On harmonic quasiconformal quasi-isometries

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.


Introduction
Let G ⊂ R 2 be a domain and let f : G → R 2 , f = (f 1 , f 2 ), be a harmonic mapping. This means that f is a map from G into R 2 and both f 1 and f 2 are harmonic functions, i.e. solutions of the two-dimensional Laplace equation ∆u = 0 . (1.1) The Cauchy-Riemann equations, which characterize analytic functions, no longer hold for harmonic mappings and therefore these mappings are not analytic. Intensive studies during the past two decades show that much of the classical function theory can be generalized to harmonic mappings (see the recent book of Duren [9] and the survey of Bshouty and Hengartner [7]). The purpose of this paper is to continue the study of the subclass of quasiconformal and harmonic mappings, introduced by Martio in [31] and further studied for example in [32,33,34,37,38,16,2,3,19,20,21,17]. The above definition of a harmonic mapping extends in a natural way to the case of vector-valued mappings f : G → R n , f = (f 1 , . . . , f n ), defined on a domain G ⊂ R n , n ≥ 2. We first recall the classical Schwarz lemma for the unit disk D = {z ∈ C : |z| < 1} :
For the case of harmonic mappings this lemma has the following counterpart.
The classical Schwarz lemma is one of the cornerstones of geometric function theory and it also has a counterpart for quasiconformal maps ( [1,26,41,45]). Both for analytic functions and for quasiconformal mappings it has a form that is conformally invariant under conformal automorphisms of D .
In the case of harmonic mappings this invariance is no longer true. In general, if ϕ : D → D is a conformal automorphism and f : D → D is harmonic, then ϕ • f is harmonic only in exceptional cases. Therefore one expects that harmonic mappings from the disk into a strip domain behave quite differently from harmonic mappings from the disk into a half-plane and that new phenomena will be discovered in the study of harmonic maps. For instance, it follows from Lemma 1.2 that holomorphic functions in plane do not increase hyperbolic distances. In general, planar harmonic mappings do not enjoy this property. On the other hand, we shall give here an additional hypothesis under which the situation will change, in the plane as well as in higher dimensions. It turns out that the local uniform boundedness property, which we are going to define, has an important role in our study.
For a domain G ⊂ R n , n ≥ 2, x, y ∈ G, let If the domain G is understood from the context, we write r instead r G . This quantity is used, for instance, in the study of quasiconformal and quasiregular mappings, cf. [45]. It is a basic fact that [43,Theorem 18 We call this property the local uniform boundedness of f with respect to r G . Note that quasiconformal mappings satisfy the local uniform boundedness property and so do quasiregular mappings under appropriate conditions; it is known that one to one mappings satisfying the local uniform boundedness property may not be quasiconformal. We also consider a weaker form of this property and say that f : G → f G with G, f G ⊂ R n satisfies the weak uniform boundedness property on G (with respect to r G ) if there is a constant c > 0 such that r G (x, y) ≤ 1/2 implies r f G (f (x), f (y)) ≤ c . Univalent harmonic mappings fail to satisfy the weak uniform boundedness property as a rule, see Example 2.7 below.
We show that if f : G → f G is harmonic then f is Lipschitz w.r.t. quasihyperbolic metrics on G and f G if and only if it satisfies the weak uniform boundedness property; see Theorem 2.19. The proof is based on a higher dimensional version of the Schwarz lemma: harmonic maps satisfy the inequality (2.15) below. An inspection of the proof of Theorem 2.19 shows that the class of harmonic mappings can be replaced by OC 1 class defined by (3.1) (see Section 3 below) and it leads to generalizations of the result; see Theorem 3.3.
Another interesting application is Theorem 2.22 which shows that if f is a harmonic K-quasiregular map such that the boundary of the image is a continuum containing at least two points, then it is Lipschitz. In Subsection 2.5, we study conditions under which a qc mapping is quasi isometry with respect to the corresponding quasihyperbolic metrics; see Theorems 2.25 and 2.31. In particular, using a quasiconformal analogue of Koebe's theorem, cf. [4], we give a simple proof of the following result, cf. [30,33]: if D and D ′ are proper domains in R 2 and h : D → D ′ is K-qc and harmonic, then it is bi-Lipschitz with respect to quasihyperbolic metrics on D and D ′ .
The results in this paper may be generalized into various directions. One direction is to consider weak continuous solutions of the p-Laplace equation div(|∇u| p−2 ∇u) = 0, 1 < p < ∞, so called p-harmonic functions. Note that 2-harmonic functions in the above sense are harmonic in the usual sense.
It seems that the case of the upper half space is of particular interest, cf. [37,33,16,3]. In Subsection 2.6, using Theorem 3.1 [23] we prove that if h is a quasiconformal p-harmonic mapping of the upper half space H n onto itself and h(∞) = ∞, then h is quasi-isometry with respect to both the Euclidean and the Poincaré distance.
2 Lipschitz property of harmonic maps w.r.t. quasihyperbolic metrics

Hyperbolic type metrics
Let B n (x, r) = {z ∈ R n : |z − x| < r}, S n−1 (x, r) = ∂B n (x, r) and let B n , S n−1 stand for the unit ball and the unit sphere in R n , respectively. Sometimes we write D instead of B 2 . For a domain G ⊂ R n let ρ : G → (0, ∞) be a continuous function. We say that ρ is a weight function or a metric density if for every locally rectifiable curve γ in G, the integral exists. In this case we call l ρ (γ) the ρ-length of γ. A metric density defines a metric d ρ : G × G → (0, ∞) as follows. For a, b ∈ G, let where the infimum is taken over all locally rectifiable curves in G joining a and b. For a fixed a, b ∈ G , suppose that there exists a d ρ -length minimizing curve γ : for all t ∈ [0, 1] . Then γ is called a geodesic segment joining a and b . It is an easy exercise to check that d ρ satisfies the axioms of a metric. For instance, the hyperbolic (or Poincaré) metric of the unit ball B n and the upper half space H n = {x ∈ R n : x n > 0} are defined in terms of the densities ρ(x) = 2/(1 − |x| 2 ) and ρ(x) = 1/x n , respectively. It is a classical fact that in both cases the length-minimizing curves, geodesics, exist and that they are circular arcs orthogonal to the boundary [6]. In both cases we have even explicit formulas for the distances: and cosh ρ H n (x, y) = 1 + |x − y| 2 2x n y n , x, y ∈ H n .

(2.2)
Because the hyperbolic metric is invariant under conformal mappings, we may define the hyperbolic metric in any simply connected plane domain by using the Riemann mapping theorem, see for example [24]. The Schwarz lemma may now be formulated by stating that an analytic function from a simply connected domain into another simply connected domain is a contraction mapping, i.e. the hyperbolic distance between the images of two points is at most the hyperbolic distance between the points. The hyperbolic metric is often the natural metric in classical function theory. For the modern mapping theory, which also considers dimensions n ≥ 3 , we do not have a Riemann mapping theorem and therefore it is natural to look for counterparts of the hyperbolic metric. So called hyperbolic type metrics have been the subject of many recent papers. Perhaps the most important of these metrics are the quasihyperbolic metric k G and the distance ratio metric j G of a domain G ⊂ R n . They are defined as follows.
2.3. The quasihyperbolic and distance ratio metrics. Let G ⊂ R n be a domain. The quasihyperbolic metric k G is a particular case of the metric d ρ when [13,12,45]). It was proved in [12] that for given x, y ∈ G , there exists a geodesic segment of length k G (x, y) joining them. The distance ratio metric is defined for x, y ∈ G by setting where r G is as in the Introduction. It is clear that Some applications of these metrics are reviewed in [46]. The recent PhD theses [27], [24], [29] study the quasihyperbolic geometry or use it as a tool.

Quasiconformal and quasiregular maps
2.5. Maps of class ACL and ACL n . For each integer k = 1, ..., n we denote R n−1 k = {x ∈ R n : x k = 0}. The orthogonal projection P k : R n → R n−1 k , is given by Let I = {x ∈ R n : a k ≤ x k ≤ b k } be a closed n-interval. A mapping f : I → R m is said to be absolutely continuous on lines (ACL) if f is continuous and if f is absolutely continuous on almost every line segment in I, parallel to the coordinate axes. More precisely, if E k is the set of all x ∈ P k I such that the function If Ω is an open set in R n , a mapping f : Ω → R m is absolutely continuous if f |I is ACL for every closed interval I ⊂ Ω. If Ω and Ω ′ are domains in R n , a If f : Ω → R m is ACL, then the partial derivatives of f exist a.e. in Ω, and they are Borel functions. We say that f is ACL n if the partials are locally integrable.

Quasiregular mappings.
Let G ⊂ R n be a domain. A mapping f : G → R n is said to be quasiregular (qr) if f is ACL n and if there exists a constant K ≥ 1 such that a.e. in G. Here f ′ (x) denotes the formal derivative of f at x , The smallest K ≥ 1 for which this inequality is true is called the outer dilatation of f and denoted by Let Ω 1 and Ω 2 be domains in R n and fix K ≥ 1 . We say that a homeomorphism f : Ω 1 → Ω 2 is a K-quasiconformal (qc) mapping if it is K-qr and injective. Some of the standard references for qc and qr mappings are [11], [26], [43], and [45]. These mappings generalize the classes of conformal maps and analytic functions to Euclidean spaces. The Kühnau handbook [25] contains several reviews dealing with qc maps. It should be noted that various definitions for qc maps are studied in [43]. The above definition of K-quasiconformality is equivalent to the definition based on moduli of curve families in [43, p. 42]. It is well-known that qr maps are differentiable a.e., satisfy condition (N) i.e. map sets of measure zero (w.r.t. Lebesgue's n-dimensional measure) onto sets of measure zero. The inverse mapping of a K-qc mapping is also K-qc. The composition of a K 1 -qc and of a K 2 -qc map is a K 1 K 2 -qc map if it is defined.

Examples
We first show that, as a rule, univalent harmonic mappings fail to satisfy the local uniform boundedness property.

Example .
The univalent harmonic mapping f : H 2 → f (H 2 ) , f (z) = arg z + i Imz, fails to satisfy the local uniform boundedness property with respect to r H 2 .

Example.
The univalent harmonic mapping f : H 2 → H 2 , f (z) = Re z Im z+i Imz, fails to satisfy the local uniform boundedness property with respect to r H 2 .
For a harmonic mapping f (z) = h(z) + g(z) , we introduce the following notation The following Proposition shows that a one to one harmonic function satisfying the local uniform boundedness property need not be quasiconformal.

Proposition .
The function f (z) = log(|z| 2 ) + 2iy is a univalent harmonic mapping and satisfies the local uniform boundedness property, but f is Moreover, f is quasiconformal on every compact subset D ⊂ Π + and λ f , Λ f are bounded from above and below on D. Therefore f is a quasi-isometry on D and by Theorem 2.19 below, f satisfies the local uniform boundedness property on D.
From now on we consider the restriction of f to We are going to show that: • f satisfies the local uniform boundedness property, but f is not quasiconformal on V .
We see that f is not quasiconformal on V , because |ν(z)| → 1 as z → ∞, z ∈ V . For s > 1, define V s = {z : 1 < x < s, 0 < y < 1}. Note that f is qc on V s and therefore f satisfies the property of local uniform boundedness on V s for every s > 1.
We consider separately two cases.
) and therefore f satisfies the property of local uniform boundedness on V 4 with respect to r V .
Case B. It remains to prove that f satisfies the property of local uniform boundedness on V \ V 4 with respect to r V .
Observe first that for z, z 1 ∈ V and |z 1 | ≥ |z| ≥ 1 , we have the estimate and therefore for z, We write Then and by the definition of f we see that Clearly and for Rew > 1 + log 2 and w ∈ f V , we find (16)

Higher dimensional version of Schwarz lemma
Before giving a proof of the higher dimensional version of the Schwarz lemma for harmonic maps we first establish some notation. Suppose that h : B n (a, r) → R n is a continuous vector-valued function, harmonic on B n (a, r), and let M * a = sup{|h(y) − h(a)| : y ∈ S n−1 (a, r)}. Proof. Without loss of generality, we may suppose that a = 0 and h(0) = 0. Let where ω n is the volume of the unit ball B n in R n . Then where dσ is the (n − 1)-dimensional surface measure on S n−1 (0, r). A simple calculation yields Hence, for 1 ≤ j ≤ n, we have Let η ∈ S n−1 be a unit vector and |ξ| = r. For given ξ, it is convenient to write K ξ (x) = K(x, ξ) and consider K ξ as a function of x.

This last inequality yields
M * 0 nω n r n−1 ω n r n = M * 0 n r and the proof is complete.
Let G ⊂ R n , be a domain, let h : G → R n be continuous. For x ∈ G let B x = B n (x, 1 4 d(x)) and If h is a harmonic mapping, then the inequality (2.13) yields We also refer to (2.15) as the inner gradient estimate.

Harmonic quasiconformal quasi-isometries
For our purpose it is convenient to have the following lemma.
2.16. Lemma. Let G and G ′ be two domains in R n , and let σ and ρ be two continuous metric densities on G and G ′ , respectively, which define the elements of length ds = σ(z)|dz| and ds = ρ(w)|dw|, respectively; and suppose that f : G → G ′ , is a C 1 -mapping. a) If there is a positive constant c 1 such that ρ(f (z)) |f ′ (z)| ≤ c 1 σ(z), z ∈ G , then d ρ (f (z 2 ), f (z 1 )) ≤ c 1 d σ (z 2 , z 1 ), z 1 , z 2 ∈ G. b) If f (G) = G ′ and there is a positive constant c 2 such that ρ(f (z)) l(f ′ (z)) ≥ c 2 σ(z), z ∈ G , then d ρ (f (z 2 ), f (z 1 )) ≥ c 2 d σ (z 2 , z 1 ), z 1 , z 2 ∈ G The proof of this result is straightforward and it is left to the reader as an exercise.

Pseudo-isometry and a quasi-isometry.
Let f be a map from a metric space (M, d M ) into another metric space (N, d N ).
• We say that f is a pseudo-isometry if there exist two positive constants a and b such that for all x, y ∈ M, • We say that f is a quasi-isometry or a bi-Lipschitz mapping if there exists a positive constant a ≥ 1 such that for all x, y ∈ M, For the convenience of the reader we begin our discusssion for the unit disk case.

Theorem. Suppose that h : D → R 2 is harmonic and satisfies the weak uniform boundedness property.
(c) Then h : Proof. The part (d) is proved in [33]. For the proof of part (c) fix x ∈ D and y ∈ B x = B(x, 1 4 d(x)). Then d(y) ≥ 3 4 d(x) and therefore r(x, y) < 1/2. By the hypotheses |h(y) − h(x)| ≤ c d(h(x)). The Schwarz lemma, applied to B x , yields in view of (2.14) The proof of part (c) follows from Lemma 2.16.
A similar proof applies for higher dimensions; the following result is a generalization of the part (c) of Theorem 2.18 .

Theorem.
Suppose that G is a proper subdomain of R n and h : G → R n is a harmonic mapping. Then the following conditions are equivalent (1) h satisfies the weak uniform boundedness property.
Proof. Let us prove that (1) implies (2). By the hypothesis (1) f satisfies the weak uniform boundedness property: for every This inequality together with Lemma 2.12 gives d(x)|f ′ (x)| ≤ c 3 d(f (x)) for every x ∈ G. Now an application of Lemma 2.16 shows that (1) implies (2).
It remains to prove that (2) implies (1). Suppose that f is Lipschitz with the multiplicative constant c 2 . Fix x, y ∈ G with r G (x, y) ≤ 1/2. Then |y − x| ≤ d(x)/2 and therefore by Lemma 2.4 Since f −1 is qc, an application of [12, Theorem 3] to f −1 and Theorem 2.19 give the following corollary: In [45,Example 11.4] (see also [44,  Therefore bounded analytic functions do not satisfy the weak uniform boundedness property in general. The situation will be different for instance if the boundary of the image domain is a continuum containing at least two points. Note that if k G is replaced by the hyperbolic metric λ G of G, then f : Let ∂G ′ be a continuum containing at least two distinct points. If f is a harmonic mapping, then f : Proof. Fix x ∈ G and let B x = B n (x, d(x)/4). If |y − x| ≤ d(x)/4, then d(y) ≥ 3d(x)/4 and therefore, Because j G (x, y) = log(1 + r G (x, y)) ≤ r G (x, y), using Lemma 2.4(a), we find By [45,Theorem 12.21] there exists a constant c 2 > 0 depending only on n and K such that and hence, using Lemma 2.4(b) and k G (y, x) ≤ 1, we see that (2.23) By (2.15) applied to B x = B n (x, d(x)/4), we have and therefore using the inequality (2.23), we have where c = e c 2 ; and the proof follows from Lemma 2.16.
The first author has asked the following Question (cf. [33]: Suppose that G ⊂ R n is a proper subdomain, f : G → R n is harmonic K-qc and G ′ = f (G). Determine whether f is a quasi-isometry w.r.t. quasihyperbolic metrics on G and G ′ . This is true for n = 2 (see Theorem 2.26 below). It seems that one can modify the proof of Proposition 4.6 in [42] and show that this is true for the unit ball if n ≥ 3 and K < 2 n−1 , cf. also [20].

Quasi-isometry in planar case
Astala and Gehring [4] proved a quasiconformal analogue of Koebe's theorem, stated here as Theorem 2.24. These concern the quantity associated with a quasiconformal mapping f : G → f (G) ⊂ R n ; here J f is the Jacobian of f ; while B x stands for the ball B(x; d(x, ∂G); and |B x | for its volume.
2.24. Theorem [4]. Suppose that G and G ′ are domains in R n : where c is a constant which depends only on K and n.
Let Ω ∈ R n and R + = [0, ∞). If f, g : Ω → R + and there is a positive constant c such that we write f ≈ g on Ω.
Our next result concerns the quantity associated with a quasiconformal mapping f : G → f (G) ⊂ R n ; here J f is the Jacobian of f ; while B x stands for the ball B(x, d(x, ∂G)/2 and |B x | for its volume.

Define
A f,G = n E f,G .

2.25.
Theorem. Suppose f : Ω → Ω ′ is a C 1 qc homeomorphism. The following conditions are equivalent: a)f is bi-Lipschitz with respect to quasihyperbolic metrics on Ω and Proof. It is known that a) is equivalent to b) (see for example [36]). In [36], using Gehring's result on the distortion property of qc maps (see [10], p.383; [43], p.63), the first author gives short proofs of a new version of quasiconformal analogue of Koebe's theorem; it is proved that By Theorem 2.24, a f ≈ d * /d and therefore b) is equivalent to c). The rest of the proof is straightforward.
If Ω is planar domain and f a harmonic qc map, then we proved that the condition d) holds.
The next theorem is a short proof of a recent result of V. Manojlovic [30],see also [33]. Proof. Without loss of generality, we may suppose that h is preserving orientation. Let z ∈ D and h = f + g be a local representation of h on B z , where f and g are analytic functions on B z , Λ h (z) = |f ′ (z)| + |g ′ (z)|, λ h (z) = |f ′ (z)| − |g ′ (z)| and k = K−1 K+1 . Since h is K-qc, we see that on B z and since log |f ′ (ζ)| is harmonic, Hence, using the right hand side of (2.27), we find Hence, a h,D (z) ≤ √ K |f ′ (z)| and in a similar way using the left hand side of (2.27), we have

Now, an application of the Astala-Gehring result gives
This pointwise result, combined with Lemma 2.16 (integration along curves), easily gives Note that in [30] the proof makes use of the interesting fact that log 1 J h is a subharmonic function; but we do not use it here. Define Suppose that G and G ′ are domains in R n : If f : G → G ′ is K-quasiconformal; by the distortion property we find m f (x, r) ≥ a(x)r 1/α . Hence, as in [20] and [36] , we get: Proof. Since G is compact J f attains minimum on G at a point x 0 ∈ G. By Lemma 2.30, m 0 = J f > 0 and therefore since f ∈ C 1,1 is a K− quasiconformal, we conclude that functions |f x k |, 1 ≤ k ≤ n are bounded from above and below on G; hence f is bi-Lipschitz with respect to Euclidean metric on G.
By Theorem 2.24, we find a f, An application of Theorem 2.25 completes the proof.

The upper half space H n .
Let H n denote the half-space in R n . If D is a domain in R n , by QCH(D) we denote the set of Euclidean harmonic quasiconformal mappings of D onto itself.
In particular if x ∈ R 3 , we use notation x = (x 1 , x 2 , x 3 ) and we denote by x k the partial derivative of f with respect to x k . A fundamental solution in space R 3 of the Laplace equation is 1 |x| . Let U 0 = 1 |x+e 3 | , where e 3 = (0, 0, 1). Define h(x) = (x 1 + ε 1 U 0 , x 2 + ε 2 U 0 , x 3 ). It is easy to verify that h ∈ QCH(H 3 ) for small values of ε 1 and ε 2 .
Using the Herglotz representation of a nonnegative harmonic function u (see Theorem 7.24 and Corollary 6.36 [5]), one can get: Lemma A. If u is a nonnegative harmonic function on a half space H n , continuous up to the boundary with u = 0 on H n , then u is (affine) linear.
In [33], the first author has outlined a proof of the following result: Theorem A. If h is a quasiconformal harmonic mapping of the upper half space H n onto itself and h(∞) = ∞, then h is quasi-isometry with respect to both the Euclidean and the Poincaré distance.
Note that the outline of proof in [33] can be justified by Lemma A. We show that the analog statement of this result holds for p-harmonic vector functions (solutions of p-Laplacian equations) using the mentioned result obtained in the paper [23], stated here as: Theorem B. If u is a nonnegative p-harmonic function on a half space H n , continuous up to the boundary with u = 0 on H n , then u is (affine) linear.

Theorem.
If h is a quasiconformal p-harmonic mapping of the upper half space H n onto itself and h(∞) = ∞, then both h : (H n , | · |) → (H n , | · |) and h : (H n , ρ H n ) → (H n , ρ H n ) are bi-Lipschitz where ρ = ρ H n is the P oincaré distance.
Since 2-harmonic mapping are Euclidean harmonic this result includes Theorem A.
Proof. It suffices to deal with the case n = 3 as the proof for the general case is similar. Let h = (h 1 , h 2 , h 3 ).
By Theorem B, we get h 3 (x) = ax 3 , where a is a positive constant. Without loss of generality we may suppose that a = 1.
3 Pseudo-isometry and OC 1 (G) In this section, we give a sufficient condition for a qc mapping f : G → f (G) to be a pseudo-isometry w.r.t. quasihyperbolic metrics on G and f (G). First we adopt the following notation.
If V is a subset of R n and u : V → R m , we define osc V u = sup{|u(x) − u(y)| : x, y ∈ V } .
The proof of Theorem 2.19 gives the following more general result: 3.3. Theorem. Suppose that G ⊂ R n , f : G → G ′ , f ∈ OC 1 (G) and it satisfies the weak property of uniform boundedness with a constant c on G. Then (e) f : (G, k G ) → (G ′ , k G ′ ) is Lipschitz.
(f) In addition, if f is K-qc, then f is pseudo-isometry w.r.t. quasihyperbolic metrics on G and f (G).
Proof. By the hypothesis f satisfies the weak property of uniform boundedness: for every x ∈ G. This inequality together with (3.1) gives d(x)|f ′ (x)| ≤ c 3 d(f (x)). Now an application of Lemma 2.16 gives part (e). Since f −1 is qc, an application of [12, Theorem 3] on f −1 gives part (f).
In order to apply the above method we introduce subclasses of OC 1 (G) (see, for example, below (3.5)).
Let f : G → G ′ be a C 2 function and B x = B(x, d(x)/2). We denote by OC 2 (G) the class of functions which satisfy the following condition: for every x ∈ G and therefore OC 2 (G) ⊂ OC 1 (G). Now the following result follows from the previous theorem.
3.7. Corollary. Suppose that G ⊂ R n is a proper subdomain, f : G → G ′ is K-qc and f satisfies the condition (3.5). Then f : (G, k G ) → (G ′ , k G ′ ) is Lipschitz.
We will now give some examples of classes of functions to which Theorem 3.3 is applicable. Let SC 2 (G) denote the class of f ∈ C 2 (G) such that |∆f (x)| ≤ ar −1 sup{|f ′ (y)| : y ∈ B n (x, r)}, for all B n (x, r) ⊂ G, where a is a positive constant. Note that the class SC 2 (G) contains every function for which d(x)|∆f (x)| ≤ a|f ′ (x)|, x ∈ G. It is clear that SC 1 (G) ⊂ OC 1 (G) and by the mean value theorem, OC 2 (G) ⊂ SC 2 (G). For example, in [39] it is proved that SC 2 (G) ⊂ SC 1 (G) and that the class SC 2 (G) contains harmonic functions, eigenfunctions of the ordinary Laplacian if G is bounded, eigenfunctions of the hyperbolic Laplacian if G = B n and thus our results are applicable for instance to these classes.