On the Characterization of p-Harmonic Functions on the Heisenberg Group by Mean Value Properties

We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1<p<\infty$, following the scheme described in Manfredi et al. (2009) for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.


Introduction
In this paper we study p−harmonic functions in the Heisenberg group in terms of an asymptotic mean value property. The corresponding characterization of the p-harmonic functions in terms of an asymptotic mean value property in the Euclidean sense was obtained in [MPR]. More precisely in [MPR], the authors show that if u is a continuous function in a domain Ω ⊂ R n and p ∈ (1, ∞], then the asymptotic expansion u(x) = p − 2 2(p + n) max Bǫ(x) u + min Bǫ(x) u + 2 + n p + n Bǫ(x) u(y) dy + o(ǫ 2 ), holds as ǫ → 0 for all x ∈ Ω in the viscosity sense if and only if u is a viscosity solution of the p-Laplace equation div |∇u(x)| p−2 ∇u(x) = 0.
Here B ǫ (x) is the Euclidean ball centered in x with radius ǫ. We want to extend this characterization to functions defined on the Heisenberg group H n . In Section 2 we present an overview of the Heisenberg group, where the geometry and analysis are different than in Euclidean space. For this introduction we anticipate a few definitions from Section 2 and refer the reader to this section for full details. For p ∈ (1, ∞) the (subelliptic) p-Laplace operator in the Heisenberg group is ∆ p,H n u = div H n |∇ H n u| p−2 ∇ H n u .
Here ∇ H n and div H n are respectively the intrinsic gradient and the intrinsic divergence in the Heisenberg group. From now on we will denote by B(P, ǫ) the intrinsic ball of radius ǫ with respect to the gauge distance, centered at the point P ∈ H n .
We point out that for p = 2, ∆ 2,H n u = ∆ H n u is the real part of the Kohn-Laplace operator, a linear degenerate second order elliptic operator whose lowest eigenvalue is always zero. In particular, to give an idea of the structure of this linear operator corresponding to the case p = 2 and n = 1 we set ∆ H u(P ) = div H (M(P )∇u(P )) = Trace(M(P )D 2 u(P )), where the 3 × 3 matrix M(P ) for P = (x, y, t) ∈ H is given by Observe that we always have min λ : λ is an eigenvalue of M(P ) = 0.
For p = 1 we get a subelliptic version of the mean curvature operator See the monograph [CDPT] for the intrinsic mean curvature operator in the Heisenberg setting. See also [CC] and [FLM] for some recent results on the the flow by mean curvature in the subelliptic setting.
Our main result is the following: Theorem 1.1. Let 1 < p < ∞ and let u be a continuous function defined in a domain Ω ⊂ H n . The asymptotic expansion holds as ǫ → 0 for every P ∈ Ω in the viscosity sense if and only if in Ω in the viscosity sense, where , and C(n) = 1 2(n + 1) We remark that α + β = 1 and that The key tool in the proof of Theorem 1.1 is Lemma 3.2 in Section 3. Roughly speaking, in Lemma 3.2 we prove that if P 0 = (x 0 , y 0 , t 0 ) ∈ H n is not characteristic for the level set {u = u(P 0 )}, then the extrema of the function u in the intrinsic ball of radius ǫ and center P 0 are attained at points (x ǫ , y ǫ , t ǫ ) that satisfy where p is the imaginary curvature of the level surface {u = u(P 0 )} at P 0 introduced in [AF1] and [AF2] and given by We recall now the definition of viscosity solution in the Heisenberg group taken from [Bi].
Observe that if u is smooth then where we have used (1.2) Definition 1.2. Fix a value of p ∈ (1, ∞) and consider the p-Laplace equation (i) A lower semi-continuous function u is a viscosity supersolution of (1.3) if for every φ ∈ C 2 (Ω) such that u − φ has a strict minimum at P 0 ∈ Ω, and ∇ H n φ(P 0 ) = 0 we have (ii) A lower semi-continuous function u is a viscosity subsolution of (1.3) if for every φ ∈ C 2 (Ω) such that u − φ has a strict maximum in P 0 ∈ Ω, and ∇ H n φ(P 0 ) = 0, we have (iii) A continuous function u is a viscosity solution of of (1.3) if it is both a viscosity supersolution and a viscosity subsolution.
As shown in [JLM] for the Euclidean case and in [Bi] for the subelliptic case, it suffices to consider smooth functions whose horizontal gradient does not vanish. In addition in those papers it is shown that the notions of viscosity and weak solutions agree for homogeneous equation −∆ p,H n u = 0.
Next we state carefully what we mean when we say that the asymptotic expansion (1.1) holds in the viscosity sense. Recall the familiar definition of "little o"for a real valued function h defined in a neighborhood of the origin. We write Let h be a real valued function defined in a neighborhood of zero. We say that h(x) ≤ o(x 2 ) as x → 0 + if any of the three equivalent conditions is satisfied: A similar definition is given for by reversing the inequalities in a) and c), requiring that g(x) ≤ 0 in b) and replacing h + by h − in c).
Definition 1.4. A continuous function defined in a neighborhood of a point P ∈ H n satisfies as ǫ → 0, and (ii) for every continuous function φ defined in a neighborhood of a point P such that u − φ has a strict maximum at P with u(P ) = φ(P ) then as ǫ → 0.
For the case p = ∞ we could consider the 1-homogeneous infinity-Laplacian 1.2, but there remain some technical difficulties. We certainly conjecture that the statement of Theorem 1.1 holds in this case.

Heisenberg group Preliminaries
For n ≥ 1 we denote by H n the set R 2n+1 endowed with the non-commutative product law given by where we have denote points in R 2n+1 as P = (x, y, t) with x, y ∈ R n and t ∈ R and x · y denote the usual inner product in R n . The pair H n ≡ (R 2n+1 , ⋆) is the Heisenberg group of order n. From now we will denote the group operation · instead of ⋆ when there is no risk of confusion.
Given P = (x, y, t) ∈ H n we write X i = (e i , 0, 2y i ) and Y i = (0, e i , −2x i ) for i = 1, . . . , n, where {e i } 1≤i≤n is the canonical basis in R n . We identify these vectors with the vector fields The commutator between the vector fields X i and Y j is 0 except when the indexes are the same i = j, in which case we have The intrinsic (or horizontal) gradient of a smooth function u at the point P is given by The horizontal tangent space at the point P is the 2n-dimensional space We build a metric in the horizontal tangent space by declaring that the set of vectors {X 1 , . . . , X n , Y i , . . . , Y n } is an orthonormal basis. Thus for every pair of horizontal vectors U = n j=1 (α 1,j X j (P ) + β 1,j Y j (P )) and V = n j=1 (α 2,j X j (P ) + β 2,j Y j (P )) we have the natural inner product In particular we get the corresponding norm The norm of the intrinsic gradient of the smooth function u in P is then given by If ∇ H n u(P ) = 0 then we say that the point P is characteristic for the surface {u = u(P )}. For every point P that is not characteristic the intrinsic normal to the surface {u = u(P )} is defined, up to orientation, by the unit horizontal normal vector A semigroup of anisotropic dilations that are group homomorphism is defined as follows: for every r > 0 and P ∈ H n let δ r (P ) = (rx, ry, r 2 t).
We then have δ r (P ) = r P . We use this property to define the gauge ball of radius r centered in 0 iby B(0, r) = {P ∈ H n : P < r}.
The Haar measure in H n turns out to be the Lebesgue measure in R 2n+1 . Moreover we have For these properties and much more see the book [CDPT].
Proof. Let us consider the case of the maximum, the case of the minimum being analogous. Let us proceed by contradiction. Assume that a sequence of positive numbers {ǫ j } j∈N ⊂ R + and a sequence of points {P j } j∈N ⊂ B(0, ǫ j ) such that ǫ j → 0, as j → +∞ and max B(0,ǫ j ) u = u(P j ).
Then for every j ∈ N, we have that ∇u(P j ) = 0 because P j is in the interior of B(0, ǫ j ). Hence we get a contradiction with the fact that by continuity of ∇u gives ∇u(0) = 0, which implies ∇ H n u(0) = 0.
Since the calculation is the same for every j = 1 . . . n, it suffices to solve the following system where ξ = x j and η = y j . Hence, it follows that if u t (P ǫ,j ) = 0 we have As a consequence solving for ξ and η, squaring and adding we get (3.4) Thus, we obtain: (3.7) Squaring and summing one more time and keeping in mind that we get ρ 2 = ρ 6 ǫ 4 + 4ρ 4 u 2 t , which implies ρ 4 ǫ 4 + 4ρ 2 u 2 t = 1. We deduce that ρ ∼ ǫ whenever ǫ → 0, since u t (0) is bounded. As a consequence it follows from (3.6) and (3.7) that On the other hand recalling (3.5) and the fact that ρ ∼ ǫ, as ǫ → 0, we get If there is a sequence of point {P ǫ j } j∈N such that u t (P ǫ j ) = 0, then either t ǫ j = 0, or λ ǫ j = 0. In the first case we get that ρ 4 = ǫ 4 , so that ρ = ǫ. Thus we get Xu(P ǫ j ) = 4λ ǫ j ξǫ 2 and Y u(P ǫ j ) = 4λ ǫ j ηǫ 2 . As a consequence, squaring and summing once again, we get |∇ H n u(P ǫ j )| 2 = 16λ 2 ǫ j ǫ 4 ρ 2 . So that since |∇ H n u(0)| = 0, this implies As a consequence, as ǫ j → 0 we get In the case that there exists a sequence of points {P ǫ j } j∈N such that such that λ ǫ j = 0, then we would get Xu(P ǫ j ) = 0 and Y u(P ǫ j ) = 0, getting a contradiction with the assumption that ∇ H n u(0) = 0. We just need to justify the sign of the limit. Using the Taylor's formula in the Heisenberg group we get: , (x ǫ,M , y ǫ,M ) + 2t∂ t u(0)) + o(ǫ 2 Hence, dividing by ǫ > 0 we get Letting ǫ → 0 we get Lemma 3.3. Let u be a smooth function defined in a open subset of the Heisenberg group. Then B(P,ǫ) u(x, y, t) = u(P ) + C(n)∆ H n u(P )ǫ 2 + o(ǫ 2 ), Proof. Without loss of generality we set P = 0. We average the Taylor expansion 2.3 to get Indeed we have that the linear terms vanish Proceeding analogously for the second order terms we get where we have denoted by ω(n) the Euclidean surface area of they unit sphere ∂B(0, 1) in R 2n , and have used the formula |B(0, ǫ)| = ǫ 2n+2 ω(2n) n 1 0 (1−t 2 ) n 2 dt for the Lebesgue measure in R 2n+1 of the gauge ball of radious ǫ.

Proof of Theorem 1.1
The first step in the proof of Theorem 1.1 is the following expansion valid for smooth functions.
Let P ∈ Ω be a point and φ be a C 2 -function defined in a neighborhood of P. We denote by P ǫ,M ∈ B(P, ǫ) and P ǫ,m ∈ B(P, ǫ) the points of maxima and minima φ(P ǫ,M ) = max B(P,ǫ) φ, and φ(P ǫ,m ) = min B(P,ǫ) φ.