Journal of Lie Theory Volume 8 (1998) 229–254 C ◯1998 Heldermann Verlag

Converse mean value theorems on trees and symmetric spaces

Enrico Casadio Tarabusi, Joel M. Cohen, Adam Korányi^{1} , and Massimo A. Picardello^{2}

Communicated by M. A. Moskowitz

Abstract. Harmonic functions satisfy the mean value property with respect to all integrable radial weights: if f is harmonic then h∗f=f ∫ h for any such weight h. But need a function f that satisfies this relation with a given (non-negative) h be harmonic? By a classical result of Furstenberg the answer is positive for every bounded f on a Riemannian symmetric space, but if the boundedness condition is relaxed then the answer turns out to depend on the weight h.

In this paper various types of weights are investigated on Euclidean

and hyperbolic spaces as well as on homogeneous and semi-homogeneous trees. If h decays faster than exponentially then the mean value property h∗f=f ∫ h does not imply harmonicity of f. For weights decaying slower than exponentially, at least a weak converse mean value property holds: the eigenfunctions of the Laplace operator which satisfy h∗f=f ∫ h are harmonic. The critical case is that of exponential decay. In this class we exhibit weights that characterize harmonicity and others that do not. 1991 Mathematics Subject Classification. Primary 43A85; Secondary 31B05, 05C05, 53C35 Key words and phrases. Mean value property, harmonic functions, Laplace operator, trees, symmetric spaces, hyperbolic spaces, convolution operators, exponential decay.

1. Introduction

A harmonic function f on (a domain of) ℝ^{n} satisfies the mean value property (MVP): the average of f on every ball or sphere equals the value at the center. Conversely, if the MVP holds for every ball or sphere then the function is harmonic. If f satisfies the MVP for only one sphere centered at each point then f need not be harmonic unless some kind of boundedness of f (e.g., positivity) and some regularity of the radius function are also assumed [2], [4], [5], [16], [25],

1 Partially supported by the NSF and a PSC-CUNY grant. 2 The author acknowledges the hospitality of the Department of Mathematics of the University of Maryland, where part of the research was carried out.

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag [24], [35], [36]. The situation is similar on hyperbolic spaces (which are, together with ℝ^{n}, the main examples of non-compact harmonic spaces) as well as on homogeneous and semi-homogeneous trees (which can be regarded as discrete analogues of hyperbolic spaces); there are also similarities with non-homogeneous trees [33]. Harmonic functions on trees have been studied intensely in the recent literature, both for their intrinsic interest (e.g., Poisson and Martin boundaries, asymptotic behavior: see [23], [8], [33], and references therein) and for their applications to Hardy spaces [31] and to unitary representations of free groups [9], [20], although we will not pursue these applications in the present paper. In general, trees can also be a good testing ground for theorems that often extend, even with similar proofs, to hyperbolic spaces or manifolds with non-positive curvature. Besides the present paper, this principle has yielded several results in the past (cf., e.g., [11] or [7]).

(The reader is referred to §2 below, especially Remark 2.1 and Definition 2.2, for the notation and terminology used in the remainder of this introduction.)

On ℝ^{n}, the characteristic function h of a ball does not characterize harmonicity, by a well-known result of Delsarte [18]: at least two different balls are necessary (for a related question on homogeneous trees see [15]). But if f is assumed positive or bounded there are positive results in a number of cases [13], [34]: condition (∗) implies harmonicity for bounded f in ℝ^{n} [12] and all non-compact Riemannian symmetric spaces [22], [6]. Thus (∗) does imply harmonicity if suitable boundedness assumptions are made on f, but does not in the general case. It is proved in [30], inspired by [1], that a specific weight h of exponential decay on ℝ^{n} and hyperbolic spaces does or does not characterize harmonicity depending on whether the dimension of the space is small or large. This result does not contradict the one recalled above, because (∗) makes sense only if f ∈ L_{h}^{1} , a restriction on the growth of f. The effect of such restriction depends on the dimension, whence the converse to the MVP also does.

The above seems to suggest the following general principle: a nonnegative radial weight h which decreases at infinity fast enough (faster than any exponential) does not characterize harmonicity; one, however, which decreases slowly enough (slower than any exponential) does. For intermediate rates of decay (exponential decay), the behavior depends on the exact nature of h. (Exponential decay is understood with respect to the distance from o in the ambient space; note that on hyperbolic spaces and trees the volume of spheres grows exponentially.) The purpose of this paper is to prove some instances of this principle for ℝ^{n}, hyperbolic spaces and (homogeneous and semi-homogeneous) trees. The analysis turns out to be not only easier but also more effective in this last environment, for which we shall prove suitable spectral synthesis results.

We show in Theorem 4.5 for trees and in Theorem 7.5 for hyperbolic spaces and ℝ^{n} that any weight h which satisfies h(r) ≤ Ce^{−A|r|^{1+α}} for some positive constants C, A, α does not (with a trivial exception) characterize harmonicity. On the other hand, if h is non-negative and decays slower than any exponential then it characterizes harmonicity in the weak sense (Theorem 4.4 and Theorem 7.4).

In the remainder of the paper exponentially decaying weights are considered. We construct examples with arbitrarily fast (exponential) decay which nevertheless characterize harmonicity, and others with arbitrarily slow (exponential) decay which do not (some of these were already known for ℝ^{n} and hyperbolic spaces). Our results are more complete for trees—all results here were in fact obtained on trees first, thereby providing inspiration for the continuous setup. This is reflected in the structure of the paper: throughout §6 we present the results for trees, and in the remainder we prove some of their counterparts on hyperbolic spaces and ℝ^{n}. Notation, e.g., as concerns the parameters for spherical transforms (cf. Remark 7.3), is so chosen as to stress the similarities among the various settings.

In view of [30] one might expect that pure exponential weights on trees would characterize harmonicity only if the homogeneity degree of the tree is sufficiently low, but instead it turns out that they always do (Theorem 3.2). The idea of the proof is quite general, as it only uses the fact that the exponential function is the resolvent of the discrete Laplacian Δ. The same approach applies (Theorem 3.3) in a variety of discrete homogeneous spaces (whose groups of isometries are free groups and free products), and to Laplacians which are not necessarily isotropic. Of particular interest are semi-homogeneous trees, studied in detail in §6. Their interest is due to the fact that, together with homogeneous trees, they arise (for suitable degrees) as the Bruhat-Tits buildings associated to rank-1 linear algebraic reductive groups on non-Archimedean local fields. The group of automorphisms of a semi-homogeneous tree is not transitive, therefore convolution is somewhat clumsy, and we replace it by semi-convolution, the summation against an automorphism-invariant kernel. In extending Theorem 3.2 we prove that a slightly more general (even in the homogeneous case) exponential-type invariant kernel characterizes harmonicity. It appears worth remarking (cf. Remark 6.6) that the expression of this kernel resembles closely that of Theorem 5.5 (described below), both having an oscillatory behavior, but the latter does not characterize harmonicity.

In §5 we consider the closed convex cone generated by positive exponentials on homogeneous trees (that is, the set of their integral averages with respect to a positive measure). Every h in this cone characterizes harmonicity in the weak sense; moreover, if h is actually a finite linear combination of positive exponentials with positive coefficients then it characterizes harmonicity in the strong sense (Theorem 5.4). Then we provide examples of linear combinations of exponentials which do not characterize harmonicity (Theorem 5.5); unfortunately this does not seem to provide inspiration for a continuous analogue.

In §§7,8,9 attention is focused on the continuous setting. Here it is natural to replace exponential weights with the resolvent of Δ at an eigenvalue for which it is positive and summable. On ℝ^{n} this leads to considering the MVP with respect to a class of exponentially decaying functions related to Bessel functions of the second kind, while on hyperbolic spaces we have MVPs related to spherical functions. The positive result of §3 again holds here, with essentially the same proof (except for minor technical details): this MVP characterizes harmonic functions also in the continuous setup. The weight h can be chosen here with an arbitrarily fast exponential rate of decay at infinity. On the other hand, on ℝ^{n}, if n ≥ 9, the radial exponential a^{|x|} does not characterize harmonicity for any 0 < a < 1, by a straightforward adaptation of the proof of [30, Proposition 2.1]. Consequently the feature of an exponentially decaying weight which determines whether or not it characterizes harmonicity is not its rate of decrease.

The continuous picture is not so complete as the discrete one when we consider MVPs with respect to convex combinations of resolvents. Following the approach for trees, we prove that such weights characterize harmonicity in the weak sense, but it is not clear whether they do in the strong sense. This question occurs often in this paper, because, in order to pass from weak to strong, some spectral synthesis result is needed. So far, in the continuous case such a result has only been proved on hyperbolic spaces for one specific weight [30]. Some considerations suggesting that spectral synthesis should hold more generally are presented in §10.

Remark 1.1. Harmonicity on higher-rank symmetric spaces or buildings cannot be expected to be characterized by any single weight h, unless f is assumed to be bounded. Indeed in those cases the spherical Fourier transform ĥ of h is an analytic function of several variables. The origin is among the points at which it attains the value ∫ h (the corresponding eigenfunction of Δ is a constant, therefore in L_{h}^{1} ), and there are others arbitrarily close. The eigenfunctions corresponding to these are counterexamples in L_{h}^{1} (cf. [3]).

We wish to thank Alessandra Gallinari for her help with some extensive preliminary computations, and John M. Horváth for useful comments and suggestions.

2. Basic notation and terminology

A tree is a connected graph without loops. A tree is homogeneous if each vertex has the same degree (number of neighbors) q+1. Each such tree is a homogeneous space of a free group or a certain free product, in either case contained in the full automorphism group of the tree itself. In the rest of the paper we shall not make explicit reference to the group structure.

On a tree assume fixed a reference vertex o. Let d be the natural integervalued distance, and set |x| = d(o, x) for every vertex x. By a function on a tree we mean a function on the set of its vertices. A function h is radial if h(x) only depends on |x| ; if so, by abuse of notation we write h(|x|) = h(x). The basic instance is the normalized equidistributed measure μ_{n} on the sphere of radius n ≥ 0 around o. In particular μ_{0} = δ_{o}, where δ_{x} denotes the Dirac delta at x, i.e., the characteristic function of {x}.

If f, g are functions, set ⟨f, g⟩ = ∑ fḡ, summation being performed over the whole tree. With regard to asymptotics for |x| → ∞, write f ⪯ g or f ≺ g if f = O(|g|) or f = o(|g|), respectively, and f ∼ g if f ⪯ g ⪯ f.

Let Δ denote the isotropic Laplacian M_{1} − id on a tree, where id is the identity and M_{1} is the equidistributed nearest-neighbor transition operator. A function is harmonic if it is annihilated by Δ—equivalently, the value at each point is the average of values at its neighbors. Harmonic functions enjoy the MVP for balls or spheres, as in ℝ^{n}.

Let w_{n} be the number of vertices of the sphere {|x| = n}. If the tree is homogeneous then

w_{n} = \{ \begin{matrix}1 & if n = 0, \\ (q + 1)q^{n−1} & if n > 0, \end{matrix}

and, since the operator M_{1} is the convolution with μ_{1} , then Δ is the convolution with μ_{1} − δ_{o}, so a function f is harmonic if and only if μ_{1} ∗ f = f.

Several of the above conventions and considerations extend to hyperbolic spaces and ℝ^{n}. On each of these spaces fix a reference point o (the origin in ℝ^{n}). If d is the distance in the space, set |x| = d(o, x) for every point x. A function (or measure) h is radial if h(x) only depends on |x| , and write h(|x|) = h(x) in this case. One example is the normalized equidistributed measure μ_{r} on the sphere of radius r ≥ 0 around o. Note that μ_{0} = δ_{o}, where δ_{x} denotes the Dirac delta at x (not a function, in these settings). For f, g functions, set ⟨f, g⟩ = ∫ fḡ, integration being performed over the whole space with respect to the standard isometry-invariant measure. For |x| → ∞, write f ⪯ g or f ≺ g if f = O(|g|) or f = o(|g|), respectively, and f ∼ g if f ⪯ g ⪯ f. With Δ denote the Laplace operator on ℝ^{n}, and the Laplace-Beltrami operator on a hyperbolic space. Also in the latter setting harmonic functions enjoy the MVP for balls or spheres.

Remark 2.1. Assume the ambient space is homogeneous (semi-homogeneous trees will be dealt with separately in §6), so that its automorphism group induces a convolution product. If f is harmonic and h a summable radial weight (function or measure) then the MVP on spheres and a straightforward integration in polar coordinates yield the MVP with respect to h, namely

(∗)

h ∗ f = f ∑ h  on homogeneous trees,
h ∗ f = f ∫ h  on ℝ^{n} or hyperbolic spaces,

provided that, for each x, the left-hand side of the equality exists, i.e., the function y ↦ h(d(x, y))f(y) is summable; with a slight abuse of notation we write f ∈ L_{h}^{1} in this case. The MVP on spheres thus reads μ_{r} ∗ f = f for all r ≥ 0.

Conversely, fixing an h as above, we may ask whether (∗) implies that f is harmonic. We shall frequently make the further assumption that h ≥ 0, although several of the results also hold without it.

Definition 2.2. We shall say that the summable weight h characterizes harmonicity (in the strong sense) if every f ∈ L_{h}^{1} that satisfies (∗) is harmonic, and that it characterizes harmonicity in the weak sense if every f ∈ L_{h}^{1} that satisfies (∗) and is an eigenfunction of Δ is harmonic.

These two notions clearly coincide whenever L_{h}^{1} has spectral synthesis, in the sense that every closed translation-invariant subspace is generated by the eigenfunctions of Δ contained in it.

Remark 2.3. The weight δ_{o} does not characterize harmonicity, since (∗) holds for every f. But if h characterizes harmonicity (in the strong or weak sense), then so does h̃ = c_{0}δ_{o} + c_{1}h for all constants c_{0}, c_{1} , if c_{1} ≠ 0, because condition (∗) for h immediately follows from condition (∗) for h̃. Thus the set of weights that characterize harmonicity is not a linear space.

3. Exponential weights on homogeneous trees

The natural definition of a radial exponential weight on a homogeneous tree is h_{a}(x) = a^{|x|} (i.e., h_{a} = ∑_{n=0}^{∞} w_{n}a^{n}μ_{n}) for a real, where we stipulate that t^{0} = 1 for any real t. Observe that h_{0} = δ_{o}, and that h_{a} is summable for |a| < 1/q.

Lemma 3.1. For 0 ≠ |a| < 1/q we have

(3.1)  μ_{1} ∗ h_{a} = (λ_{a} + 1)h_{a} + cδ_{o},

where

(3.2)  λ_{a} = \frac{qa + 1/a}{q + 1} − 1

and c = −λ_{a} ∑ h_{a}. Up to a constant factor, h_{a} is thus the resolvent of μ_{1} at the eigenvalue λ_{a} + 1 (or, equivalently, of Δ at λ_{a}) for the convolution product. Furthermore

∑ h_{a} = \frac{a + 1}{1 − qa}.

Proof. We have

μ_{1} ∗ h_{a}(x) = \frac{1}{q + 1} ∑_{d(y,x)=1} a^{|y|} = \{ \begin{matrix}a & if x = o, \\ (λ_{a} + 1)a^{|x|} & if x ≠ o, \end{matrix}

whence (3.1) holds for some constant c. Applying to both sides of (3.1) the convolution homomorphism L^{1} → ℂ given by g ↦ ∑ g we obtain ∑ h_{a} = (λ_{a} + 1) ∑ h_{a} + c. Finally

∑ h_{a} = ∑_{n=0}^{∞} w_{n}a^{n} = \frac{q + 1}{q} ∑_{n=0}^{∞}(qa)^{n} − \frac{1}{q} = \frac{a + 1}{1 − qa}.

For our first result we do not need any spherical transform calculus.

Theorem 3.2. On a homogeneous tree of degree q + 1, for 0 ≠ |a| < 1/q the weight h = h_{a} characterizes harmonicity. Proof. If f ∈ L_{h_{a}}^{1} satisfies (∗) with h = h_{a}, then

\begin{aligned}μ_{1} ∗ f ∑ h_{a} & = μ_{1} ∗ (h_{a} ∗ f) = (μ_{1} ∗ h_{a}) ∗ f = ((λ_{a} + 1)h_{a} + cδ_{o}) ∗ f \\ & = ((λ_{a} + 1) ∑ h_{a} + c)f = f ∑ h_{a} \end{aligned}

(associativity holds because μ_{1} is finitely supported and f ∈ L_{h_{a}}^{1} ).

The above argument has a wider range of application—e.g., to other discrete setups—in that it only requires general properties of convolution operators. The general statement is as follows. (For convenience, λ will denote the eigenvalue with respect to μ, instead of to the associated Laplacian μ − δ_{e} ∑ μ as above.)

Theorem 3.3. Let μ be a finitely supported function on a discrete group G, and assume that its resolvent (for the convolution product) R_{μ,λ} at some eigenvalue λ exists and is summable. Then h = R_{μ,λ} characterizes μ-harmonicity, that is, h ∗ f = f ∑ h implies μ ∗ f = f ∑ μ.

Proof. As before, we have ∑ μ·∑ R_{μ,λ} = λ ∑ R_{μ,λ} +1, whence ∑ R_{μ,λ} ≠ 0. The remainder of the argument is the same.

Note that Remark 2.3 also holds in this setting, where the Dirac delta is at the identity e of G. Assume μ is non-negative. The summability condition for the resolvent is fulfilled if the eigenvalue is greater than ∑ μ (as is the case for λ_{a} +1 in Theorem 3.2) and if G is a free group or a free product of two finite groups and μ is supported on words of distance 1 from e (cf. [20], [21], [10]), or if G is the free product of finitely many copies of the same finite group, and μ is equidistributed on the set of words of block length 1 (that is, on the union of the factor groups, see [28]). Therefore, the theorem applies to homogeneous or semi-homogeneous trees with a (not necessarily radial) “Laplace operator” μ supported on words of length 0 or 1, or to symmetric graphs (in the terminology of [28]) which arise as Cayley graphs of free products of two finite groups, or of several copies of the same finite group, with μ equidistributed on the factor groups; the latter case will be considered in detail in a forthcoming paper. In §§8,9 we shall show how the same argument can be adapted to Euclidean spaces and rank-1 symmetric spaces.

4. Decay of weights and their spherical transforms on homogeneous trees

A crucial role in the sequel will be played by spherical functions, so we briefly recall their asymptotics. Our main source for background is [20, Chapter 3], to which the reader is referred for further details. Consider a homogeneous tree of degree q + 1. If μ is a finitely supported radial function then there exists a polynomial P such that μ = P (μ_{1} − δ_{o}) (powers with respect to convolution are intended here, the unit constant being consequently δ_{o}). In fact for each n there exists a polynomial P_{n} of degree n such that μ_{n} = P_{n}(μ_{1} −δ_{o}), recursively given by

P_{n}(t) = \{ \begin{matrix}1 & if n = 0, \\ t + 1 & if n = 1, \\ \frac{q + 1}{q}(t + 1)P_{n−1}(t) − \frac{1}{q}P_{n−2}(t) & if n > 1 \end{matrix}

(the recurrence relation also holds for n = 1 if we set P_{−1} = P_{1} ). Replacing t with μ_{1} − δ_{o} we obtain

(4.1)  μ_{n} ∗ μ_{1} = \{ \begin{matrix}μ_{1} & for n = 0, \\ \frac{qμ_{n+1} + μ_{n−1}}{q + 1} & for n > 0. \end{matrix}

For λ ∈ ℂ denote by ϕ_{λ} (a spherical function) the unique radial eigenfunction of Δ of eigenvalue λ (i.e., μ_{1} ∗ ϕ_{λ} = (λ + 1)ϕ_{λ}) such that ϕ_{λ}(o) = 1. Recalling §2, for each n one has

\begin{aligned}ϕ_{λ}(n) & = ∑_{|y|=n} ϕ_{λ}(y)/w_{n} = ∑_{y} μ_{n}(y)ϕ_{λ}(y) = μ_{n} ∗ ϕ_{λ}(o) \\ & = P_{n}(μ_{1} − δ_{o}) ∗ ϕ_{λ}(o) = P_{n}(λ)ϕ_{λ}(o) = P_{n}(λ). \end{aligned}

Spherical functions are indexed in [20] by the parameter z that satisfies λ = λ_{q^{−z}} = γ(z) − 1, where

(4.2)  γ(z) = \frac{q^{z} + q^{1−z}}{q + 1} = \frac{2\sqrt{q}}{q + 1} cosh((z − 1/2) log q).

(Each of λ, z will be used frequently in the sequel, but λ, the eigenvalue of Δ, appears to be a better choice since γ is not one-to-one.) By [20, Theorem 3.3.3] the spectral radius of μ_{1} in L^{2} is ρ = 2\sqrt{q}/(q+1). In the parameter z, from [20, Theorem 3.2.2] we have a closed expression for P_{n}(λ) which turns out to be a linear combination of the exponentials h_{q^{−z}}, h_{q^{z−1}} , namely

(4.3)  ϕ_{λ}(n) = \{ \begin{matrix}c(z)q^{−nz} + c(1 − z)q^{n(z−1)} & if γ(z) ≠ ±ρ, \\ (±1)^{n}(1 + n\frac{q − 1}{q + 1})q^{−n/2} & if γ(z) = ±ρ, \end{matrix}

where c(z) = (q^{1−z} − q^{z−1})/(q + 1)(q^{−z} − q^{z−1}). The asymptotic behavior for fixed λ as n → ∞ (see also [14, Theorem 3]) is

(4.4) ϕ_{λ}(n) ⪯ \{ \begin{matrix}nq^{−n/2} & if Re z = 1/2, \\ q^{−n min\{Re z,1−Re z\}} = q^{−n(1/2−|Re z−1/2|)} & if Re z ≠ 1/2, \end{matrix}

uniformly on {z ∈ ℂ: |Re z − 1/2| ≥ c} for each c > 0. For real values of the parameter λ we obtain (4.5)

|P_{n}(λ)| ⪯ \{ \begin{matrix}nq^{−n/2} & if |λ + 1| ≤ ρ, \\ (\frac{(q + 1) |λ + 1| + \sqrt{(q + 1)^{2} |λ + 1|^{2} − 4q}}{2q} )^{n} & if |λ + 1| > ρ. \end{matrix}

The above inequalities are the so-called majorization principle [20, §3.5]. In particular, the following asymptotic monotonicity properties hold for real λ, λ′, λ′′:

(4.6) |λ + 1| < ρ < |λ′ + 1| < |λ′′ + 1| implies P_{n}(λ) ≺ P_{n}(ρ − 1) ≺ P_{n}(λ′) ≺ P_{n}(λ′′).

The majorization principle generalizes to non-real eigenvalues. The value of λ is not real unless either Im z is an integer multiple of π/ log q or Re z = 1/2; in general

|P_{n}(γ(z) − 1)| ≤ |P_{n}(γ(Re z) − 1)| .

(This follows immediately [20, Remark 3.2.4] from the integral expression

ϕ_{λ}(x) = ∫_{Ω} K(x, ω)^{z} dω,

where dω is the measure on the Poisson boundary Ω of the tree with respect to o, and K is the corresponding Poisson kernel.) Since the map γ is onto ℂ, this gives an asymptotic estimate of P_{n}(λ) for every complex λ. The inequality means that for every ζ ∈ ℝ the values of λ on the ellipse ∂E_{ζ}, boundary of

(4.7) E_{ζ} = {λ ∈ ℂ: (Re λ + 1)^{2} + (Im λ)^{2} coth^{2}((ζ − 1/2) log q) ≤ γ(ζ)^{2}}

(cf. [20, Proposition 3.3.1]) satisfy |P_{n}(λ)| ≤ |P_{n}(γ(ζ) − 1)| . Observe that the greater of the two real points of ∂E_{ζ} is γ(ζ) − 1.

If h is a function on the tree write ĥ(λ) = ⟨h, ϕ̄_{λ}⟩ whenever the righthand side is defined. For h ∈ L^{1} the value ĥ(γ(z) − 1) coincides with the spherical transform of h at z in [20, Chapter 3]. The map h ↦ ĥ is an injective homomorphism of the commutative Banach convolution algebra of radial L^{1} functions on the tree into the algebra of analytic functions, and takes μ_{1} − δ_{o} to the identity, because

μ̂_{1}(λ) = ⟨μ_{1}, ϕ_{λ}⟩ = μ_{1} ∗ ϕ_{λ}(o) = (λ + 1)ϕ_{λ}(o) = λ + 1.

Note that if h is a summable radial weight then ĥ(0) is the constant ∑ h of (∗).

Definition 4.1. Let h be a summable function on a homogeneous tree, and set h(n) = ⟨h, μ_{n}⟩ for all n (this extends the common abuse of notation for radial functions). We say that, for n → ∞, the function h decays:

(1) faster than exponentially if |h(n)| ≺ a^{n} for every a > 0;

(2) exponentially (like a^{n}) if 0 < a < 1/q is such that b^{n} ≺ |h(n)| ≺ c^{n} whenever 0 < b < a < c;

(3) slower than exponentially if a^{n} ≺ |h(n)| whenever 0 < a < 1/q.

Remark 4.2. The requirement that a summable function decay slower than exponentially is rather restrictive, since the measure of the spherical surface of radius n grows as the reciprocal q^{n} of the ‘critical’ decay.

As customary for Fourier transforms, a function will be said to be analytic in a subset of ℂ if it is analytic in its interior and continuous up to the boundary.

Proposition 4.3. Let h be a summable function on a homogeneous tree. Then:

(1) if h decays faster than exponentially then ĥ is entire;

(2) if h decays exponentially like a^{n} then ĥ is analytic on its domain E_{1/qa} of (4.7);

(3) if h decays slower than exponentially then ĥ is analytic on E_{1} .

Theorem 4.4. A non-negative summable radial weight h on a homogeneous tree that decays slower than exponentially (in the sense of Definition 4.1) characterizes harmonicity in the weak sense.

Proof. By Proposition 4.3(3), the spherical transform ĥ is defined only in E_{1} , and ĥ(λ) = ∑_{n=0}^{∞} h(n)w_{n}P_{n}(λ). If (∗) holds for some non-zero f ∈ L_{h}^{1} such that Δf = λf for some λ, then: first, translating if necessary, we can assume that f(o) ≠ 0; then, radializing and normalizing at o, we may assume that f = ϕ_{λ} (because the radializing operator M given by Mf(x) = ⟨f, μ_{|x|}⟩ satisfies h∗Mf = M(h∗f)). Then h∗ϕ_{λ} = ĥ(λ)ϕ_{λ} implies ĥ(λ) = ∑ h = ĥ(0). But every λ ∈ ∂E_{1} can be written as γ(it)−1 (or, equivalently, γ(it+1)−1) for some real t, hence |P_{n}(λ)| ≤ P_{n}(0) = 1 by the majorization principle. On the other hand, by (4.3), for each λ ∈ ∂E_{1} the sequence |P_{n}(λ)| is not constantly 1 on E_{1} unless λ = 0. Observe that if P_{n}(λ) = 1 for every n then ϕ_{λ}, being constant, is harmonic, so λ = 0. By the maximum principle |ĥ(λ)| < ĥ(0) in E_{1} except at λ = 0.

For functions decaying sufficiently fast we have the following result.

Theorem 4.5. Let h be a radial weight on a homogeneous tree such that for some positive constants A, α we have |h(n)| ⪯ e^{−An^{1+α}} for n → ∞. Then h does not characterize harmonicity, unless h = c_{0}δ_{o} + c_{1}μ_{1} with constants c_{0}, c_{1} , and c_{1} ≠ 0.

Proof. Since the change of parameter γ in (4.2) is periodic with imaginary period (namely 2πi/ log q), then so is the spherical transform ĥ(γ(z) − 1), which is therefore bounded on {z ∈ ℂ: u_{z} ≤ 1}, where u_{z} = 1/2 + |Re z − 1/2| . On the other hand, on {u_{z} > 1} we have (replacing the constant A with A log q) that

|ĥ(γ(z) − 1)| ≤ C ∑_{n=0}^{∞} w_{n}q^{−An^{1+α}} |P_{n}(γ(z) − 1)| ≤ C′ ∑_{n=0}^{∞} q^{−n(An^{α}−u_{z})}

by estimates (4.4) and the fact that w_{n} ∼ q^{n}. Split the last sum as I_{1} + I_{2} , where I_{1} is the sum over the index set D_{1} = {n ∈ ℕ: An^{α} > 2u_{z}}, and I_{2} over its complement D_{2} . On D_{1} one has An^{α} − u_{z} ≥ u_{z} ≥ 1, thus

I_{1} ≤ C′ ∑_{n∈D_{1}} q^{−n} ≤ C′ ∑_{n=0}^{∞} q^{−n} = C_{q}.

On D_{2} one has

I_{2} ≤ C′ ∑_{n∈D_{2}} q^{nu_{z}} ≤ C′q^{u_{z} max D_{2}} max D_{2} ≤ C′′u_{z}^{1/α}Q^{u_{z}^{1+1/α}},

where C′′ > 0 and Q > 1.

Since u_{z} ≤ |z| + 1, then ĥ ∘ (γ − 1) is of finite order (in the variable z), and this is immediately seen to imply that ĥ itself is (in the variable λ). If ĥ(λ) = ĥ(0) only at λ = 0, Hadamard’s product theorem [32, Theorem 1.13] yields ĥ(λ) − ĥ(0) = Bλ^{k}e^{P (λ)} for all λ ∈ ℂ with some constant B ≠ 0, some k ≥ 1, and some polynomial P of degree not larger than the order of ĥ. If P is not constant (i.e., if the support of h is infinite) then e^{P ∘(γ−1)} does not have finite order, and neither does ĥ ∘ (γ − 1). Therefore P is constant, that is, h has finite support. Hadamard’s product theorem is not really needed for this case, since every finitely supported weight equals Q(μ_{1}) for some polynomial Q. Up to a constant factor and to addition of a multiple of δ_{o} (cf. Remark 2.3) we may assume that ĥ(λ) = λ^{k}, that is, h = (μ_{1} − δ_{o})^{k}. This weight characterizes harmonicity if and only if k = 1. Indeed f given by f(x) = (|x| (q^{2} − 1) + 2q^{1−|x|})/(q − 1)^{2} satisfies Δf ≡ 1, whence it is not harmonic, but Δ^{k}f ≡ 0 for k ≥ 2. Note that, since the support of h is finite, every function belongs to L_{h}^{1} .

Remark 4.6. The above argument fails if we only require h(n) ≤ e^{−An log n}. In this case ĥ(λ) is still of finite order, but ĥ(γ(z) − 1) in general is not, so one cannot conclude that the polynomial P is constant.

Remark 4.7. The simplest examples of non-constant weights that do not characterize harmonicity on a homogeneous tree are μ_{1}^{2} and μ_{1} ∗ μ_{2} = (qμ_{3} + μ_{1})/(q + 1). For instance in the former case the eigenfunctions of μ_{1} with eigenvalue −1 (that is, of Δ with eigenvalue −2) are non-harmonic functions which nevertheless fulfill (∗). The spherical function ϕ_{−2} is the alternating function (−1)^{|x|}, which is bounded. This shows that Furstenberg’s result quoted in the introduction (that every weight characterizes harmonicity for bounded functions) has no analogue on trees.

5. The convex cone generated by exponentials on homogeneous trees

In order to examine more general exponentially decaying radial weights than h_{a}(x) = a^{|x|} let us first compute its spherical transform. We follow the notation of §3; in particular λ_{a} is given by (3.2).

Lemma 5.1. For 0 ≠ |a| < 1/q, the spherical transform of the radial exponential h_{a} on a homogeneous tree is

(5.1)  ĥ_{a}(λ) = \frac{a − 1/a}{(q + 1)(λ − λ_{a})}.

Consequently

(5.2)  ĥ_{a}(λ) − ĥ_{a}(0) = λ\frac{a + 1}{(qa − 1)(λ − λ_{a})}.

If λ ∈ ℝ and |λ + 1| < λ_{|a|}+1 (i.e., for λ = γ(z)−1, if |a| < q^{−z} < 1/q|a| ) then the series defining ĥ_{a}(λ) converges absolutely to the expression given in (5.1); otherwise, it diverges.

Proof. From Lemma 3.1 we get λĥ_{a}(λ) = λ_{a}ĥ_{a}(λ) + c, with c = −λ_{a}(a + 1)/(1−qa) = (a−1/a)/(q+1). We can then solve for ĥ_{a} and infer (5.1), provided λ ≠ λ_{a} and the series defining ĥ_{a}(λ) converges.

Since a ↦ qa + 1/a is decreasing in (0, 1/q), we have λ_{|a|} > 0. By the monotonicity properties (4.6) and by (4.5), if |λ + 1| < λ_{|a|} + 1 (notice that λ_{−a} + 1 = −(λ_{a} + 1)) then asymptotically |P_{n}(λ)| < |P_{n}(λ_{|a|})| ∼ (q |a|)^{−n}, and the series ∑_{n=0}^{∞} w_{n} |a|^{n} |P_{n}(λ)| converges if and only if |λ + 1| < λ_{|a|} + 1.

Consider now the convex cone generated by h_{a} for 0 < a < 1/q and closed with respect to pointwise convergence.

Lemma 5.2. Let ν be a positive measure on (0, 1/q) such that the function a ↦ ∑ h_{a} = (a + 1)/(1 − qa) is integrable. Then the radial weight h = ∫_{0}^{1/q} h_{a} dν(a) is summable, and its spherical transform is

ĥ(λ) = ∫_{0}^{1/q} \frac{a − 1/a}{(q + 1)(λ − λ_{a})} dν(a).

The spherical function ϕ_{λ} is in L_{h}^{1} if and only if |λ + 1| < λ_{a_{0}} + 1, where a_{0} = max supp ν. Proof. The result follows from Lemma 5.1, because h_{a_{0}} is the slowest decreasing in the one-parameter family of functions h_{a} for a ∈ supp ν, so it is the one that determines summability against ϕ_{λ}; indeed λ_{a_{0}} = min{λ_{a}: a ∈ supp ν}, since a ↦ qa + 1/a (hence a ↦ λ_{a}) is decreasing in (0, 1/q). It may be worth showing this directly, however. We have h(n)ϕ_{λ}(n) = w_{n}P_{n}(λ) ∫_{0}^{a_{0}} a^{n} dν(a), so

∑ |hϕ_{λ}| = ∑_{n=0}^{∞} w_{n} |P_{n}(λ)| ∫_{0}^{a_{0}} a^{n} dν(a) ≤ \frac{q + 1}{q}ν([0, a_{0}]) ∑_{n=0}^{∞}(qa_{0})^{n} |P_{n}(λ)| .

On the other hand, if 0 < a′ < a_{0} then

∑_{n=0}^{∞} w_{n} |P_{n}(λ)| ∫_{0}^{a_{0}} a^{n} dν(a) ≥ C ∑_{n=0}^{∞}(qa′)^{n} |P_{n}(λ)|

for some positive constant C, since ∫_{a′}^{a_{0}} dν(a) > 0.

Proposition 5.3. The radial weight h of Lemma 5.2 characterizes harmonicity in the weak sense. Proof. As in Theorem 4.4 we can assume f = ϕ_{λ} (which belongs to L_{h}^{1} by assumption). Then evaluating (∗) at o yields ĥ(λ) = ĥ(0). We have to prove that λ = 0 is the only root of this equation satisfying |λ + 1| < λ_{a_{0}} +1. Suppose λ ≠ 0 is another such root. From (5.2) we have

0 = ĥ(λ) − ĥ(0) = −λ ∫_{0}^{1/q} \frac{a + 1}{(1 − qa)(λ − λ_{a})} dν(a).

The imaginary part of the integrand is (a + 1)(Im λ)/(1 − qa) |λ − λ_{a}|^{2} , whose factors (except possibly Im λ) are all positive; therefore λ must be real. Hence the integrand itself is real, and its only factor that can change sign is λ − λ_{a}; in fact it must do so, in order for the integral to vanish. Therefore for some a ∈ supp ν this factor is positive, contradicting |λ + 1| < λ_{a_{0}} + 1 ≤ λ_{a} + 1.

Proposition 5.3 shows that if h belongs to the convex cone generated by exponentials then its ĥ(0)-eigenspace does not contain any eigenfunction of Δ except harmonic functions. For finite convex combinations of exponentials we can improve this by showing that the ĥ(0)-eigenspace of h and the kernel of Δ coincide:

Theorem 5.4. On a homogeneous tree of degree q + 1, the radial weight h = ∑_{j=1}^{k} c_{j}h_{a_{j}}, where c_{1}, …, c_{k} are positive constants and 0 < a_{1}, …, a_{k} < 1/q, characterizes harmonicity. Proof. Since

ĥ(λ) − ĥ(0) = λ ∑_{j=1}^{k} c_{j} \frac{a_{j} + 1}{(qa_{j} − 1)(λ − λ_{a_{j}})},

if P (λ) = ∏_{j=1}^{k}(λ − λ_{a_{j}}) then P (λ)(ĥ(λ) − ĥ(0)) is a polynomial of degree k in λ, with roots λ_{(1)} = 0 and λ_{(2)}, …, λ_{(k)} > 0 (cf. the end of the proof of Proposition 5.3), thus it equals Cλ ∏_{j=2}^{k}(λ − λ_{(j)}) for some non-zero constant C. Suppose f ∈ L_{h}^{1} satisfies (∗). As before, we can assume it radial. We have 0 = P (μ_{1} − δ_{o}) ∗ (h − δ_{o} ∑ h) ∗ f = C ∏_{j=2}^{k}(μ_{1} − (λ_{(j)} + 1)δ_{o}) ∗ Δf. If Δf ≠ 0, let k_{0} be the smallest integer (with 2 ≤ k_{0} ≤ k) such that ∏_{j=2}^{k_{0}}(μ_{1} −(λ_{(j)} +1)δ_{o})∗Δf = 0. Then ∏_{j=2}^{k_{0}−1}(μ_{1} −(λ_{(j)} +1)δ_{o})∗Δf is a radial eigenfunction of Δ of non-zero eigenvalue λ_{(k_{0})} , does not vanish identically, and belongs to L_{h}^{1} . Yet it satisfies (∗), a contradiction by Proposition 5.3.

Until now we have only produced exponentially decaying weights which characterize harmonicity (in the weak or strong sense). We now exhibit a linear combination with positive coefficients of two exponentials that does not.

Theorem 5.5. On a homogeneous tree of degree q + 1, for 0 < a < 1/q and 0 < c ≤ 1 the non-negative radial weight h = h_{a} + ch_{−a} does not characterize harmonicity. Proof. From Lemma 5.1 we have

ĥ(λ) − ĥ(0) = −λ(\frac{1 + a}{(1 − qa)(λ − λ_{a})} + c\frac{1 − a}{(1 + qa)(λ − λ_{−a})}).

This quantity (using the equality λ_{−a} + 1 = −(λ_{a} + 1)) can be seen to vanish if λ equals either λ_{(1)} = 0 or

λ_{(2)} = −\frac{(1 + a)(1 + qa) − c(1 − a)(1 − qa)}{(1 + a)(1 + qa) + c(1 − a)(1 − qa)}(λ_{a} + 1) − 1.

For c > 0 the absolute value of the coefficient of λ_{a} + 1 is less than 1, that is, |λ_{(2)} + 1| < λ_{a} + 1. Thus ϕ_{λ_{(2)}} is in L_{h_{a}}^{1} = L_{h_{−a}}^{1} , whence also in L_{h}^{1} , and satisfies (∗), but is not harmonic, because λ_{(2)} + 1 < 0.

The same proof works for c > 1—although h is not positive—except for c = (1 + a)^{2}(1 + qa)^{2}/(1 − a)^{2}(1 − qa)^{2} (since in this case λ_{(2)} = 0). Theorem 5.5 shows that Theorem 5.4 does not generalize to negative exponentials.

As an immediate consequence of Theorem 5.5 we see that the property of characterizing harmonicity is unstable under small perturbations of the weight in the L^{1} norm:

Corollary 5.6. For c → 0^{+} the weights h_{a} +ch_{−a}, which do not characterize harmonicity, tend in L^{1} to h_{a}, which, instead, does.

6. Exponential-type bi-weights on semi-homogeneous trees

On a tree, let ϵ be the parity function with respect to o, given by ϵ(x) = (−1)^{|x|}. A function f is alternating if it factors through ϵ. In this case, letting c_{±} (or c^{±}) equal f(o_{±}), where o_{+} = o, and o_{−} is a fixed neighbor of o, we have f(x) = c_{ϵ(x)} , so we write c_{ϵ} instead of f. Obviously f is constant if c_{+} = c_{−}. Since ϵ is radial, we will write ϵ(n) = (−1)^{n}; thus ϵ(x) = ϵ(d(x, y))ϵ(y) for any two vertices x, y.

If h is a summable radial weight on a homogeneous tree, we have set h ∗ f(x) = ∑_{y} k(x, y)f(y) whenever the integrand is summable for each x (i.e., if f ∈ L_{h}^{1} ), where the summation kernel k(x, y) = h(d(x, y)) is invariant under the diagonal action of the group of automorphisms, i.e., k(g · x, g · y) = k(x, y) for any two vertices x, y and any automorphism g. In fact, the group acts transitively on the tree, and, for every x, the isotropy group acts transitively on each sphere centered at x. Conversely, given such a kernel k the corresponding radial weight is determined by h(n) = k(x, y) for any x, y such that d(x, y) = n.

A tree is semi-homogeneous if its degree q + 1 = q_{ϵ} + 1 is alternating. Summation against a kernel, automorphism-invariant in the sense explained above, can substitute for convolution on a semi-homogeneous, but not homogeneous tree, as is typical in non-homogeneous settings. Since the group of automorphisms in this case has exactly two orbits and the isotropy subgroups are transitive on spheres, the kernel k is determined by a pair (called summable radial bi-weight) h^{ϵ} = (h^{+}, h^{−}) of independent, summable weights on the tree— namely: h^{+} , radial around o_{+} = o; and h^{−}, radial around a fixed neighbor o_{−} of o—via the relation k(x, y) = h^{ϵ(x)}(d(x, y)); conversely, k determines the biweight h^{ϵ} through h^{±}(n) = k(x, y) for any two vertices x, y such that ϵ(x) = ±1 and d(x, y) = n. The semi-convolution of the bi-weight h^{ϵ} with a function f is given by

h^{ϵ} ∗′ f(x) = ∑_{y} h^{ϵ(x)}(d(x, y))f(y),

if the integrand is summable for every x, which we shall indicate by f ∈ L_{h^{ϵ}}^{1} . If a tree is homogeneous, in particular it is also semi-homogeneous, hence semiconvolution with a bi-weight h^{ϵ} makes sense. The corresponding summation kernel k(x, y) is now invariant under parity-preserving automorphisms only. The space L_{h^{ϵ}}^{1} contains L_{h^{+}}^{1} ∩ L_{h^{−}}^{1} . This convolution coincides with ordinary convolution by h^{+} if and only if h^{+}(n) = h^{−}(n) for all n.

On a semi-homogeneous tree, using semi-convolution define the MVP with respect to the bi-weight h^{ϵ} as

(∗′)  h^{ϵ} ∗′ f(x) = f(x) ∑ h^{ϵ(x)}  for all x,

for f ∈ L_{h^{ϵ}}^{1} . Note that the function x ↦ ∑ h^{ϵ(x)} is alternating (non-constant, in general). Terminology of Definition 2.2 carries over in the obvious way.

To prove the analogue of Remark 2.1 we need to show that harmonic functions enjoy the (ordinary) MVP with respect to spheres of any radius. Let w_{n}^{±} be the number of vertices at distance n from a vertex of parity ±1. Setting q = \sqrt{q_{+}q_{−}}, it is easy to verify that

(6.1)  w_{n}^{±} = \{ \begin{matrix}1 & if n = 0, \\ (q_{±} + 1)q^{n−1}\sqrt{q_{∓}/q_{±ϵ(n)}} & if n > 0. \end{matrix}

Let μ_{n}^{±} be the normalized equidistributed measure on the sphere of radius n centered at o_{±}, and let M_{n} be the operator of semi-convolution with μ_{n}^{ϵ}, so that M_{n}f(x) = (1/w_{n}^{ϵ(x)}) ∑_{d(y,x)=n} f(y). Now, f is harmonic if M_{1}f = μ_{1}^{ϵ} ∗′ f = f. It turns out that for every odd n the operator M_{n} does not preserve the L^{1} norm of positive functions, that is, it is not an L^{1} -isometry when restricted to the cone. Instead it is an isometry in this cone for the weighted L^{1} space with alternating weight (q_{ϵ} + 1)/(q_{−ϵ} + 1). As a generalization of relations (4.1), one gets

Lemma 6.1. We have

(6.2)  M_{n}M_{1} = \{ \begin{matrix}M_{1} & if n = 0, \\ \frac{q_{ϵ(n)ϵ}M_{n+1} + M_{n−1}}{q_{ϵ(n)ϵ} + 1} & if n > 0. \end{matrix}

Therefore if f is harmonic then M_{n}f = f for all n. Proof. The procedure for (6.2) is similar to that followed for (3.1). If y is a vertex at distance n from x, then the number of neighbors of y farther away from x is

v_{n}(x) = \{ \begin{matrix}q_{ϵ(x)} + 1 & if n = 0, \\ q_{ϵ(n)ϵ(x)} & if n > 0. \end{matrix}

Thus for n > 0

M_{n}M_{1}f(x) = \frac{1}{w_{n}^{ϵ(x)}} ∑_{d(y,x)=n} \frac{1}{q_{ϵ(y)} + 1} ∑_{d(z,y)=1} f(z)

= \frac{1}{(q_{ϵ(n)ϵ(x)} + 1)w_{n}^{ϵ(x)}} ( ∑_{d(z,x)=n+1} +v_{n−1}(x) ∑_{d(z,x)=n−1})f(z)

= \frac{w_{n+1}^{ϵ(x)}M_{n+1} + v_{n−1}(x)w_{n−1}^{ϵ(x)}M_{n−1}}{(q_{ϵ(n)ϵ(x)} + 1)w_{n}^{ϵ(x)}} f(x).

Now replace each factor w_{m}^{ϵ(x)} with its expression as given in (6.1).

The second statement holds for n = 0 with any f, and for n = 1 by hypothesis. Proceed by induction on n, using (6.2).

Corollary 6.2. For any summable radial bi-weight h^{ϵ}, if f ∈ L_{h^{ϵ}}^{1} is harmonic then (∗′) holds. Proof. Integrating in polar coordinates around x we have

h^{ϵ} ∗′ f(x) = ∑_{n=0}^{∞} h^{ϵ(x)}(n) ∑_{d(y,x)=n} f(y) = f(x) ∑_{n=0}^{∞} h^{ϵ(x)}(n)w_{n}^{ϵ(x)} = f(x) ∑ h^{ϵ(x)},

the middle equality following from Lemma 6.1.

Remark 6.3. Exactly as in Remark 2.3, the bi-weight δ_{o_{ϵ}} = (δ_{o_{+}}, δ_{o_{−}}) does not characterize harmonicity, and, if the bi-weight h^{ϵ} characterizes harmonicity (in the strong or weak sense), then so does h̃^{ϵ} = c_{0}^{ϵ}δ_{o_{ϵ}} + c_{1}^{ϵ}h^{ϵ} for all constants c_{0}^{+}, c_{0}^{−}, c_{1}^{+}, c_{1}^{−}, with c_{1}^{+}, c_{1}^{−} ≠ 0.

A natural analogue of the exponential weight h_{a} is the bi-weight h_{a}^{ϵ} given by h_{a}^{±}(n) = a^{n}. We shall consider one which is slightly more general, given by

(6.3)  h_{a,b}^{±}(n) = a^{n}b^{±ϵ(n)}

for 0 ≠ |a| < 1/q (so h_{a,b}^{±} is summable) and b ≠ 0; the corresponding summation kernel is therefore k(x, y) = a^{d(x,y)}b^{ϵ(y)} . Corresponding to Lemma 3.1 we have:

Lemma 6.4. The operator H of semi-convolution with h^{ϵ} = h_{a,b}^{ϵ} satisfies the relation

(6.4)  M_{1} \frac{H}{∑ h^{ϵ}} = λ_{ϵ} \frac{H}{∑ h^{ϵ}} + c_{ϵ} id,

with

λ_{±} = \frac{(q_{±}a + 1/a) ∑ h^{±}}{(q_{±} + 1) ∑ h^{∓}} .

and c_{±} = 1 − λ_{±}. Explicitly

∑ h^{±} = \frac{(q_{∓}a + 1/a)b^{±1} + (q_{±} + 1)b^{∓1}}{1/a − q^{2}a} .

Proof. We have

\begin{aligned}M_{1} \frac{H}{∑ h^{ϵ}} f(x) & = \frac{1}{q_{ϵ(x)} + 1} ∑_{d(y,x)=1} \frac{1}{∑ h^{ϵ(y)}} ∑_{z} a^{d(y,z)}b^{ϵ(z)}f(z) \\ & = ∑_{z} \frac{b^{ϵ(z)}f(z) ∑_{d(y,x)=1} a^{d(y,z)}}{(q_{ϵ(x)} + 1) ∑ h^{−ϵ(x)}} = λ_{ϵ(x)} \frac{H}{∑ h^{ϵ}} f(x) + c(x)f(x) \end{aligned}

for some function c, because

\frac{∑_{d(y,x)=1} a^{d(y,z)}}{q_{ϵ(x)} + 1} = \{ \begin{matrix}a & if z = x, \\ \frac{q_{ϵ(x)}a + 1/a}{q_{ϵ(x)} + 1}a^{d(x,z)} & if z ≠ x. \end{matrix}

To finish the proof of (6.4) we are only left to show that c is itself alternating: applying the identity to a non-zero constant function we indeed obtain λ_{ϵ}+c ≡ 1.

Finally

∑ h^{±} = ∑_{n=0}^{∞} a^{n}b^{±ϵ(n)}w_{n}^{±} = ∑_{n=0}^{∞} a^{n}b^{±ϵ(n)}(q_{±} + 1)q^{n−1}\sqrt{q_{∓}/q_{±ϵ(n)}} − b^{±1}/q_{±}.

Splitting the series according to the parity of the index n, one has two geometric series with ratio qa; the remainder is an immediate algebraic verification.

We are now in a position to prove the generalization of Theorem 3.2.

Theorem 6.5. On a semi-homogeneous tree, the bi-weight h^{ϵ} = h_{a,b}^{ϵ} characterizes harmonicity if 0 ≠ |a| < 1/q and b ≠ 0.

Proof. If f satisfies (∗′), then by (6.4)

M_{1}f = M_{1} \frac{H}{∑ h^{ϵ}} f = λ_{ϵ} \frac{H}{∑ h^{ϵ}} f + c_{ϵ}f = (λ_{ϵ} + c_{ϵ})f = f.

Observe that, although ∑ h^{ϵ} appears in relation (6.4) and in the expression of the ‘bi-eigenvalue’ λ_{ϵ}, its explicit value is never needed for the purpose of proving Theorem 6.5.

Remark 6.6. On a homogeneous tree, the weight of Theorem 5.5 equals, up to a multiplicative constant, the bi-weight h^{±}(n) = a^{n}b^{ϵ(n)} for some b > 1. In spite of the similarity with the expression (6.3) of h_{a,b}^{ϵ}, the former does not characterize harmonicity, whereas the latter does. The respective semi-convolution kernels k(x, y) are a^{d(x,y)}b^{ϵ(d(x,y))} and a^{d(x,y)}b^{ϵ(x)ϵ(d(x,y))} = a^{d(x,y)}b^{ϵ(y)} . Observe that a^{d(x,y)}b^{ϵ(x)} is equivalent to a^{d(x,y)} in the sense of Remark 6.3.

7. Decay of weights in the continuous setting

Let us recall the asymptotics of spherical functions on hyperbolic spaces, i.e., rank-1 symmetric spaces of non-compact type. The picture is essentially identical to that of trees given in §4, and well known; notation is as in [30] and references therein. On a hyperbolic space ℍ there are intrinsic polar coordinates as follows: let K be the stabilizer subgroup of o in the connected component G of the isometry group of ℍ containing the identity e. Then ℍ = G/K, and every point of ℍ can be uniquely written as ka_{t} · o, where k ∈ K and {a_{t}} is a one-parameter group of transvections based at o. Therefore functions on ℍ can be lifted to K -invariant functions on G, and we shall do so whenever appropriate without further notice. A quantity that occurs often is ρ = p/2 + q, where p, q are the multiplicities of short, respectively, long roots. If dk denotes the normalized Haar measure on K then the intrinsic volume element on ℍ (with respect to which the scalar product of functions is defined—cf. §2) is dv = (2 sinh r)^{p+q}(2 cosh r)^{q} dr dk. The K -invariant surface measure w_{r} of the spherical surface of radius r grows as e^{2ρr} for r → ∞.

The radial component of the Laplace-Beltrami operator Δ is

∂^{2}/∂r^{2} + ((p + q) coth r + q tanh r)∂/∂r.

The spherical function ϕ_{λ} is the bi-K -invariant (i.e., radial) eigenfunction of Δ of eigenvalue λ such that ϕ_{λ}(e) = 1, and, using the parameter z that satisfies γ̃(z) = λ, where γ̃(z) = z(z + 2ρ), is given by

(7.1)  ϕ_{λ}(g) = ∫_{K} P (g, k)^{z} dk,

where P (g, k) = e^{r(g^{−1}k)} is the Poisson kernel, and r(g^{−1}k) stands for the value of r such that g^{−1}k = k′a_{r} for some k′ ∈ K. As for trees, it follows that

(7.2)  |ϕ_{γ̃(z)}| ≤ |ϕ_{γ̃(Re z)}| .

The spherical transform ĥ of a function h is defined on {λ ∈ ℂ: ϕ_{λ} ∈ L_{h}^{1}} as ĥ(λ) = ⟨h, ϕ̄_{λ}⟩.

The asymptotic behavior of spherical functions is almost the same as on trees (deeper analogies between these two setups are described in [9]; see also [20]). With our choice of parameters, for r → ∞ we have

|ϕ_{λ}(r)| \{ \begin{matrix}⪯ e^{(|\sqrt{λ+ρ^{2}}|−ρ)r}(1 + r) & if ℝ ∋ λ > −ρ^{2}, \\ ∼ e^{(|Re \sqrt{λ+ρ^{2}}|−ρ)r} & otherwise. \end{matrix}

As done on trees in Definition 4.1, we define the rate of decay on hyperbolic spaces (as well as Euclidean spaces).

Definition 7.1. Let h be a summable function on a hyperbolic space ℍ, and write h(r) = ⟨h, μ_{r}⟩ = ∫_{K} h(ka_{r} · o) dk for all r ≥ 0. We say that, for r → ∞, the function h decays:

(1) faster than exponentially if |h(r)| ≺ a^{r} for every a > 0;

(2) exponentially (like a^{r}) if 0 < a < e^{−2ρ} is such that b^{r} ≺ |h(r)| ≺ c^{r} whenever 0 < b < a < c;

(3) slower than exponentially if a^{r} ≺ |h(r)| whenever 0 < a < e^{−2ρ}.

While, if h is a summable function on ℝ^{n}, set h(r) = ⟨h, μ_{r}⟩ for all r ≥ 0. We say that, for r → ∞, the function h decays:

(1) faster than exponentially if |h(r)| ≺ a^{r} for every a > 0;

(2) exponentially (like a^{r}) if a > 0 is such that b^{r} ≺ |h(r)| ≺ c^{r} whenever 0 < b < a < c;

(3) slower than exponentially if a^{r} ≺ |h(r)| for every a > 0.

As observed for trees in Remark 4.2, the requirement that a summable function on a hyperbolic space decay slower than exponentially is rather restrictive, because the measure of the spherical surface of radius r grows itself as e^{2ρr}. For the domain of the spherical transform we have the following (with the terminologic convention before Proposition 4.3 about analyticity on non-open sets).

Proposition 7.2. Let h be a summable function on a hyperbolic space ℍ, and set S_{t} = \{λ ∈ ℂ: |Re \sqrt{λ + ρ^{2}}| ≤ −ρ + t\}. Then:

(1) if h decays faster than exponentially then ĥ is entire;

(2) if h decays exponentially like a^{r} then ĥ is analytic on its domain S_{− log a};

(3) if h decays slower than exponentially then ĥ is analytic on its domain S_{2ρ}.

Remark 7.3. The parameter λ is chosen, for all the three ambient spaces, to be the eigenvalue of Δ. Consequently, if h is a summable radial weight then the constant ∫ h of (∗) equals ĥ(0) in any of these setups.

The results of Theorem 4.4 and Theorem 4.5 for slowly as well as for fast decaying weights carry over to hyperbolic spaces and Euclidean spaces.

Theorem 7.4. A non-negative summable radial weight h on a hyperbolic space, or on ℝ^{n}, that decays slower than exponentially (in the sense of Proposition 7.2) characterizes harmonicity in the weak sense. Proof. The proof for hyperbolic spaces is the same as for trees given in Theorem 4.4. By the majorization principle (7.2), for λ ∈ S_{2ρ} we have the inequality |ĥ(λ)| ≤ ∫ h = ĥ(0), which is strict in the interior of S_{2ρ} by the maximum principle, and is an equality for λ = 0. On the rest of ∂S_{2ρ} the inequality is also strict since |ϕ_{λ}(x)| < ϕ_{0}(x) = 1 for x in a set of positive measure, because of cancellations in the integral ⟨h, ϕ_{λ}⟩ due to the fact that the corresponding exponent z in (7.1) has a non-zero imaginary part.

Finally, on ℝ^{n} the statement is obvious: the Fourier transform ĥ(λ) converges only for real λ, and |ĥ(λ)| < ∫ h = ĥ(0) for λ ≠ 0, because h is a non-atomic positive measure.

Theorem 7.5. Let h be a radial weight on a hyperbolic space, or on ℝ^{n}, such that for some positive constants A, α we have |h(r)| ⪯ e^{−Ar^{1+α}} for r → ∞. Assume in addition that ∫ h ≠ 0. Then h does not characterize harmonicity.

Proof. We first give the proof for ℝ^{n}. It is enough to show that ĥ(λ) = ĥ(0) for some λ ≠ 0 in ℂ^{n}. Note first that ĥ(λ) → 0 as |λ| → ∞ in {|Im λ| < 1}. In fact, given ϵ > 0 one can find R > 0 and ϕ ∈ C^{∞}(ℝ^{n}) with support {|x| ≤ R} such that

∫_{|x|>R} |h(x)| e^{|x|} dx < ϵ/3,
∫_{|x|≤R} |h(x) − ϕ(x)| dx < ϵ/3e^{R}.

Now, by the Paley-Wiener theorem, when |λ| is sufficiently large and |Im λ| < 1 we have |ϕ̂(λ)| < ϵ/3. But also, if |Im λ| < 1 then

\begin{aligned}|ĥ(λ) − ϕ̂(λ)| & = |∫_{|x|>R} h(x)e^{iλ·x} dx + ∫_{|x|≤R}(h(x) − ϕ(x))e^{iλ·x} dx| \\ & ≤ ∫_{|x|>R} |h(x)| e^{|x|} dx + ∫_{|x|≤R} |h(x) − ϕ(x)| e^{|x|} dx ≤ 2ϵ/3, \end{aligned}

thus |ĥ(λ)| < ϵ.

Next we show that the entire function ĥ has finite order, that is, as |λ| → ∞ we have |ĥ(λ)| ⪯ e^{^{|λ|}^{B}} for some B. Indeed,

|ĥ(λ)| ≤ C ∫_{ℝ^{n}} e^{−|x|(A|x|^{α}−|Im λ|)} dx.

Split the integral as I_{1} +I_{2} , where I_{1} is the value of the integral over the domain D_{1} = {x ∈ ℝ^{n}: A |x|^{α} > 2 |Im λ| }, and I_{2} over its complement D_{2} ; it is enough to consider |Im λ| ≥ 1. On D_{1} one has A |x|^{α} − |Im λ| ≥ |Im λ| ≥ 1, hence

I_{1} ≤ ∫_{D_{1}} e^{−|x|} dx ≤ ∫_{ℝ^{n}} e^{−|x|} dx = C_{n}.

On D_{2} one has

I_{2} ≤ ∫_{D_{2}} e^{^{|x|}^{|Im λ|}} dx ≤ e^{(2/A)^{1/α}|Im λ|^{1+1/α}} Vol D_{2}
≤ C_{n}(2 |Im λ| /A)^{n/α}e^{(2/A)^{1/α}|Im λ|^{1+1/α}}.

Thus ĥ, hence ĥ − ĥ(0), is of finite order. If ĥ(λ) = ĥ(0) only at λ = 0, then, for any fixed unit vector v ∈ ℝ^{n}, again Hadamard’s product theorem yields ĥ(sv) − ĥ(0) = B′s^{k}e^{P (s)} for all s ∈ ℂ with some B′ ≠ 0, some k ≥ 1, and some polynomial P of degree not larger than the order of ĥ. It follows that |ĥ(sv) − ĥ(0)| tends to 0 or ∞ as s → ∞ through real numbers. This contradicts the Riemann-Lebesgue lemma, which implies that the limit must be |ĥ(0)| .

An obvious modification of the proof for ℝ^{n} works for hyperbolic spaces, adapting the argument used in Theorem 4.5 for trees, the bounds of whose spherical functions are essentially the same. The Riemann-Lebesgue lemma needed here is [26, Exercise B.6 of the Introduction, with solution], with the condition |Re \sqrt{λ + ρ^{2}}| < ρ (under which the spherical function ϕ_{λ} is bounded).

8. The resolvent of the Laplacian on Euclidean spaces

In order to find a positive radial weight h of exponential decay on ℝ^{n} which characterizes harmonicity, let us proceed as in §3, and look for a fundamental solution R_{ζ} of the equation ΔR_{ζ} = ζR_{ζ}−δ_{o}. For r > 0 the equation ΔR_{ζ} = ζR_{ζ} in polar coordinates is the Bessel equation, hence for ζ > 0 a positive radial fundamental solution exists, given by

(8.1)  R_{ζ}(x) = c_{n} |x|^{1−n/2}K_{n/2−1}(\sqrt{ζ} |x|),

where K_{n/2−1} is the Bessel function of the second kind, and c_{n} is a suitable constant. For r = |x| → 0, the solution R_{ζ} has the characteristic singularity cr^{2−n} if n is odd, and cr^{2−n} + d log r if n is even, for c, d ≠ 0 [17]. Positivity follows from the integral representations of K_{ν} [19, (7.12.21–23)].

Lemma 8.1. We have ∫ R_{ζ} = 1/ζ. Proof. Formally this is immediate: 0 = ⟨R_{ζ}, Δ1⟩ = ⟨ΔR_{ζ}, 1⟩ = ζ⟨R_{ζ}, 1⟩ − ⟨δ_{o}, 1⟩ = ζ ∫ R_{ζ} − 1. To make this rigorous, denote by B_{r} the ball of radius r centered at o. Then, for 0 < r_{1} < r_{2} , Green’s identity yields

ζ ∫_{B_{r_{2}} \B_{r_{1}}} R_{ζ} = ∫_{B_{r_{2}} \B_{r_{1}}} ΔR_{ζ} = ∫_{∂B_{r_{2}}} \frac{∂R_{ζ}}{∂r} − ∫_{∂B_{r_{1}}} \frac{∂R_{ζ}}{∂r} .

On the right-hand side the first integral tends to 0 as r_{2} → ∞, because ∂R_{ζ}/∂r decays exponentially, while the second tends to −1 for r_{1} → 0, due to the characteristic singularity of R_{ζ}.

The argument of Theorem 3.2 applies to this continuous setup:

Theorem 8.2. On ℝ^{n}, for ζ > 0 the resolvent h = R_{ζ} of Δ at the eigenvalue ζ characterizes harmonicity. Proof. The fact that f ∈ L_{h}^{1} implies that f ∗ h(x) exists as a Lebesgue integral for all x. Also, f ∗ h is locally integrable. Hence for any ϕ ∈ D(ℝ^{n}) the function ϕ(x + y)f(x)h(y) is in L^{1}(ℝ^{2n}), and [27, Condition (Γ)] is satisfied. So by [27, Proposition 18] and (∗) we have in the distribution sense Δf/ζ = Δ(h ∗ f) = Δh ∗ f = (ζh − δ_{o}) ∗ f = ζh ∗ f − f = 0.

This shows that the resolvent of the Laplacian in ℝ^{n} characterizes harmonicity in much the same way as on trees. For the convex cone generated by resolvents we can prove the analogue of Proposition 5.3.

Proposition 8.3. Let ν be a positive measure on (0, ∞) such that the radial weight h = ∫_{0}^{∞} R_{ζ} dν(ζ) on ℝ^{n} is finite and summable. Then h characterizes harmonicity in the weak sense. Proof. We follow the approach of [30, Proposition 2.1]. Let f be such that (∗) and Δf = λf hold, with λ ∈ ℂ. As before, f may be assumed radial. Radial eigenfunctions of Δ with eigenvalue λ which are finite (and non-zero) at the origin are multiples, for z = ±\sqrt{λ}/2πi, of a function j_{z} (related to the Bessel function of the first kind), given by

j_{z}(x) = ∫_{K} e^{2πz⟨k·x,v⟩} dk,

where dk is the normalized Haar measure on K = SO(n), v ∈ ℝ^{n} is a fixed unit vector, and ⟨ · , · ⟩ is here the standard scalar product in ℝ^{n}. The asymptotic behavior for |x| → ∞ is j_{z}(x) ∼ |x|^{−(n+1)/2}e^{2π|Im z||x|} (see [30] for references). On the other hand, if ζ_{0} = min supp ν we see, as for Lemma 5.2, that (8.1) implies h(x) ∼ |x|^{(1−n)/2}e^{−\sqrt{ζ_{0}}|x|}. Therefore the condition j_{z} ∈ L_{h}^{1} amounts to

(8.2)  2π |Im z| < \sqrt{ζ_{0}}.

As in [30], h ∗ j_{z} = ĥ(zv)j_{z}, hence (∗) gives ĥ(zv) = ∫ h = ĥ(0).

Since ΔR_{ζ} = ζR_{ζ} − δ_{o}, for every λ ∈ ℝ^{n} the Fourier transform of R_{ζ} satisfies − |λ|^{2}R̂_{ζ}(λ) = ζR̂_{ζ}(λ) − 1, so that R̂_{ζ}(λ) = 1/(ζ + |λ|^{2}). Therefore

0 = ĥ(zv) − ĥ(0) = −z^{2} ∫_{0}^{∞} \frac{1}{(ζ + z^{2})ζ} dν(ζ).

The imaginary part of the integrand is −(Im z^{2})/ζ |ζ + z^{2}|^{2} , whence z^{2} is real. If z ≠ 0 is a solution of the equation ĥ(zv) = ĥ(0) then the integral vanishes, therefore at some ζ ∈ supp ν the integrand must be negative, so z^{2} < −ζ ≤ −ζ_{0} ≤ 0. Then |Im z| = |z| > \sqrt{ζ_{0}} , contradicting (8.2). So the only possible solution is z = 0, corresponding to the function j_{0} , a constant, hence harmonic.

9. The resolvent of the Laplacian on hyperbolic spaces

In this section we extend to hyperbolic spaces the results proved for ℝ^{n} in the previous section. With notation as in §7, the spherical function ϕ_{λ} (the radial eigenfunction of Δ with eigenvalue λ such that ϕ_{λ}(o) = 1) can be expressed as a linear combination of Φ_{λ}^{+}, Φ_{λ}^{−}, where Φ_{λ}^{±}(r) is singular at r = 0, behaves as e^{(±\sqrt{λ+ρ^{2}}−ρ)r} for r → ∞, and is real-valued if and only if ℝ ∋ λ > −ρ^{2} .

For ζ > 0 the resolvent R_{ζ} of Δ at the eigenvalue ζ (i.e., the Green kernel of Δ − ζ id) is min{Φ_{ζ}^{+}, Φ_{ζ}^{−}}, cf. [29] (observe the analogy with trees, where, as briefly outlined in §4, spherical functions decompose as linear combinations of two exponentials, the smaller of which is the resolvent of Δ which was used in Theorem 3.2—see [20, Chapter 3] for more details). Hence for r → ∞

R_{ζ}(r) ∼ e^{−(ρ+\sqrt{ζ+ρ^{2}})r}.

Lemma 9.1. We have ∫_{ℍ} R_{ζ} = 1/ζ. Proof. The argument is the same as for Lemma 8.1. The only difference is that for r_{2} → ∞ the volume of the sphere of radius r_{2} now grows exponentially, namely as e^{2ρr_{2}} , but this is compensated by (∂R_{ζ}/∂r)(r_{2}) ∼ e^{−(ρ+\sqrt{ζ+ρ^{2}})r_{2}} ≺ e^{−2ρr_{2}} . Hence in Green’s identity the integral over such sphere tends to 0.

As in §8, the relevant part of [27] works in this setting, so we can infer

Theorem 9.2. On a hyperbolic space, for ζ > 0 the resolvent h = R_{ζ} of Δ at the eigenvalue ζ characterizes harmonicity.

Also the analogue of Proposition 8.3 holds here.

Proposition 9.3. Let ν be a positive measure on (0, ∞) such that the radial weight h = ∫_{0}^{∞} R_{ζ} dν(ζ) on a hyperbolic space is finite and summable. Then h characterizes harmonicity in the weak sense. Proof. We proceed as for Proposition 8.3. By Proposition 7.2(2), for given ζ > 0 the spherical transform R̂_{ζ}(λ) = ∫ R_{ζ}ϕ_{λ} is analytic in |Re \sqrt{λ + ρ^{2}}| < \sqrt{ζ + ρ^{2}} , therefore ĥ(λ) is analytic in the strip

(9.1)  |Re \sqrt{λ + ρ^{2}}| < \sqrt{ζ_{0} + ρ^{2}},

where ζ_{0} = min supp ν. In order that ϕ_{λ} be in L_{h}^{1} , inequality (9.1) must hold.

Taking the spherical Fourier transform of the resolvent identity ΔR_{ζ} = ζR_{ζ} − δ_{o} we obtain R̂_{ζ}(λ) = 1/(ζ − λ), so

0 = ĥ(λ) − ĥ(0) = λ ∫_{0}^{∞} \frac{1}{(ζ − λ)ζ} dν(ζ)

(cf. Remark 7.3). Taking the imaginary part of the integrand, one sees that λ is real. Unless λ itself vanishes, we get λ > ζ ≥ ζ_{0} ≥ 0 for some ζ ∈ supp ν, as long as ĥ(λ) = ĥ(0). This implies |Re \sqrt{λ + ρ^{2}}| = \sqrt{λ + ρ^{2}} > \sqrt{ζ_{0} + ρ^{2}} , contradicting (9.1).

10. A conjecture on the spectral resolution of the Laplace-Beltrami operator

In the continuous setting (ℝ^{n} and ℍ) we have shown that the convolution eigenspace M_{h} = {f ∈ L_{h}^{1} : f satisfies (∗)} of an exponentially decaying weight h in a large class does not contain any eigenfunction of Δ except its kernel. We have also shown that Δ(h ∗ f) = Δh ∗ f. Then, under mild assumptions on f, convolution by h commutes with Δ: in fact Δh ∗ f = h ∗ Δf by integration by parts, provided that f grad h and h grad f vanish at infinity. But in our classes it is easy to see that grad h ∼ h, therefore it is enough to assume that hf vanishes at infinity (because then so does h grad f, by integration by parts); in this case M_{h} is an invariant subspace for Δ. In order to show that h characterizes harmonicity, it would thus be sufficient to prove that M_{h} is a direct sum of eigenspaces of Δ|_{M_{h}}.

In the discrete setting of trees, a similar result is required to prove that weights in the closed convex cone generated by exponentials characterize harmonicity. The statement for finite positive combinations was proved in Theorem 5.4 by identifying the spectrum of Δ|_{M_{h}} with the set of roots of a suitable polynomial. Here no vanishing at infinity is needed to ensure that M_{h} is invariant for the Laplacian, because it is finitely supported and commutes with every radial convolutor. A similar approach has been followed in [30] with a specific weight h for which the equation ĥ = ∫ h has finitely many solutions, showing that the spectrum of Δ|_{M_{h}} consists exactly of such solutions.

It seems reasonable to believe that M_{h} is indeed a direct sum of eigenspaces of Δ|_{M_{h}} in all the cases considered. For assume that we have a spectral resolution Δ|_{M_{h}} = ∫_{σ} λ dE(λ), with σ denoting the spectrum, and ∫_{σ} dE(λ) = id. If the radial eigenvector ϕ_{λ} of Δ|_{M_{h}} with eigenvalue λ is in L_{h}^{1} , then h ∗ ϕ_{λ} = ĥ(λ)ϕ_{λ}. This suggests that the operator H of convolution by h satisfies H = ∫_{σ} ĥ(λ) dE(λ). On the other hand, H on M_{h} equals ĥ(0) times the identity, hence its spectral decomposition with respect to the operator-valued measure E(λ) should be ∫_{σ} ĥ(0) dE(λ). The support of E(λ) should thus be the set {λ: ĥ(λ) = ĥ(0)}, discrete because ĥ is analytic.

References

[1] Ahern, P., M. Flores, and W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993), 380–397.

[2] Akcoglu, M. A., and R. W. Sharpe, Ergodic theory and boundaries, Trans. Amer. Math. Soc. 132 (1968), 447–460.

[3] Arazy, J., and G. Zhang, Invariant mean value and harmonicity in Cartan and Siegel domains, in “Interaction between functional analysis, harmonic analysis, and probability,” Columbia 1994 (N. Kalton et al., eds.), Lecture Notes in Pure and Appl. Math. 175, M. Dekker, New York, 1996, pp. 19–40.

[4] Baxter, J. R., Restricted mean values and harmonic functions, Trans. Amer. Math. Soc. 167 (1972), 451–463.

[5] —, Harmonic functions and mass cancellation, Trans. Amer. Math. Soc. 245 (1978), 375–384.

[6] Ben Natan, Y., Y. Benyamini, H. Hedenmalm, and Y. Weit, Wiener’s Tauberian theorem in L^{1}(G//K) and harmonic functions in the unit disk, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 43–49.

[7] Berenstein, C. A., and E. Casadio Tarabusi, An inversion formula for the horocyclic Radon transform on the real hyperbolic space, in “Tomography, impedance imaging, and integral geometry,” South Hadley 1993 (E. T. Quinto et al., eds.), Lectures in Appl. Math. 30, Amer. Math. Soc., Providence, 1994, pp. 1–6.

[8] Cartier, P., Fonctions harmoniques sur un arbre, in “Calcolo delle probabilità,” 1971, Sympos. Math. 9, Academic Press, London, 1972, pp. 203– 270.

[9] —, Harmonic analysis on trees, in “Harmonic analysis on homogeneous spaces,” Williamstown 1972 (C. C. Moore, ed.), Proc. Sympos. Pure Math. 26, Amer. Math. Soc., Providence, 1973, pp. 419–424.

[10] Cartwright, D. I., and P. M. Soardi, Harmonic analysis on the free product of two cyclic groups, J. Funct. Anal. 65 (1986), 147–171.

[11] Casadio Tarabusi, E., Inversion of the X-ray transform: continuous vs. discrete, in “Integral geometry and tomography,” Arcata 1989 (E. Grinberg, E. T. Quinto, eds.), Contemp. Math. 113, Amer. Math. Soc., Providence, 1990, pp. 31–39.

[12] Choquet, G., and J. Deny, Sur l’équation de convolution μ = μ ∗ σ, C. R. Acad. Sci. Paris 250 (1960), 799–801.

[13] —, Noyaux de convolution et balayage sur tout ouvert, in “Théorie du potentiel et analyse harmonique,” Strasbourg 1973 (J. Faraut, ed.), Lecture Notes in Math. 404, Springer, Berlin, 1974, pp. 60–112.

[14] Cohen, J. M., Operator norms on free groups, Boll. Un. Mat. Ital. B (6) 1 (1982), 1055–1065.

[15] Cohen, J. M., and M. A. Picardello, The 2-circle and 2-disk problems on trees, Israel J. Math. 64 (1988), 73–86.

[16] Cornea, A., and J. Veselý, Martin compactification for discrete potential theory and the mean value property, Potential Anal. 4 (1995), 547–569.

[17] Courant, R., and D. Hilbert, “Methods of mathematical physics, II: Partial differential equations,” Wiley Classics Lib., Interscience-Wiley, New York, 1953.

[18] Delsarte, J., Note sur une propriété nouvelle des fonctions harmoniques, C. R. Acad. Sci. Paris 246 (1958), 1358–1360.

[19] Erdélyi, A., et al., “Higher transcendental functions, II,” California Institute of Technology Bateman Manuscript Project, McGraw-Hill, New York, 1953.

[20] Figà-Talamanca, A., and M. A. Picardello, “Harmonic analysis on free groups,” Lecture Notes in Pure and Appl. Math. 87, M. Dekker, New York, 1983.

[21] Figà-Talamanca, A., and T. Steger, “Harmonic analysis for anisotropic random walks on homogeneous trees,” Mem. Amer. Math. Soc. 531 (110), 1994.

[22] Furstenberg, H., A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386.

[23] —, Random walks and discrete subgroups of Lie groups, (P. Ney, ed.), Adv. Probab. Related Topics 1, M. Dekker, New York, 1971, pp. 1–63.

[24] Hansen, W., and N. Nadirashvili, A converse to the mean value theorem for harmonic functions, Acta Math. 171 (1993), 139–163.

[25] Heath, D., Functions possessing restricted mean value properties, Proc. Amer. Math. Soc. 41 (1973), 588–595.

[26] Helgason, S., “Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions,” Pure and Appl. Math. 113, Academic Press, Orlando, 1984.

[27] Horváth, J., Sur la convolution des distributions, Bull. Sci. Math. (2) 98 (1974), 183–192.

[28] Iozzi, A., and M. A. Picardello, Spherical functions on symmetric graphs, in “Harmonic analysis,” Cortona 1982 (G. Mauceri et al., eds.), Lecture Notes in Math. 992, Springer, Berlin, 1983, pp. 344–386.

[29] Karpelevič, F. I., The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Trans. Moscow Math. Soc. 14 (1967), 51–199.

[30] Korányi, A., On a mean value property for hyperbolic spaces, in “Representation theory and harmonic analysis,” Cincinnati 1994 (T. Ton-That et al., eds.), Contemp. Math. 191, Amer. Math. Soc., Providence, 1995, pp. 107–116.

[31] Korányi, A., M. A. Picardello, and M. H. Taibleson, Hardy spaces on non-homogeneous trees, in “Analisi armonica, spazi simmetrici e teoria della probabilità,” Cortona 1984, Sympos. Math. 29, Academic Press, London, 1987, pp. 205–254.

[32] Levin, B. Ja., “Distribution of zeros of entire functions,” Transl. Math. Monographs 5, Amer. Math. Soc., Providence, 1964.

[33] Picardello, M. A., and W. Woess, A converse to the mean value property on homogeneous trees, Trans. Amer. Math. Soc. 311 (1989), 209–225.

[34] Raugi, A., Fonctions harmoniques sur les groupes localement compacts à base dénombrable, in “Fonctions harmoniques et théorèmes limites pour les marches aléatoires sur les groupes,” Bull. Soc. Math. France Mém. 54, Soc. Math. France, Paris, 1977, pp. 5–118.

[35] Veech, W. A., A zero-one law for a class of random walks and a converse to Gauss’ mean value theorem, Ann. of Math. (2) 97 (1973), 189–216.

[36] —, A converse to the mean value theorem for harmonic functions, Amer. J. Math. 97 (1975), 1007–1027.

Dipartimento di Matematica “G. Castelnuovo” Università di Roma “La Sapienza” Piazzale A. Moro 2 00185 Roma Italy casadio@alpha.science.unitn.it Department of Mathematics University of Maryland College Park, MD 20742 USA jmc@math.umd.edu

Department of Mathematics H. H. Lehman College CUNY Bronx, NY 10468 USA adam@alpha.lehman.cuny.edu Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Roma Italy picard@mat.uniroma2.it

Received February 22, 1997 and in final form March 4, 1998