n−1 2 2 2 n−1 u(y)dH (y) − u(x) = cnTr(A(x)D u(x))r + o(r ), H (∂E(x, r)) ∂E(x,r) (18) as r → 0, then we can define the following integral: Z +∞ Z ! 1 n−1 −s n−1 u(y)dH (y) − u(x) r dr. (19) 0 H (∂E(x, r)) ∂E(x,r) Moreover by coarea formula we get ! Z +∞ 1 Z u(y)dHn−1(y) − u(x) r−sdr Hn−1(∂E(x, r)) 0 ∂E(x,r) (20) Z η(x − y)−2α−1|A−1(x) · (x − y)| = n−1 (u(y) − u(x))dy, n H (∂E(x, η(x − y))) R where η(x − y) = phA(x)−1(x − y), (x − y)i. Hence Z α L u(x) = (u(y) − u(x))Kα(x − y)dy. n R

The two results obtained in this section put in evidence two approaches to construct nonlocal linear operators. The main diﬀerences is that in the non-divergence case we do not have, in general, exact mean value formulas associated with the operator. We will introduce an analogous problem later in the nonlinear setting. In order to be precise in this construction, we treat in the following section a particular case of the divergence type, the sublaplacians in Carnot groups.

4. Carnot groups. In what follows we briefly recall some standard facts on Carnot groups, see e.g. [5,6] for an exhaustive source of definitions and results about the subject. A finite dimensional Lie algebra g is said to be stratified of step k ∈ N if there exists V1,...,Vk subspaces of g with linear dimension vk := dim Vk such that: g = V1 ⊕ · · · ⊕ Vk, where [V1,Vi] = Vi+1 i = 1, . . . , k and [V1,Vk] = {0}. A connected and simply connected Lie group G is a Carnot group if its Lie algebra g MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 95

Pi is finite dimensional and stratified. We also denote by h0 := 0, hi := j=1 vj and m := hk. The number k is called step of the group. For each λ > 0 and each x ∈ G we denote by δλ : G −→ G and τx : G −→ G the mappings defined respectively by:

σ1 σk δλ(x) = δλ(x1, . . . , xm) := (λ x1, . . . , λ xm), and τy(x) := y · x, where σi ∈ N is called the homogeneity of the variable xi in G and it is deﬁned by σj := i whenever hi−1 < j ≤ hi. We endow G with a norm and a quasi-distance deﬁning

k 1 2k! 2k! (1) (k) X (j) j ||x||G := ||(x , . . . , x )||G := kx k , (21) j=1 −1 (j) (j) and dG(x, y) := ||y · x||G. Here x := (xhj−1+1, . . . , xhj ) and kx k denotes the standard Euclidean norm in Rhj −hj−1 . We deﬁne the gauge ball centered at x ∈ G of radius R > 0 by

B(x, R) := {y ∈ G : dG(x, y) < R}. m Pm Let G = (R , ·) be a Carnot group. The number Q := j=1 σj, is called the homogeneous dimension of the group G. m Let X = (X1,...,Xm) be a Jacobian basis of G = (R , ·), see [15, Proposition 2.2] or [5, Corollary 1.3.19], for the definition. For any function f ∈ C1(Rm) we define the horizontal gradient by h X1 ∇V1 f := (Xif)Xi. i=1 Moreover, we define the horizontal laplacian of f : G −→ R and we denote it by ∆Gf , the following function h X1 ∆Gf := XiXif. (22) i=1 n n The usual Euclidean space (R , +) is trivially a Carnot group of step 1, | · |R is the classical Euclidean norm and ∆ is the Laplace operator. The Heisenberg group Hn (see [6,5] for the definition) is the simplest example of non-commutative Carnot group. A basis for the first stratum H = V1 of the algebra g in the Heisenberg n ∂ ∂ ∂ ∂ n group is Ui = + 2vi , Vi = − 2ui , for i = 1, . . . , n. In we have, H ∂ui ∂t ∂vi ∂t H see definition (21),

2n 1 X 2 2 2 2 4 n || (u, v, t) ||H = (ui + vi ) + t . (23) i=1 Deﬁnition 4.1. Let us deﬁne Z +∞ α −1−2α −(−∆G) φ(x) = (MG[φ](x, r) − φ(x)) r dr, 0 where Z m−1 MG[φ](x, r) = φ(σ)KG(x, σ)dH (σ), (24) ∂B(x,r) 2 −Q+1 |∇H dG(x, σ)| KG(x, σ) = cdG(x, σ) , (25) |∇dG(x, σ)| R m−1 c is the constant such that ∂B(x,r) KG(x, σ)dH (σ) = 1. 96 FAUSTO FERRARI

The properties of the kernel (25) are described in [5], see Theorem 5.5.4. We can show the proof of Theorem 1.2. Proof of Theorem 1.2. By the coarea formula and previous deﬁnition (4.1) we get: Z +∞ α −1−2α −(−∆G) φ(x) = (MG[φ − φ(x)](x, r)) r dr 0 Z +∞ Z ! m−1 −1−2α = (φ(y) − φ(x))KG(x, σ)dH (σ) r dr 0 ∂B(x,r) Z −Q+1−1−2α 2 = c dG(x, y) |∇H dG(x, y)| (φ(y) − φ(x))dy G Z |∇ d (x, y)|2 = c (φ(y) − φ(x)) H G dy. d (x, y)Q+2α G G (26)

Corollary 1. In the Heisenberg group we get Z 2 s |∇H d n (x, y)| n H −(−∆H ) φ(x) = c (φ(y) − φ(x)) Q+2α dy n d n (x, y) H H and 2 2 2 |∇ d n (x, y)| | u − ξ | + | v − η | H H = , (27) n Q+2α n Q+2+2α dH (x, y) dH (x, y) where H denotes the ﬁrst layer in the Heisenberg group and x = (u, v, t), y = (ξ, η, τ). Proof. By straightforward computation we get 2 2 2 | u − ξ | + | v − η | n | ∇H dH (x, y) | = −1 2 , || x ◦ y || n H −1 n n where x = (u, v, t), y = (ξ, η, τ), dH (x, y) =|| x ◦ y ||H , x ◦ y = (u + ξ, v + η, t + τ + 2(hξ, vi − hη, ui)) , y−1 = (−ξ, −η, −τ) and

p4 n 2 2 2 2 || x ||H = (| u | + | v | ) + t ; (28) see (23) and [5] for the detailed deﬁnitions. As a consequence Z 2 2 n s | u − ξ | + | v − η | −(−∆ ) φ(x) = c (φ(y) − φ(x)) Q+2+2α dy. H d n (x, y) G H

Remark 3. We want to point out here that, by changing the mean value operator we obtain different operators. Indeed, let us repeat the construction starting from an equivalent distance in the Heisenberg group, the so called Carnot-Carathéodory distance. As usual, we suggest the reference [5] for further details about the definitions. Here we introduce only the essential information to introduce the Carnot-Carathé- odory distance in the Heisenberg group. Indeed, let H = span{U1,..., Un, V1,..., Vn} the family of vector spaces generated by {U1,..., Un, V1,..., Vn}. Then for each MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 97

n Pn couple of vector fields W1(x), W2(x) ∈ H(x), x ∈ H , W1(x) = i=1 a1i(x)Ui(x) + Pn b1i(x)Vi(x),W (x) = i=1 a2i(x)Ui(x) + b2i(x)Vi(x), we define the inner product n X hW1, W2iH(x) = (a1i(x)a2i(x) + b1i(x)b2i(x)). i=1 Moreover, let 1 n 0 P(x, y) = {γ ∈ C ([0, 1], H ): γ(0) = x, γ(1) = y γ (s) ∈ H(γ(s))} be the set of horizontal paths joining x with y. We define the Carnot-Carathéodory distance between x ∈ Hn and y ∈ Hn in the Heisenberg group Hn, as Z 1 q 0 0 dCC (x, y) = inf{ hγ (s), γ (s)iH(γ(s))ds : γ ∈ P(x, y)}. 0

n It can be proved that the distance arising from the gauge, dH , and the Carnot-Ca- rathéodory distance in the Heisenberg group are equivalent. Hence it is worthwhile to consider the mean value operator on the balls generated by the Carnot-Carathéodory distance in the Heisenberg group. Thus if we define Z +∞ α −1−2α −(−∆C ) φ(x) = (MC [φ](x, r) − φ(P )) r dr, 0

n then recalling that |∇H dCC | = 1, see [30], we get: Z φ(y) − φ(x) − (−∆ )αφ(x) = c dS. (29) C d (x, y)Q+2α G CC

It is interesting to remark that, if φ satisﬁes the equation ∆Hφ = 0, in H, that is u α is harmonic in the intrinsic meaning in H, then φ also satisﬁes −(−∆H) φ(x) = 0 in H, because the exact mean value formula holds. On the contrary we can not suppose that ∆Hφ = 0 in G implies, for example α −(−∆C ) φ(x) = 0, because in general it is not true that

MC [φ](x, r) = φ(x).

5. Nonlinear setting. In the previous approach we investigated the relation between mean value formulas and global linear operators deﬁned on that mean value properties. In this section we want to give a glance to the nonlinear setting.

5.1. Shared fundamental solutions. Let us introduce in Rn the p−Laplace operator, p > 1 div(|∇u|p−2∇u) = 0. (30) Let us compare the fundamental solution of the p−Laplace operator, namely p−n |x| p−1 , see [26], and the α fractional fundamental solution |x|−m+2α, in Rm, see Proposition3 in the Appendix. p−n We deduce from p−1 = −m + 2α that m − 2α + n p = . m − 2α + 1 Moreover, if we assume that the dimensions coincide, i.e. m = n, then n − α n − α p = 2 = 2 . n + 1 − 2α n − α + (1 − α) 98 FAUSTO FERRARI

Hence for α ∈]0, 1[, 2n n − α < p = 2 < 2. n + 1 n + 1 − 2α This implies that for every α ∈]0, 1[, the fractional operator −(−∆)α is related with p(α)−2 n−α the nonlinear operator div(|∇·| ∇), where p(α) = 2 n−α+(1−α) , since they have the same fundamental solution. 5.2. Examples of nonlinear fractional operators. It is known that it is possible to characterise subsolutions and supersolutions of the p−Laplace operator by using approximated mean value formulas, see [28]. Indeed, in [28] the authors proved that viscosity solutions of the equation (30) in Ω ⊆ Rn are characterised by asymptotic mean value formulas of the following type max u + min u Z u(x) = γ B(x,r) B(x,r) + β u(y)dy + o(r2), (31) 2 B(x,r) p−2 2+n as r → 0, where γ = p+n and β = p+n . It is not difficult to repeat the proof that characterises the previous representation formula and obtain an analogous formula where now the sphere centred in x appears. It is sufficient to use spherical mean approximation of C2 functions, that is Z n−1 1 2 2 u(σ)dH (σ) = u(x0) + ∆u(x0)r + o(r ) ∂B(x0,r) 2n as r → 0. Indeed we obtain M r(x) + mr (x) Z u(x) =γ ˜ u u + β˜ u(y)dy + o(r2), 2 ∂B(x,r) r r p−2 as r → 0, where now Mu(x) = max∂B(x,r) u, mu(x) = min∂B(x,r) u, γ˜ = p+n−2 , and ˜ n β = p+n−2 . In this case we can select those functions such that for every x ∈ Rn M r(x) + mr (x) Z u(x) =γ ˜ u u + β˜ u(σ)dHn−1(σ), (32) 2 ∂B(x,r) for every r > 0. That is we can define a mean value operator defined on the boundary of the balls B(x, r), r > 0 for continuous function φ as follows: M r(x) + mr (x) Z φ φ ˜ n−1 Mh[φ](x, r) =γ ˜ + β φ(σ)dH (σ). 2 ∂B(x,r) If u satisfies (32) we say that u is p− spherical harmonious functions. We gave the previous definition just to put in evidence that in literature, see [29], p− harmonious functions are defined as the functions that satisfy the following equality max u + min u Z u(x) = γ B(x,r) B(x,r) + β u(σ)dHn−1(σ), (33) 2 B(x,r) for every r > 0. In our case, we can define a fractional harmonious operator for every α ∈]0, 1[ in the following way M r (x)+mr (x) Z +∞ u u ˜ Z α 2 − u(x) β u(y) − u(x) Lp u(x) =γ ˜ 1+2α dr + n+2α dy, r nω n | x − y | 0 n R MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 99

p−2 ˜ n whereγ ˜ = p+n−2 and β = p+n−2 . We say that u is α− spherical harmonious, α ∈]0, 1[, if for every x ∈ Rn M r (x)+mr (x) Z +∞ u u ˜ Z 2 − u(x) β u(y) − u(x) γ˜ 1+2α dr + n+2α dy = 0, r nω n | x − y | 0 n R or equivalently Z +∞ r r Mu(x) + mu(x) ˜ 1 γ˜ − u(x) + β (M[u](x, r) − u(x)) 1+2α dr = 0. 0 2 r In the case when n = 1, we get that Z +∞ r r Mu(x) + mu(x) ˜ u(x + r) + u(x − r) 1 γ˜ − u(x) + β − u(x) 1+2α dr 0 2 2 r

r r −1+2α −1+2α −1+2α Mu(x)+mu(x) ||x|−r| +||x|+r| In particular, if u(x) = |x| , then we get 2 = 2 at least when |x|= 6 r. Hence if u(x) = |x|−1+2α, then Z +∞ r r α hp − 2 Mu(x) + mu(x) L1 u(x) = − u(x) 0 p − 1 2 1 u(x + r) + u(x − r) i 1 + − u(x) dr p − 1 2 r1+2s Z +∞ u(x + r) + u(x − r) 1 = − u(x) 1+2s dr = 0, 0 2 r for every x ∈ R\{0}, because we reduce to the case of Proposition2, in the Appendix see also Lemma 6.1 in the Appendix. Indeed, in dimension n = 1 harmonious functions coincide with the harmonic functions and moreover for every α ∈]0, 1[, d2 Lα = −(− )α, 1 dx2 because in R : M r(x) + mr (x) u(x + r) + u(x − r) u u = . 2 2 Analogous definition can be given recalling that in [12] and in [13] more general characterisations shave been obtained respectively in the Heisenberg group and more in general in Carnot groups. In particular, in [12] has been proved the following result. Theorem 5.1. Let 1 < p < ∞ and let u be a continuous function defined in a domain Ω ⊂ Hn. The asymptotic expansion ! Z γ n 2 H n u(P ) = min u + max u + βH u(x, y, t) + o( ), (34) 2 B n (P,) B n (P,) B n (P,) H H H holds as → 0 for every P ∈ Ω in the viscosity sense if and only if p−2 n n n n ∆p,H u = divH |∇H u| ∇H u = 0, in Ω in the viscosity sense, where 2(p − 2)C(n) 1 γ n = , β n = , H 2(p − 2)C(n) + 1 H 2(p − 2)C(n) + 1 100 FAUSTO FERRARI and 1 n+1 R 2 2 1 0 (1 − s ) dt C(n) = 1 n . 2(n + 1) R 2 2 0 (1 − s ) dt We have to point out that approximating formulas like (31) or (34) are not unique, see e.g. [21], [20] and [27] or [13]. However, having in mind the idea that we want to construct a fractional operator following our procedure in a different framework, for example in Carnot groups, we need a mean value operator on spherical surfaces. One of the possible choices is to use Theorem 5.5.7 in [5], see also the seminal paper [7], where for a given Carnot 2 n group G, endowed with a gauge norm | · |G, for every C (Ω, R ) function and for every x ∈ Ω ⊆ G we have, M [u](x, r) − u(x) lim G = a ∆ u(x), (35) r→+0 r2 G G where aG is a positive constant and the mean value operator MG[u](x, r) is the same already introduced in (24). Arguing as in [13] we can obtain analogous approximation formulas useful to define the G− spherical harmonious functions using the boundary set originated by the gauge norm. Indeed using the formula (35) instead of the direct integration of the adapted Taylor formula on the gauge balls as done in [12] and [13], we get the following 2 representation formula for all the C functions, true in the points where ∇Gu(P ) 6= 0 : γ0 u(P ) = G min u + max u + β0 M [u](x, r) ∂B (P,r) ∂B (P,r) G G 2 G G ∇ u(P ) ∇ u(P ) − a β0 (p − 2)hD2∗u(P ) H , H i + ∆ u(P ) r2 + o(r2), G G H G |∇H u(P )| |∇H u(P )| (36) as r → 0, where one more time γ0 + β0 = 1 and γ0 , β0 , are constants depending G G G G γ0 G on a and p and that can be straightforwardly computed since a β0 = p − 2. G G With this notation, if we define the new mean value operator stillG defined for all continuous functions u, as γ0 M u(P ) = G M r (P ) + mr (P ) + β0 M [u](x, r), G,p,r 2 u,G u,G G G r r M (P ) = max∂B (P,r) u, m (x) = min∂B (P,r) u, we can define the − spherical u,G G u,G G G harmonious functions as those continuous functions such that

u(P ) = MG,p,ru(P ), for every r > 0. We deﬁne for every α ∈]0, 1[, p > 1, the fractional α−G−p spherical harmonious operator as Z +∞ M r (x) + mr (P ) α 0 u,G u,G L n u(x) = γ n − u(x) H ,p H 0 2 (37) + β0 (M [u](x, r) − u(x)) r−1−2αdr, G G MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 101 that is, by coarea formula, Z +∞ M r (x) + mr (x) α 0 u,G u,G −1−2α L n u(x) = γ n − u(x) r dr H ,p H 2 0 (38) Z |∇ d (x, y)|2 + β0 H G (u(y) − u(x))dy. G d (x, y)Q+2α G G In the particular case of the Heisenberg group we get, recalling (27), that Z +∞ M r (x) + mr (x) α 0 u,G u,G −1−2α L n u(x) = γ n − u(x) r dr H ,p H 2 0 (39) Z 2 2 0 | κ − ξ | + | µ − η | + β n (u(y) − u(x)) Q+2+2α dy, H n d n (x, y) H H that is Z +∞ M r (x) + mr (x) α 0 u,G u,G −1−2α L n u(x) = γ n − u(x) r dr H ,p H 0 2 Z 2 2 (40) 0 | κ − ξ | + | µ − η | + β n (u(y) − u(x)) dy, H Q+2+2α n || −x ◦ y || n H H where x = (ξ, η, t) ∈ Hn and y = (κ, µ, τ) ∈ Hn.

6. Appendix. In this section we recall some important results, more or less well known in literature and mainly described in [34] and [25], see also [2], even if they are not always explicitly available in literature. We collect them below since they are useful for our approach. 6.1. Classical motivations. In particular the method of the analytical continuation is useful to justify the metric approach dealt with in this paper. The details of the analytical continuation method, in this framework, can be found in [25]. Let u be a solution of Lu = µ. Assume that there exists −(−L)−1 such that u = −(−L)−1µ − α −1 α and it is well deﬁned Iα = (−L) 2 , where I2 = (−L) , and we deﬁne (−L) 2 such that α α −1 α−2 −(−L) 2 u = (−L) 2 (−L) µ := (−L) 2 µ = I2−αµ in case Iα is a convolution operator with kernel

Iαµ = kα ∗ µ we obtain that

α Z Z −(−L) 2 u = k2−α ∗ Lu = k2−α(y − x)Lu(y)dy = k2−α(y)Lu(x + y)dy.

This is justiﬁed by the fact that

kγ ∗ kβ = kγ+β. If y → φ(x + y) is in S, then as a distribution α −(−L) 2 φ(x) = hLk2−α|φ(x + ·)i. 1 In particular if K−α = Lk2−α ∈ Lloc, it is well deﬁned Z φ(x + y)K−α(y)dy. 102 FAUSTO FERRARI

−n+α In the classical case of the Laplace operator, the kernel kα(y) = c|y| . Hence for every y ∈ Rn \{0}, −n−α ∆k2−α(y) = c|y| −n−α n and since |y| ∈ Lloc(R ) if and only if α < 0, we can not represent hLk2−α|φ(x+ ·)i as an integral for Re(α) ≥ 0. Nevertheless, since Z η(x, α) = c φ(x + y)|y|−n−αdy is an analytic function in the half plane Re(α) < 0 we can extend the function η, in the variable α, with the method of the analytic continuation to all the plane C. In particular we conclude that we can extend analytically the function η for 0 < α < 2 as Z +∞ Z ! n−1 −1−α η(x, α) = cnωn φ(y)dH (y) − φ(x) t dt, 0 ∂B(x,t) that is Z φ(y) − φ(x) η(x, α) = cnωn n+α dt. n |x − y| R 6.2. The fundamental solution for the fractional Laplace operator. It is well known that | x |2α−n is the fundamental solution of the α−fractional Laplace operator, α ∈]0, 1[, −(−∆)α. Indeed as a distribution 2α−n −2α F | x | = cα,n | ξ | . Hence h(−∆)α | x |2α−n| φi −1 2α 2α−n −1 −1 2α −2α = hF (| ξ | F | x | ) | φi = cα,nhF (| ξ | | ξ | ) | φi = Cα,nδ0(φ). (41) We want to give a direct prove that | x |2α−n is a solution, as a function, of our fractional Laplace operator for every x ∈ Rn \{0}. This result is implicitly already hidden in [14]. Indeed, in Theorem 1.1 [14], we have that for radial functions u, regular enough, and for every α ∈]0, 1[, α (−∆) u(r) = cα,n Z +∞ r × r−2α u(r) − u(rτ) + (u(r) − u( ))τ −n+2α τ(τ 2 − 1)−1−2αH(τ)dτ, 1 τ (42) n where cn,α > 0 is a constant, r = |x| > 0, x ∈ R , and π n−3 Z 2α 2 n−2 p 2 2 π H(τ) = 2παn sin θ τ − sin θ + cos θ dθ, τ ≥ 1, αn = n−1 . 0 Γ( 2 ) Thus, assuming that the result can be applied to u(x) =| x |2α−n, then we need to calculate, for every r > 0 and for every τ ≥ 1, the value of r u(r) − u(rτ) + (u(r) − u( ))τ −n+2α (43) τ discovering that substituting in (43) the function u(r) = u(| x |) =| x |2α−n we magically obtain that, for every r > 0 and for every τ ≥ 1, r u(r) − u(rτ) + (u(r) − u( ))τ −n+2α = 0. τ MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 103

This implies that for every x ∈ Rn \{0}, (−∆)s | x |2α−n= 0. Here we are dealing with with our operator defined using the mean value operator. Indeed, recalling our operator, for α ∈]0, 1[, and beginning from the case of dimension n = 1 we have 2 Z +∞ d α u(x + r) + u(x − r) −1−2α −(− 2 ) u(x) = − u(x) r dr. dx 0 2 Thus, we would like to calculate, for every x 6= 0, the value of 2 Z +∞ −1+2α −1+2α d α −1+2α | x+r | +| x − r | 1−2α −1−2α −(− 2 ) | x | = − | x | ) r dr. dx 0 2 (44) Lemma 6.1. For every α ∈]0, 1[, Z +∞ | 1 + s |−1+2α + | 1 − s |−1+2α − 1) s−1−2αdr = 0. 0 2 Proof. We split the integral collecting differently the addends, so that Z +∞ | 1 + s |−1+2α + | 1−s |1−2α −1) s−1−2αds 0 2 1 Z +∞ 1 Z +∞ = | 1+s |−1+2α −1 | s |−1−2α ds+ | 1−s |−1+2α −1 | s |−1−2α ds. 2 0 2 0 (45) In the first integral we perform the change of variable ρ = 1 + s, while in the second one we set t = 1 − s. Thus 1 Z +∞ 1 Z 1 = | ρ |−1+2α −1 | 1−ρ |−1−2α dρ + | t |−1+2α −1 | 1−t |−1−2α dt, 2 1 2 −∞ moreover with a further change of variable in both the integral, this time the same, 1 1 that is ρ = s in the first and t = s in the second we conclude the proof obtaining: 1 Z 1 = s−1+2α | s |1−2α −1 | 1 − s |−1−2α ds 2 0 1 Z 1 − s−1+2α | s |1−2α −1 | 1 − s |−1−2α ds = 0. 2 0

Now we can prove that for every α ∈]0, 1[ the function | x |−1+2α is the funda- d2 α mental solution of −(− dx2 ) with pole in 0. Indeed, we have the following result. Proposition 2. For every α ∈]0, 1[ and for every x 6= 0, d2 −(− )α | x |−1+2α= 0. dx2 Proof. In particular d2 − (− )α | x |−1+2α dx2 Z +∞ | x + s |−1+2α + | x − s |−1+2α ! (46) = | x |−1 |x| |x| − 1) s−1−2αds 0 2 104 FAUSTO FERRARI

d2 α −1+2α In order to evaluate −(− dx2 ) | x | it is suﬃcient to calculate the integral Z +∞ | x + s |−1+2α + | x − s |−1+2α ! |x| |x| − 1) s−1−2αds. 0 2 On the other hand by symmetry it is suﬃcient calculate the case x = 1. That is 2 Z +∞ −1+2α −1+2α d α −1+2α | 1 + s | + | 1−s | −1−2α −(− 2 ) | x | =| x | −1) s ds dx 0 2 d2 =−| x |−1 (− )α(| x |−1+2α)(1). dx2 (47) The result follows recalling previous Lemma 6.1, because we proved there that Z +∞ | 1 + s |−1+2α + | 1 − s |−1+2α − 1) s−1−2αds = 0. 0 2

Previous arguments can be applied even for n > 1. Indeed the following result holds. Proposition 3. For every α ∈]0, 1[ and for every x 6= 0, −(−∆)α | x |−n+2α= 0. 6.3. α-Kelvin transformation. Let us consider the α−Kelvin transformation, with respect to the origin in the ball of radius 1, deﬁned as x K u(x) =| x |α−n u(T (x)) =| x |α−n u( ), α | x |2 x where T (x) = |x|2 . Then as a distribution α −2α−n α (−∆) (K2αu)(x) =| x | (−∆) u(T (x)). (48) It is well known that | x |2α−n is the fundamental solution of the α−fractional Laplace operator. Indeed as a distribution 2α−n −2α F | x | = cα,n | ξ | . Hence α 2α−n −1 −1 2α −2α h(−∆) | x | | φi = cα,nhF (| ξ | | ξ | ) | φi = Cα,nδ0(φ). (49) Let us consider the α−Kelvin transform, with respect to the origin in the ball of radius 1 is deﬁned as x K u(x) =| x |α−n u(T (x)) =| x |α−n u( ), α | x |2 x where T (x) = |x|2 . Then as a distribution α −2α−n α (−∆) (K2αu)(x) =| x | (−∆) u(T (x)). (50) More precisely, previous results is true from the distribution point of view. Indeed, for every φ ∈ S(Rn), α α 2α−n α h(−∆) (K2αu)(x)|φi = h(K2αu(x)|(−∆) φi = h| x | u(T (x))|(−∆) φi. (51) Moreover, we can prove that this result is true considering our operator point wise in Rn \{0}. Indeed, MEAN VALUE PROPERTIES OF FRACTIONAL SECOND ORDER OPERATORS 105

Proposition 4. For every α ∈]0, 1[, for every ∈ Rn \{0} α −2α−n α (−∆) (K2αu)(x) =| x | (−∆) u(T (x)). (52) Proof. Z 2α−n 2α−n α | y | u(T (y))− | x | u(T (x)) (−∆) (K2αu)(x) = n+2α dy n | y − x | R (53) Z Z 2α−n 2α−n 2α−n u(T (y)) − u(T (x)) | y | − | x | = | y | n+2α dy + u(T (x)) n+2α dy n | y − x | n | y − x | R R On the other hand, with a change of variable, Z Z α u(y) − u(T (x)) u(T (η)) − u(T (x)) −2n (−∆) u(T (x)) = n+2α dy = n+2α | η | dη. n | y − T (x) | n | T (η) − T (x) | R R (54) Hence recalling that | η − y | | T (η) − T (x) |= , | η || y | from (54) we deduce that Z α n+2α −n+2α u(T (η)) − u(T (x)) (−∆) u(T (x)) =| x | | η | n+2α dη. (55) n | η − x | R Now having in mind that | x |−n+2α is the fundamental solution of −(−∆)α, we conclude, comparing (54) and (55), that (48) holds.

Acknowledgments. The author wants to thank Juha Kinnunen for the support during the program Evolutionary problems held at Mittag-Leﬄer Institut in 2013.

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