1 Introduction

A complete Riemannian manifold (X,g) of dimension n+1 is called harmonic if the volume density function in normal coordinates around a point depends only on the distance from this point. Rank one symmetric spaces are harmonic, and Lichnerowicz conjectured that a simply connected harmonic space must be flat or rank one symmetric. For compact simply connected spaces this is true by a theorem of Szabo [22]. However, certain 3-step solvmanifolds, constructed by Damek and Ricci [6], provide examples of non-compact non-symmetric homogeneous harmonic spaces. Heber [11] proved that there exist no other simply connected homogeneous harmonic spaces. Recently, Knieper [14] showed for non-compact simply connected harmonic spaces that (i) having a purely exponential volume density function, (ii) being Gromov-hyperbolic, and (iii) having an Anosov geodesic flow, are all equivalent conditions.

Non-compact harmonic spaces have no conjugate points. Moreover, they are Einstein and therefore analytic by the Kazdan–De Turck theorem. It was shown by Willmore [26] that harmonic manifolds can also be characterized as those analytic spaces for which all harmonic functions f satisfy the mean value property, namely, that the average of f over any geodesic sphere equals the value of f at its center. It is well known that in a harmonic space every function satisfying the mean value property at all points for all radii must be harmonic. This is no longer true if a function only satisfies the mean value property at all points for a single radius r 1>0. A simple example of a non-harmonic function satisfying the mean value property for the radius r 1=2π is the cosine function on the real line \(X = \mathbb {R}\). We will show in Theorem 4.3 that in arbitrary non-compact harmonic spaces, the mean value property for two generically chosen radii r 1,r 2 implies harmonicity of the function. Similarly, by Theorem 4.2, the vanishing of the integral of a function over all spheres, or over all balls, of radii r 1,r 2 implies vanishing of the function, if and only if the pair (r 1,r 2) lies in the generic subset of \(\mathbb {R}^{+}\times \mathbb {R}^{+}\) given in Proposition 4.1. In the example of the real line, this set is the set of pairs with irrational quotient.

The paper is organized as follows. In Sect. 2, we introduce basic notions and define convolutions in harmonic spaces following ideas of Szabo [22], and prove useful properties of them.

In Sect. 3, we derive fundamental results for the Abel transform and the spherical Fourier transform, in particular, that the Abel transform and its dual are topological isomorphisms (Theorem 3.8), using finite propagation speed of the wave equation and a D’Alembert type formula for the Klein–Gordon equation.

In Sect. 4, we prove the above-mentioned integral geometric results for all non-compact harmonic manifolds. The arguments there are analogous to our earlier paper [20], where we studied two radius results for Damek–Ricci spaces. This realizes the proposed research direction indicated in [2, Sect. 10]. A crucial step is the reduction of the problem to a classical result of L. Schwartz [21] on mean periodic functions. For a modern treatment of mean periodic functions in symmetric spaces, see [25].

Finally, in Sect. 5, we present some results related to the Cheeger constant (Theorem 5.1) and to the heat kernel (Theorem 5.6) of non-compact harmonic manifolds.

2 Radial Eigenfunctions and Convolutions

Henceforth, (X,g) denotes a non-compact, complete, simply connected harmonic space, θ(r) the density function of a geodesic sphere of radius r>0, H≥0 the mean curvature of all horospheres, and x 0∈X a particular reference point. Let r(x):=d(x 0,x). The closed ball of radius r>0 around x∈X is denoted by B r (x)⊂X. For the inner product, we use the notation

$$\langle f, g \rangle= \int_X f(x) g(x) dx. $$

Let \(\mathcal {E}(X)\), resp., \(\mathcal {D}(X)\) denote the vector space of smooth functions on X, resp., smooth functions with compact support, equipped with the topology of uniform convergence of all derivatives on compact sets, see [12, Chap. II Sect. 2], for instance.

Definition 2.1

For every x∈X, the spherical projector \(\pi_{x}: \mathcal {E}(X) \to \mathcal {E}(X)\) is defined by

$$(\pi_x f) (y) := \frac{1}{\operatorname {vol}(S_r(x))} \int_{S_r(x)} f \quad \text{with $r=d(x,y)$,} $$

where S r (x) denotes the geodesic sphere around x with radius r.

Let \(\mathcal {E}_{0}(X,x) := \pi_{x}(\mathcal {E}(X))\) and \(\mathcal {D}_{0}(X,x) := \pi_{x}(\mathcal {D}(X))\). Functions in these spaces are called radial functions about x. We simply write \(\pi, \mathcal {E}_{0}(X), \mathcal {D}_{0}(X)\) for \(\pi_{x_{0}}, \mathcal {E}_{0}(X,x_{0}), \mathcal {D}_{0}(X,x_{0})\). A radial function \(f\in \mathcal {E}_{0}(X)\) is of the form

$$ f(x)=\tilde{f}\bigl(r(x)\bigr)=\tilde{f} \bigl(d(x,x_0)\bigr) $$
(1)

with some even function \(\tilde{f}\in \mathcal {E}_{0}(\mathbb {R})\). We often do not distinguish between f and \(\tilde{f}\) in our notation, i.e., we simply write f(x)=f(r(x)).

We now present basic properties of the spherical projector. The first property below is obvious, and the second identity can be found, e.g., in [17, Lemma 2].

Lemma 2.2

For x∈X, the operator π x has the following properties:

(2)
(3)

The Laplacian \(\Delta= {\rm div} \circ \operatorname {grad}\), applied to a radial function \(f \in \mathcal {E}_{0}(X)\), can be written as

$$ (\Delta f) \bigl(r(x)\bigr) = f'' \bigl(r(x)\bigr) + \frac{\theta'(r(x))}{\theta(r(x))} f'\bigl(r(x)\bigr). $$
(4)

It is well known that θ′(r)/θ(r) is the mean curvature of a geodesic sphere S r (x), and that θ′(r)/θ(r) is a monotone decreasing function converging to H (see [16, Cor. 2.1]).

Concerning eigenvalues, we follow the sign convention in [4], and call \(f \in \mathcal {E}(X)\) an eigenfunction to the eigenvalue \(\mu\in \mathbb {C}\) if Δf+μf=0.

We now prove uniqueness and existence of radial eigenfunctions of the Laplacian. For positive eigenvalues this is shown in [22].

Proposition 2.3

For each \(\lambda\in \mathbb {C}\) there is a unique smooth function \(\varphi_{\lambda}\in \mathcal {E}_{0}(X)\) such that

$$ \Delta\varphi_\lambda+ \biggl( \lambda^2 + \frac{H^2}{4} \biggr) \varphi_\lambda= 0, \quad\text{\textit{and}}\quad \varphi_\lambda(x_0)=1. $$
(5)

We obviously have φ −λ =φ λ and φ iH/2=φ −iH/2=1. Also φ λ (r) is holomorphic in λ.

Proof

We fix \(\lambda\in \mathbb {C}\) and abbreviate \(L:= - ( \lambda^{2} + \frac{H^{2}}{4} )\). The eigenvalue equation (5) translates to

$$ \varphi_\lambda''+ \frac{\theta '}{\theta}\varphi_\lambda' = \frac{(\theta\varphi_\lambda')'}{\theta }=L \varphi_\lambda\quad\text{and}\quad\varphi_\lambda(0)=1. $$
(6)

Integrating twice we get that this is equivalent to

(7)

where

$$q(r,r_1)=\int_{r_1}^r \frac{\theta(r_1)}{\theta(r_2)}dr_2. $$

By [16, Prop. 2.2] the function θ≥0 increases, hence 0≤q(r,r 1)≤r−r 1, and the Volterra integral equation of the second kind (7) has a unique solution (see [13, Thm. 5]). In order to obtain a power series in L for φ λ , we use (7) iteratively, starting with the constant function 1, and obtain

$$\varphi_\lambda(r)=1+\sum_{k=1}^\infty a_k(r) L^k $$

with coefficients

$$a_k(r)=\int_{r\geq r_1\geq r_2\geq\cdots r_k\geq0} q(r,r_1)q(r_1,r_2) \cdots q(r_{k-1},r_k) dr_1\cdots dr_k. $$

Since

$$0 \le a_k(r) \le\int _{r\geq r_1\geq r_2\geq\cdots r_k\geq0} (r-r_1) (r_1-r_2) \cdots (r_{k-1}-r_k) dr_1\cdots dr_k = \frac{r^{2k}}{(2k)!}, $$

the power series above converges for all \(L\in \mathbb {C}\). □

The following lemma can be found in [22, Lemma 1.1]:

Lemma 2.4

We have

$$\pi_x \circ\Delta= \Delta\circ\pi_x. $$

An immediate consequence of Lemma 2.4 is the fact that if Δf+μf=0, then \(g:= \pi_{x} f \in \mathcal {E}(X,x)\) is also an eigenfunction of Δ with eigenvalue μ.

The displacement of a radial function \(f \in \mathcal {E}_{0}(X,x)\) at a point y∈X is denoted by \(f_{y} \in \mathcal {E}_{0}(X,y)\) and defined by

$$f_y(z) := \widetilde{f}\bigl(d(y,z)\bigr), $$

with \(\widetilde{f}(d(x,y)) = f(y)\). We have f y (z)=f z (y).

Lemma 2.5

The displacement (φ λ ) x of an eigenfunction φ λ is again an eigenfunction to the same eigenvalue and

$$\pi\bigl((\varphi_\lambda)_x\bigr) = \varphi_\lambda(x) \varphi_\lambda. $$

Proof

Because of the representation (4) of the Laplacian in polar coordinates, which is independent of the center, the displacement (φ λ ) x is also an eigenfunction to the eigenvalue μ=(λ 2+H 2/4). From Δ∘π=π∘Δ (Lemma 2.4) we conclude that π((φ λ ) x ) is a radial eigenfunction about x 0 to the eigenvalue μ and, by uniqueness, a multiple of φ λ . We have

$$\bigl(\pi\bigl((\varphi_\lambda)_x\bigr)\bigr) (x_0) = (\varphi_\lambda)_x(x_0) = (\varphi_\lambda)_{x_0}(x) = \varphi_\lambda(x). $$

 □

A smooth function \(F: X \times X \to \mathbb {C}\) is called a radial kernel function if there is a function \(\widetilde{f}: [0,\infty) \to \mathbb {C}\) such that

$$F(x,y) = \widetilde{f}\bigl(d(x,y)\bigr) \quad\text{for all $x,y \in X$.} $$

F is called of compact support if there is a radius R>0 such that F(x,y)=0 for all d(x,y)≥R.

Proposition 2.6

(See [22, Prop. 2.1])

Let \(F,G: X \times X \to \mathbb {C}\) be two radial kernel functions, one of them of compact support. Then the convolution

$$F*G(x,y) := \int_X F(x,z) G(z,y) dy $$

is, again, a radial kernel function, i.e., F∗G(x,y) depends only on the distance d(x,y).

Radial functions \(f\in \mathcal {E}_{0}(X)\) are in one-one correspondence with radial kernel functions F:C ∞(X×X) via

where \(\widetilde{f}\) was introduced in (1). This correspondence leads to a natural convolution f∗g of radial functions \(f,g \in \mathcal {E}_{0}(X)\), and even to an extension of this notion if only one of the two functions is radial:

Definition 2.7

Let \(f, g \in \mathcal {E}(X)\), one of them with compact support, and one of them radial about x 0. If f is the radial function, the convolution \(f * g \in \mathcal {E}(X)\) is defined as

$$f*g(y) := \langle f_y, g \rangle= \int_X f_y(z) g(z) dz. $$

Similarly, if g is the radial function, we define

$$f*g(y) := \langle f, g_y \rangle= \int_X f(z) g_y(z) dz. $$

Remark 1

The convolution f∗g is well defined, since if both f,g are radial, we have

$$\int_X f_y(z) g(z) dz = F*G(y,x_0) \quad \text{and}\quad \int_X f(z) g_y(z) dz = F*G(x_0,y), $$

where F and G are the radial kernel functions associated with f,g. By Proposition 2.6, we have F∗G(y,x 0)=F∗G(x 0,y). Moreover, the definition immediately implies commutativity of the convolution.

The following two lemmas are further consequences of Proposition 2.6. The proofs are straightforward, once the statements are reformulated in terms of radial kernel functions.

Lemma 2.8

Let \(f, g \in \mathcal {D}_{0}(X)\). Then \(f * g \in \mathcal {D}_{0}(X)\).

Lemma 2.9

Let \(f \in \mathcal {D}(X)\) and \(g, h \in \mathcal {D}_{0}(X)\). Then we have

$$ f*(g*h) = (f*g)*h. $$
(8)

Let \({\mathcal {D}}'(X)\) and \({\mathcal {E}}'(X)\) be the dual spaces of \(\mathcal {E}(X)\) and \(\mathcal {D}(X)\), the space of distributions and the space of distributions with compact support. For their topologies we refer, again, to [12, Chap. II Sect. 2]. Let \(\mathcal {E}_{0}'(X)\) and \(\mathcal {D}_{0}'(X)\) be the corresponding subspaces of radial distributions. The spherical projector and the convolution are continuous and \(\mathcal {D}_{0}(X)\) embeds canonically into \(\mathcal {E}_{0}'(X)\) via f↦T f , where 〈T f ,g〉:=∫ X f(x)g(x)dx. For \(T \in \mathcal {E}'(X)\) and \(f \in \mathcal {E}_{0}(X)\), T∗f can be interpreted as a function in \(\mathcal {E}(X)\), i.e.,

$$T*f(x) = \langle T, f_x \rangle. $$

Since \(\mathcal {D}_{0}(X)\) lies dense in \(\mathcal {E}_{0}'(X)\) (this follows by using a Dirac sequence \(\rho_{\epsilon}\in \mathcal {D}_{0}(X)\) and T∗ρ ϵ →T), all the above properties for functions carry over to distributions (as, for instance, the fact that the convolution of two radial distributions is radial, or the associativity of the convolution of radial distributions).

Since \(\int_{X} g_{y}(z) f(z)dz = \int_{X} g(z) (\pi_{y} f)_{x_{0}}(z)\) for \(g \in \mathcal {D}_{0}(X)\) and \(f \in \mathcal {E}(X)\), the convolution of \(T\in \mathcal {E}_{0}'(X)\) and \(f\in \mathcal {E}(X)\) is given by

$$ T*f(y) := \bigl\langle T, (\pi_yf)_{x_0} \bigr\rangle = \biggl\langle T, z \mapsto\frac{1}{\operatorname {vol}(S_{r(z)}(y))} \int _{S_{r(z)}(y)}f(x)dx \biggr\rangle. $$
(9)

As an example, consider the distribution \(T_{r}\in \mathcal {E}_{0}'(X)\), given by \(\langle T_{r},f\rangle=\int_{S_{r}}f\). If \(f\in \mathcal {E}(X)\) we obtain

$$ T_r*f(y)= \biggl\langle T_r, z \mapsto\frac{1}{\operatorname {vol}(S_{r(z)}(y))} \int_{S_{r(z)}(y)} f \biggr\rangle= \int _{S_r(y)}f. $$
(10)

The convolution of two radial distributions \(S, T \in \mathcal {E}_{0}'(X)\) lies in \(\mathcal {E}_{0}'(X)\) and can be written as follows: If \(f \in \mathcal {E}_{0}(X)\), we have

$$\langle S*T, f \rangle= \bigl\langle S, x \mapsto\langle T,f_x \rangle\bigr\rangle. $$

For general \(f \in \mathcal {E}(X)\), we have

$$\langle S*T, f \rangle= \langle S*T, \pi f \rangle\ \bigl(= \bigl\langle S, x \mapsto\bigl\langle T,(\pi f)_x \bigr\rangle\bigr\rangle\bigr). $$

Proposition 2.10

For \(T\in \mathcal {E}_{0}'(X)\) and \(f\in \mathcal {E}(X)\) we have

$$\pi(T*f)=T*(\pi f). $$

Proof

In view of (9), the claim means that

is equal to

$$\bigl[T*(\pi f)\bigr](y)=\bigl\langle T, (\pi_y\pi f)_{x_0}\bigr\rangle $$

for all y∈X. To see this, we show that for all y,z∈X we have

$$ \pi\bigl[u\mapsto(\pi_u f)_{x_0}(z) \bigr](y) = \bigl[(\pi_y\pi f)_{x_0}\bigr](z). $$
(11)

We first show (11) if f is an eigenfunction of the Laplacian. So assume

$$f\in \mathcal {E}(X)\quad \text{with }\Delta f+\mu f = 0. $$

Then Δπ u f+μπ u f=0 and \(\pi_{u} f\in \mathcal {E}(X,u)\) is radial. Choose \(\lambda\in \mathbb {C}\) such that μ=λ 2+H 2/4. By the uniqueness of the radial eigenfunctions we get

$$(\pi_u f)_{x_0}=f(u) \bigl((\varphi_\lambda)_u \bigr)_{x_0}=f(u)\varphi_\lambda. $$

Therefore,

and

$$\bigl[(\pi_y\pi f)_{x_0}\bigr](z)=\bigl[\bigl( \pi_y\bigl[f(x_0)\varphi_\lambda \bigr] \bigr)_{x_0}\bigr](z)=f(x_0)\varphi_\lambda(y) \varphi_\lambda(z). $$

In order to show (11) for arbitrary functions f, note that for fixed y,z∈X the values of both sides of (11) depend on the restriction of f to a compact subset K⊂X only. Let K′ be a compact subset of X with smooth boundary containing K in its interior. Since f is smooth, we find linear combinations of Dirichlet eigenfunctions of the Laplacian on K′ approximating f uniformly on K. Since both sides of (11) are continuous in f with respect to uniform convergence, this establishes (11) for all functions f. □

3 Abel and Spherical Fourier Transformation

We first introduce some fundamental notions. Let SX be the unit tangent bundle of (X,g). The Busemann function associated with a unit tangent vector \(v_{0} \in S_{x_{0}}X\) is defined by

$$b(x) = b_{v_0}(x) := \lim_{s \to\infty} \bigl( d \bigl(c(s),x\bigr) - s\bigr), $$

where \(c: \mathbb {R}\to X\) is the geodesic with c(0)=x 0, c′(0)=v 0. Δb=H implies that b is an analytic function. The level sets of b are smooth hypersurfaces and are called horospheres. They are denoted by

$$\mathcal {H}_s := b^{-1}(s). $$

These horospheres foliate X and we have \(x_{0} \in \mathcal {H}_{0}\). We also need the smooth unit vector field

$$N(x) = - \operatorname {grad}b(x), $$

orthogonal to the horospheres \(\mathcal {H}_{s}\) and satisfying N(x 0)=v 0. We choose an orientation of \(\mathcal {H}_{0}\) and orientations of \(\mathcal {H}_{s}\) such that the diffeomorphisms

$$\varPsi_s: \mathcal {H}_0 \to \mathcal {H}_s, \quad \varPsi_s(x) := \exp_x\bigl(-s N(x)\bigr), $$

are orientation preserving. Since HN is the mean curvature vector of the horospheres and thus the variation field of the area functional we have

Proposition 3.1

$$(\varPsi_s)^* \omega_s = e^{s H} \omega_0. $$

We combine the diffeomorphisms \(\varPsi_{s}: \mathcal {H}_{0} \to \mathcal {H}_{s}\) to construct a global diffeomorphism

$$\varPsi: \mathbb {R}\times \mathcal {H}_0 \to X, \quad\varPsi(s,x) := \varPsi_s(x). $$

We have \(D \varPsi(\frac{\partial}{\partial s}) = -N\).

We choose an orientation on \(\mathbb {R}\times \mathcal {H}_{0}\) such that every oriented base v 1,…,v n of \(\mathcal {H}_{0}\) induces an oriented base \(\frac{\partial}{\partial s}, v_{1}, \dots, v_{n}\) on \(\mathbb {R}\times \mathcal {H}_{0}\). This yields also an orientation on X by requiring that Ψ is orientation preserving. An immediate consequence of Proposition 3.1 is

Corollary 3.2

Let ω denote the volume form of the harmonic space (X,g). Then we have

$$\varPsi^* \omega= e^{s H}ds \wedge\omega_0. $$

Next, we fix a unit vector \(v_{0} \in S_{x_{0}}X\), and denote the associated Busemann function \(b_{v_{0}}\) by b, for simplicity. We first consider the following important transform:

Definition 3.3

Let \(j : \mathcal {E}_{0}(\mathbb {R}) \to \mathcal {E}(X)\) be defined as

$$(jf) (x) = e^{-\frac{H}{2} b(x)} f\bigl(b(x)\bigr). $$

The transformation \(a: \mathcal {E}_{0}(\mathbb {R}) \to \mathcal {E}_{0}(X)\) is then defined as

$$a = \pi\circ j. $$

The Abel transform \(\mathcal {A}: \mathcal {E}_{0}'(X) \to \mathcal {E}_{0}'(\mathbb {R})\) is defined as the dual of a, i.e., we have for all \(T \in \mathcal {E}_{0}'(X)\) and \(f \in \mathcal {E}_{0}(\mathbb {R})\):

$$\langle \mathcal {A}T, f \rangle_\mathbb {R}= \langle T, a f \rangle_X. $$

The functions \(\psi_{\lambda}(s) = \frac{1}{2} (e^{i\lambda s} + e^{-i\lambda s}) = \cos(\lambda s)\) are the radial eigenfunctions of the Laplacian Δf=f″ on the real line.

Lemma 3.4

We have

$$a \psi_\lambda= \varphi_\lambda. $$

Proof

We first observe that, under the diffeomorphism \(\varPsi: \mathbb {R}\times \mathcal {H}_{0} \to X\), the Laplacian has the form

$$ \Delta= \frac{\partial^2}{\partial^2 s} + H \frac{\partial}{\partial s} + A_s, $$
(12)

where A s is a differential operator with derivatives tangent to \(\mathcal {H}_{0}\). Consequently, the functions f α =e −αb are eigenfunctions of Δ with

$$\Delta f_\alpha= - \alpha(H - \alpha) f_\alpha. $$

Choosing \(\alpha= \frac{H}{2} \pm i \lambda\), we obtain

$$\Delta f_\alpha= - \biggl( \lambda^2 + \frac{H^2}{4} \biggr) f_\alpha, $$

and, by uniqueness, πf α must be a multiple of φ λ . Since f α (x 0)=1, we conclude that φ λ =πf α . Let \(\alpha_{\pm}= \frac{H}{2} \pm i \lambda\). Then one easily checks that

$$j \psi_\lambda= \frac{1}{2} \bigl(e^{-\alpha_- b} + e^{-\alpha_+ b} \bigr), $$

and, consequently,

$$a \psi_\lambda= \frac{1}{2} (\pi f_{\alpha_-} + \pi f_{\alpha_+} ) = \varphi_\lambda. $$

 □

Let \(\psi_{\lambda,k}(s) = \frac{d^{k}}{d\lambda^{k}} \psi_{\lambda}(s) = \frac{s^{k}}{2} (i^{k} e^{i\lambda s} + (-i)^{k} e^{-i\lambda s})\) and \(\varphi_{\lambda,k} = \frac{d^{k}}{d\lambda^{k}} \varphi_{\lambda}\). Lemma 3.4 implies that we also have

$$ a \psi_{\lambda,k} = \varphi_{\lambda,k}, $$
(13)

for all k≥1.

Proposition 3.5

The Abel transform of a function \(f \in \mathcal {D}_{0}(X) \subset \mathcal {E}_{0}'(X)\) is \(\mathcal {A}f\in \mathcal {D}_{0}(\mathbb {R})\subset \mathcal {E}_{0}'(\mathbb {R})\) given by

$$ (\mathcal {A}f) (s) = e^{-\frac{H}{2} s} \int_{\mathcal {H}_s} f(z) \omega_s(z) = e^{\frac{H}{2} s} \int_{\mathcal {H}_0} f\bigl( \varPsi_s(z)\bigr) \omega_0(z). $$
(14)

Proof

Let \(f \in \mathcal {D}_{0}(X)\) and

$$g(s) = e^{-\frac{H}{2} s} \int_{\mathcal {H}_s} f(z) \omega_s(z). $$

Since f has compact support, there is T>0 such that \(\mathcal {H}_{s} \cap \operatorname {supp}f = \emptyset\) for all |s|≥T, i.e., g also has compact support. Moreover, by Proposition 3.1, we obtain

Next, we show 〈g,h〉=〈f,ah〉 for all \(h \in \mathcal {E}_{0}(\mathbb {R})\):

Finally, we show that g is an even function: Using 〈g,h〉=〈f,ah〉 and Lemma 3.4, we derive

$$\int_{-\infty}^\infty g(s) e^{\pm i \lambda s} ds = \langle f, \varphi_\lambda\rangle, $$

which implies that

$$\int_{-\infty}^\infty e^{i \lambda s} \bigl(g(s) - g(-s)\bigr) ds = 0, $$

for all \(\lambda\in \mathbb {C}\). This yields g(s)=g(−s), i.e., g is an even function. This finishes the proof of the proposition. □

Lemma 3.6

For \(f\in \mathcal {E}'_{0}(X)\), we have

$$\mathcal {A}(\Delta f) = \biggl(\frac{d^2}{ds^2}-\frac{H^2}{4} \biggr) \mathcal {A}f. $$

Proof

Let \(u\in \mathcal {E}_{0}(\mathbb {R})\). Since \(ju\in \mathcal {E}(X)\) is constant on the horospheres, from (12) we compute

$$[\Delta j u](x) = \biggl(\frac{\partial^2}{\partial s^2}+H\frac{\partial}{\partial s} \biggr) e^{-s H/2} u(s)\biggl \vert _{s=b(x)} = \biggl[j \biggl( \frac{d^2}{d s^2}-\frac{H^2}{4} \biggr)u \biggr](x); $$

hence,

 □

The next result will be used in the proof of Theorem 3.8 below; namely, to establish the local injectivity of \(a: \mathcal {E}_{0}(\mathbb {R}) \to \mathcal {E}_{0}(X)\).

Lemma 3.7

Let \(g\in \mathcal {D}(\mathbb {R})\). Then the Klein–Gordon equation

(15)
(16)

has a solution of the form

$$ v(t,s)=\frac{g(s-t)+g(s+t)}{2}+\int _{s-t}^{s+t} W\bigl(t,s-s'\bigr)g \bigl(s'\bigr)ds', $$
(17)

with \(W\in \mathcal {E}(\mathbb {R}^{2})\). This solution is unique in the sense that if \(\tilde{v}\) is another solution of (15) and (16) and so that for all t the function \(\tilde{v}_{t}\colon s\mapsto\tilde{v}(t,s)\) has compact support, then \(\tilde{v}=v\).

Proof

The function W is explicitly given by

$$ W(t,s) = t\sum_{k=0}^\infty \biggl(\frac{-H^2}{16} \biggr)^{k+1}\frac {(t^2-s^2)^k}{k!(k+1)!}, $$
(18)

but we will only need that W is smooth. A straightforward computation shows that (17) actually solves (15) and (16). The function W is even in the second argument s, and solves the equations

$$W(t,t)=-\frac{H^2}{16}t,\qquad W_{tt}=W_{ss}- \frac{H^2}{4}W. $$

Uniqueness of the solution follows from conservation of energy as, for instance, in [23, p. 145]. To see this, assume that \(\tilde{v}\) is another solution of (15) and (16). Then the difference \(\omega=v-\tilde{v}\) solves (15) and (16) with g replaced by 0. We now look at the energy

$$E_\omega(t):=\int_{-\infty}^\infty \omega_s(t,s)^2+\omega_t(t,s)^2+ \frac {H^2}{4}\omega(t,s)^2ds, $$

and compute, integrating by parts,

because ω satisfies (15). Since E ω (0)=0 we have E ω (t)=0 for all t, which forces ω=0. □

Theorem 3.8

The maps \(a: \mathcal {E}_{0}(\mathbb {R}) \to \mathcal {E}_{0}(X)\) and \(\mathcal {A}= a'\) are topological isomorphisms.

Proof

We first show local injectivity of a. For all R≥0 the map a induces a well-defined map

$$a^R\colon \mathcal {E}_0\bigl([-R,R]\bigr)\to \mathcal {E}_0 \bigl(B_R(x_0)\bigr), $$

i.e., for x∈X the value of au(x) depends only on the restriction of u to [−r(x),r(x)]. Local injectivity now is the fact that for all R≥0, the maps a R are injective. Thus for \(u\in \mathcal {E}_{0}(\mathbb {R})\) we have

$$a u|_{B_R(x_0)} = 0 \quad\Longrightarrow\quad u|_{[-R,R]} = 0. $$

The proof is based on the fact that \(\mathcal {A}\colon \mathcal {E}_{0}'(X)\to \mathcal {E}_{0}'(\mathbb {R})\) transforms the fundamental solution of the radial wave equation to the fundamental solution of the Klein–Gordon equation. Since we need \(\mathcal {A}\) on compactly supported distributions, finite propagation speed of the solution of the wave equation is essential here.

For all ϵ>0 we choose a function \(q_{\epsilon}\in \mathcal {E}_{0}(X)\) so that

$$q_\epsilon(x)=0\quad \text{if }r(x)>\epsilon\quad \text{and}\quad \int _Xq_\epsilon= 1. $$

Let \(w\in \mathcal {E}_{0}(\mathbb {R}\times X)\), w=w(t,r(x))=w t (r(x)), be the solution of the wave equation starting with q ϵ , i.e.,

By the finite propagation speed of the wave equation, the support of w t is compact (in fact, contained in B ϵ+t (0)).

Let \(v\in \mathcal {E}(\mathbb {R}^{2})\) be so that \(v_{t}:=\mathcal {A}w_{t}\), i.e.,

$$v(t,s)=e^{-sH/2}\int_{\mathcal {H}_s}w\bigl(t,r(x)\bigr)d \omega_s(x). $$

Then by Lemma 3.6

$$\frac{\partial^2}{\partial t^2}v_t=\frac{\partial^2}{\partial t^2}\mathcal {A}w_t=\mathcal {A}\frac{\partial^2}{\partial t^2}w_t=\mathcal {A}\Delta w_t= \biggl( \frac {\partial^2}{\partial s^2}-\frac{H^2}{4} \biggr)v_t. $$

It follows that v solves the Klein–Gordon equation (15) with initial conditions (in place of (16))

$$v_0=g_\epsilon:=\mathcal {A}q_\epsilon\text{ and } \frac{\partial}{\partial t}v(0,s)=0. $$

If |s|>ϵ then v(0,s)=g ϵ (s)=0. Also

$$\int_{-\infty}^\infty e^{sH/2} g_\epsilon(s)ds=\int_{-\infty}^\infty\int _{\mathcal {H}_s}q_\epsilon\bigl(r(x)\bigr)d\omega_s ds=\int_Xq_\epsilon=1. $$

Now, to prove local injectivity of a, let R≥0 and \(u\in \mathcal {E}_{0}(\mathbb {R})\) with \(a u|_{B_{R}(x_{0})} = 0\). For all t∈[−(R−ϵ),R−ϵ] we then have from (17)

Since W,u and g ϵ are smooth, we can take the limit ϵ→0 here to get the identity

$$\frac{u(t)+u(-t)}{2}=-\int_{-t}^{t} W(t,s)u(s)ds. $$

Since u and W are even (in s), we can write this as a fixed-point equation,

$$u(t)=-2\int_0^t W(t,s)u(s)ds. $$

Since πf(x 0)=f(x 0) for all \(f\in \mathcal {E}(X)\) we have u(0)=0. Let

$$M:=\max_{t\in[0,R],s\in[0,t]}\bigl \vert W(t,s)\bigr \vert . $$

Hence, if u|[0,R]≠0 there is some T∈[0,R] with the following properties: (i) u(T)≠0 and (ii) \(T<\frac{1}{4M}\) or \(u|_{[0,T-\frac{1}{4M}]}=0\). Now for all t∈[0,T] we estimate

contradicting our assumption about T and u(T)≠0.

Now we prove surjectivity of a: Let \(f\in \mathcal {E}_{0}(X)\) and R>0 be fixed. Choose a function \(\phi\in \mathcal {D}_{0}(X)\) with ϕ=1 on B R (x 0), 0≤ϕ≤1 on X, and ϕ=0 on X∖B R+1(x 0). We will first show that \(\phi f|_{B_{R}(x_{0})}\) is in \(a(\mathcal {E}_{0}(\mathbb {R}))|_{B_{R}(x_{0})}\).

Let φ k be an orthonormal basis of Dirichlet eigenfunctions of the Laplacian on B R+1(x 0)⊂X with corresponding eigenvalues 0≤μ k ↗∞. We have \(\pi\varphi_{k} = \varphi_{k}(x_{0}) \varphi_{\lambda_{k}}\) with \(\lambda_{k} \in \mathbb {C}\) such that \(\mu_{k} = \lambda _{k}^{2} + \frac{H^{2}}{4}\). Let

$$\phi f = \sum_{k=0}^\infty a_k \varphi_k $$

be the Fourier expansion of ϕf. Therefore,

$$\phi f = \pi(\phi f) = \sum_{k=0}^\infty a_k \varphi_k(x_0) \varphi_{\lambda_k}. $$

Our first goal is to show that the series

$$g_R(s)=\sum_{k=0}^\infty a_k\varphi_k(x_0)\cos(\lambda_k s) $$

converges uniformly with all its derivatives to a smooth function \(g_{R} \in \mathcal {E}_{0}([-R,R])\). Then we have \(a^{R}(g_{R}) = \phi f \vert_{B_{R}(x_{0})}\), by the continuity of a. We prove this by showing that

$$ \sum_{k=0}^\infty\bigl|a_k\bigr| \bigl|\varphi_k(x_0)\bigr| |\lambda_k|^m < \infty \quad\text{for all $m \in \mathbb {N}$}. $$
(19)

By the Sobolev imbedding theorem, there is a constant C 0 such that for all u∈C(B R+1(x 0)) we have

$$\|u\|_\infty\leq C_0 \bigl(\|u\|_2 + \bigl\| \Delta^{n+1} u\bigr\|_2 \bigr), $$

where ∥⋅∥∞ denotes the supremum norm and ∥⋅∥2 denotes the L 2-norm, and n+1 is the dimension of X. This implies that

$$\bigl\|\varphi_k(x_0)\bigr| \le\|\varphi_k \|_\infty\le C_0\bigl(1+ \mu_k^{n+1} \bigr). $$

By Weyl’s law, the eigenvalues μ k grow with an exponent 2/(n+1), which implies that there is a \(k_{0} \in \mathbb {N}\) and a C 1>0 such that, for all k≥k 0:

$$ \bigl|\varphi_k(x_0)\bigr| \le\| \varphi_k\|_\infty\le C_1 k^2. $$
(20)

The Fourier expansion of Δν(ϕf) is given by

$$\Delta^{\nu} (\phi f) = \sum_{k=0}^\infty a_k \mu_k^{\nu} \varphi_k, $$

and, since Δν(ϕf)∈L 2(B R+1(x 0)), we have

$$\sum_{k=1}^\infty|a_k|^2 |\mu_k|^{2\nu} < \infty, $$

which implies that

$$ \sum_{k=1}^\infty|a_k|^2 |\lambda_k|^{4 \nu} < \infty, $$
(21)

for every \(\nu\in \mathbb {N}\). We have

The required finiteness (19) now follows from (21) and Weyl’s law, for the choice l=n+1.

Hence g R defines a smooth function with a R g R =f on B R (x 0). Now for a given \(f\in \mathcal {E}_{0}(X)\) and each \(N\in \mathbb {N}\), construct a function \(g_{N}\in \mathcal {E}_{0}([-N,N])\) as above. We will have \(a^{N}(g_{N}) = f|_{B_{N}(x_{0})}\). By local injectivity of a, g N+1|[−N,N]=g N , and the functions g N patch together to define a function \(g\in \mathcal {E}_{0}(\mathbb {R})\) with a(g)=f.

This shows that a is a bijective linear continuous map. By the open mapping theorem [24, Thm. 17.1], a is a topological isomorphism. Using the corollary of Proposition 19.5 in [24], we conclude that its dual \(\mathcal {A}: \mathcal {E}_{0}'(X) \to \mathcal {E}_{0}'(\mathbb {R})\) is also a topological isomorphism. □

Definition 3.9

The spherical Fourier transformation \(\mathcal {F}T\) of a radial distribution \(T \in \mathcal {E}_{0}'(X)\) is the function \(\mathcal {F}T:\mathbb {C}\to \mathbb {C}\) with

$$\mathcal {F}T(\lambda)=\langle T, \varphi_\lambda\rangle\quad \text{for all }\lambda \in \mathbb {C}. $$

Next, we will see that there is a close relationship between the Abel transform \(\mathcal {A}\) and the spherical Fourier transform \(\mathcal {F}\). By the classical Paley–Wiener theorem for distributions (see, e.g., [8, p. 211] and [9, Thm. 5.19]), the Euclidean Fourier transform \(\mathcal {E}_{0}'(\mathbb {R}) \ni S \mapsto\hat{S}\), with \(\hat{S}(\lambda) := \langle S, \psi_{\lambda}\rangle\) is a topological isomorphism \(\mathcal {E}_{0}'(\mathbb {R}) \to \mathbf {E}_{0}'\), where \(\mathbf {E}_{0}'\) is the space of all even entire functions \(f: \mathbb {C}\to \mathbb {C}\) of exponential type which are polynomially bounded on \(\mathbb {R}\), endowed with a suitable topology.

Proposition 3.10

We have

$$\widehat{\mathcal {A}T} = \mathcal {F}T. $$

Proof

We have for \(\lambda\in \mathbb {C}\),

$$\widehat{\mathcal {A}T}(\lambda) = \langle \mathcal {A}T, \psi_\lambda\rangle = \langle T, a \psi_\lambda\rangle= \langle T, \varphi_\lambda \rangle= \mathcal {F}T( \lambda). $$

 □

Proposition 3.11

For \(S, T \in \mathcal {E}_{0}'(X)\) we have

$$\mathcal {A}(S * T) = \mathcal {A}S * \mathcal {A}T. $$

Proof

Note that \(\mathcal {A}S, \mathcal {A}T, \mathcal {A}(S*T) \in \mathcal {E}_{0}'(\mathbb {R})\). It was shown in [24, formula (30.1)] that we have, for these distributions on the real line,

$$\widehat{\mathcal {A}S} \cdot\widehat{\mathcal {A}T} = \widehat{\mathcal {A}S * \mathcal {A}T}. $$

We now show that \(\widehat{\mathcal {A}(S*T)} = \widehat{\mathcal {A}S} \cdot \widehat{\mathcal {A}T}\):

Putting both results together, we conclude that

$$\widehat{\mathcal {A}(S*T)} = \widehat{\mathcal {A}S * \mathcal {A}T}. $$

Since the Euclidean Fourier transform \(\mathcal {E}_{0}'(\mathbb {R}) \to \mathbf {E}_{0}'\), \(T \mapsto\hat{T}\), is a topological isomorphism, we finally obtain

$$\mathcal {A}(S*T) = \mathcal {A}S * \mathcal {A}T, $$

finishing the proof. □

An immediate consequence of Paley–Wiener for radial distributions in Euclidean space, Theorem 3.8, and Propositions 3.10 and 3.11 is the following:

Theorem 3.12

(Paley–Wiener for radial distributions)

The spherical Fourier transform

$$\mathcal {F}T(\lambda) = \langle T, \varphi_\lambda\rangle $$

defines a topological isomorphism

$$\mathcal {F}: \mathcal {E}_0'(X) \to \mathbf {E}_0'. $$

Furthermore, for radial distributions \(S,T \in \mathcal {E}_{0}'(X)\), we have

$$\mathcal {F}(S*T) = \mathcal {F}S \cdot \mathcal {F}T. $$

Proposition 3.13

The following diagram commutes:

(22)

Proof

Since \(a: \mathcal {E}_{0}(\mathbb {R}) \to \mathcal {E}_{0}(X)\) and \(\mathcal {A}: \mathcal {E}_{0}'(X) \to \mathcal {E}_{0}'(\mathbb {R})\) are topological isomorphisms, in view of Proposition 3.11, it only remains to show that \(a( \mathcal {A}T *_{\mathbb {R}}f) = T *_{X} a f\): For \(g,h \in \mathcal {D}_{0}(X)\) we obtain

Since \(\mathcal {D}_{0}(X)\) is dense in \(\mathcal {E}_{0}'(X)\) and \(a, \mathcal {A}\) are continuous, we conclude the required identity. □

4 Spectral Analysis/Synthesis and Two Radius Theorems

In this section, we discuss the proofs of the integral geometric results mentioned in the Introduction. Since the proofs are very similar to the ones given in [20] for Damek–Ricci spaces, we give the ideas and outlines of the proofs, and refer to that paper for more details.

The following proposition is a consequence of the holomorphicity of the map λ↦φ λ (r), and guarantees that the integral geometric results hold for two generic radii, as claimed in the Introduction.

Proposition 4.1

For each r 1>0 there is an at most countable set of r 2>0 such that there exists \(\lambda\in \mathbb {C}\) with φ λ (r 1)=φ λ (r 2)=0. In particular, the set

$$\bigl\{(r_1,r_2)\in \mathbb {R}^+\times \mathbb {R}^+\mid\forall\lambda \in \mathbb {C}: \phi_\lambda (r_1)\neq0 \ \textit{or}\ \phi_\lambda(r_2)\neq0\bigr\} $$

is generic.

Proof

Let r 1>0. By Proposition 2.3, the set \(S_{r_{1}}=\{ \lambda\in \mathbb {C}\mid\varphi_{\lambda}(r_{1})=0\}\) is at most countable. The zero set \(\varphi_{\lambda}^{-1}(0)\) of φ λ , is also at most countable because φ λ satisfies the differential equation (6) \(\varphi_{\lambda}''=- (\lambda^{2}+\frac {H^{2}}{4} )\varphi_{\lambda}-\frac{\theta'}{\theta} \varphi_{\lambda}\), the solution of which is determined by the values of φ λ (r 0) and \(\varphi_{\lambda}'(r_{0})\) for any r 0>0. Thus φ λ cannot have a limit point of zeros. It follows that the set

$$\bigl\{r_2>0\mid\exists\lambda\in \mathbb {C}: \varphi_\lambda(r_1)= \varphi_\lambda (r_2)=0\bigr\}=\bigcup _{\lambda\in S_{r_1}}\varphi_\lambda^{-1}(0) $$

is an at most countable union of at most countable sets, hence itself at most countable. □

The same reasoning applies to the function φ λ −1 (we must exclude here \(\lambda= \pm i \frac{H}{2}\), since then φ λ ≡1), and the function Φ λ given by

$$ \varPhi_\lambda(r)=\int_0^r \theta(\rho)\varphi_\lambda(\rho)d\rho $$
(23)

where θ denotes the volume density function.

Analogously to [20], we have spectral analysis and spectral synthesis in \(\mathcal {E}_{0}(X)\). Let us briefly explain this.

A variety \(V\subset \mathcal {E}_{0}(X)\) is a closed subspace satisfying \(\mathcal {E}_{0}'(X) * V \subset V\), which is proper (\(V \neq \mathcal {E}_{0}(X)\)) and contains a non-zero function.

It follows from Propositions 3.13 and 3.11, together with the isomorphism Theorem 3.8, that the transformation a maps varieties of \(\mathcal {E}_{0}(\mathbb {R})\) to varieties of \(\mathcal {E}_{0}(X)\), and that every variety in \(\mathcal {E}_{0}(X)\) is of the form a(W) with a variety \(W\subset \mathcal {E}_{0}(\mathbb {R})\). By Schwartz’s theorem on varieties in \(\mathcal {E}_{0}(\mathbb {R})\) (see Thm. 2.4 in [20]), we have

$$W=\overline{\mathrm{span}\{\psi_{\lambda,k}\in W\}}, $$

that is, each variety is the closure of the span of its spectrum, where the spectrum of a variety is the set of all those ψ λ,k contained in V. Because of (13) this carries over to X, i.e., we have for any variety \(V\subset \mathcal {E}_{0}(X)\) that

$$V=\overline{\mathrm{span}\{\phi_{\lambda,k}\in V\}}. $$

This property is called spectral synthesis. For varieties \(V \subset \mathcal {E}_{0}(X)\), it was shown in [20, Lemma 4.2] that φ λ,k ∈V implies φ λ,l ∈V, for all 0≤l≤k. There X was a Damek–Ricci space, but the arguments carry over verbatim for general harmonic manifolds. Therefore, spectral synthesis implies that every variety \(V \subset \mathcal {E}_{0}(X)\) contains a radial eigenfunction φ λ . This latter property is called spectral analysis.

As an immediate application, we can prove the analogues of the Two-Radius theorems in [20] for general non-compact harmonic spaces.

Theorem 4.2

Let (X,g) be a simply connected, non-compact harmonic manifold. Then we have the following facts.

  1. (1)

    Let r 1,r 2>0 be such that the equations

    $$\varphi_\lambda(r_j)=0,\quad j=1,2, $$

    have no common solution \(\lambda\in \mathbb {C}\). Suppose f∈C(X) and

    $$\int_{S_r(x)} f = 0 $$

    for r=r 1,r 2 and all x∈X. Then f=0.

  2. (2)

    For \(\lambda\in \mathbb {C}\), let Φ λ be given by (23). Let r 1,r 2>0 be such that the equations

    $$\varPhi_\lambda(r_j)=0,\quad j=1,2, $$

    have no common solution \(\lambda\in \mathbb {C}\). Suppose f∈C(X) and

    $$\int_{B_r(x)} f = 0 $$

    for r=r 1,r 2 and all x∈X. Then f=0.

Proof

Let us start with the proof of the first assertion. Let g∈C(X) be a non-zero function, satisfying \(\int_{S_{r_{i}}(x)} g = 0\) for all x∈X and i=1,2. We choose our reference point x 0∈X such that g(x 0)≠0. Consider the distributions \(T_{r}\in \mathcal {E}'_{0}(X)\), given by \(T_{r}f=\int_{S_{r}(x_{0})}f\). First recall (10), namely, \(\int_{S_{r}(x)}f=(T_{r}*f)(x)\). This implies that we have \(T_{r_{1}}*g = T_{r_{2}}*g = 0\). Using the extension of Proposition 2.10 to continuous functions f∈C(X), we conclude that g 0:=πg also satisfies \(T_{r_{1}}*g_{0} = T_{r_{2}}*g_{0} = 0\). Without loss of generality, we can assume that g 0 is a smooth radial function, since every continuous radial function g 0 can be approximated, uniformly on compact sets, by functions \(g_{0}^{\epsilon}:= g_{0} * \rho_{\epsilon}\in \mathcal {E}_{0}(X)\) (via a Dirac sequence \(\rho_{\epsilon}\in \mathcal {D}_{0}(X)\)), such that we still have \(T_{r_{1}}*g_{0}^{\epsilon}= T_{r_{2}}*g_{0}^{\epsilon}= 0\). Therefore, we can now assume that \(g_{0} \in \mathcal {E}_{0}(X)\). Then

contains g 0 and is a variety in \(\mathcal {E}_{0}(X)\), since for all \(T \in \mathcal {E}_{0}'(X)\) and all f∈V:

$$T_{r_i}*(T*f) = T*(T_{r_i}*f) = 0, \quad\text{with}\ i=1,2, $$

by commutativity and associativity of the convolution for radial distributions. By spectral analysis, V must contain a φ λ . But \(T_{r}(\varphi_{\lambda})=\operatorname {vol}(S_{r}(x_{0}))\varphi _{\lambda}(r)\), hence we have φ λ (r 1)=0=φ λ (r 2).

For the second assertion we work with the distributions \(T_{r}\in \mathcal {E}'_{0}(X)\) given by \(T_{r}f=\int_{B_{r}(x_{0})}f\). As before, \(T_{r}*f(x)=\int_{B_{r}(x)}f\) for all x∈X. Now, the proof proceeds as above, with the variety

Again, we conclude the existence of a φ λ satisfying

$$0 = \int_{B_{r_i}(x)}\varphi_\lambda(x)dx = \omega_n \int_0^{r_i} \varphi_\lambda(\rho)\theta(\rho)d\rho= \omega_n \varPhi_\lambda(r_i), $$

where \({\rm dim}(X) = n+1\) and ω n is the volume of the standard unit sphere of dimension n. As before, this contradicts the choice of the r i . □

Also, harmonicity of a function follows from the mean value property for two suitably chosen radii:

Theorem 4.3

Let r 1,r 2>0 be such that the equations

$$\varphi_\lambda(r_j)=1,\quad j=1,2, $$

have no common solution \(\lambda\in \mathbb {C}\backslash\{\pm iH/2\}\).

Then f∈C ∞(X) is harmonic if and only if

$$\frac{1}{{\rm vol}(S_r(x))} \int_{S_r(x)} f = f(x) $$

for r=r 1,r 2 and all x∈X.

Proof

We now use the distributions \(T_{r} f = \frac{1}{\operatorname {vol}(S_{r}(x_{0}))} ( \int_{S_{r}(x_{0})} f ) - f(x_{0})\) and assume, as above, the existence of a function \(g \in \mathcal {E}_{0}(X)\) with \(T_{r_{1}} * g = T_{r_{2}} * g = 0\) and Δg≠0 (i.e., g not harmonic). As in the proof of Theorem 1.3 of [20], we consider the variety \(V_{0}^{g} = \overline{\{ T * g \mid T \in \mathcal {E}_{0}'(X) \}} \subset \mathcal {E}_{0}(X)\), and show that the only non-zero functions \(\varphi_{\lambda,k} \in V_{0}^{g}\) are φ ±iH/2=1. Therefore, \(V_{0}^{g}\) consists only of constant functions, contradicting that \(g \in V_{0}^{g}\) and Δg≠0. □

5 Cheeger Constant and Heat Kernel

The Cheeger constant h(X) of a non-compact, complete n-dimensional Riemannian manifold (X,g) is defined as

$$ h(X) := \inf_{K \subset X} \frac{\operatorname {area}(\partial K)}{\operatorname {vol}(K)}, $$
(24)

where K ranges over all connected, open submanifolds of X with compact closure and smooth boundary. The volume growth exponent of X is defined by

$$ \mu(X) := \limsup_{r \to\infty} \frac{\log \operatorname {vol}(B_r(x))}{r}. $$
(25)

One easily checks that μ(X) does not depend on the choice x∈X. The following result states that several fundamental constants of non-compact harmonic spaces agree.

Theorem 5.1

Let (X,g) be a non-compact, simply connected harmonic space and H≥0 be the mean curvature of its horospheres. Then we have the equalities

$$h(X) = H = \mu(X) = \lim_{r \to\infty} \frac{\log \operatorname {vol}(B_r(x))}{r}. $$

Proof

Our first goal is to prove h(X)≥H. The proof is very similar to the proof of Theorem 3 in [19]. We refer the reader to this reference for more details. Let \(\varPsi: \mathbb {R}\times \mathcal {H}_{0} \to X\) be the diffeomorphism introduced in Sect. 3. We work in the space \(X' = \mathbb {R}\times \mathcal {H}_{0}\) with the induced Riemannian metric g′=Ψ ∗ g. We know from Corollary 3.2 that the volume element on X′ is given by e sH dt∧ω 0.

Without loss of generality, we can assume H>0, for otherwise there is nothing to prove. Let \(P: X' \to \mathcal {H}_{0}\) be the canonical projection and K⊂X′ be an admissible set of (24). Let U be the projection of K without the critical points of P| ∂K . By Sard’s theorem, U has full measure in P(K). For x∈U, let f ±(x) be the maximum, resp., minimum of the set \(\{ t \in \mathbb {R}\mid(t,x) \in K \}\). Let \(\widetilde{K} := \{ (x,t) \mid x \in U, f^{-}(x) \le t \le f^{+}(x) \}\). Then

$$\operatorname {vol}(K) \le \operatorname {vol}(\widetilde{K}) = \frac{1}{H} \int_U \bigl( e^{f^+(x) H} - e^{f^-(x) H} \bigr) \omega_0(x). $$

Now we introduce the sets ∂K ±:={(u,f ±(u))∣u∈U}. Obviously, we have \(\operatorname {area}(\partial K) \ge \operatorname {area}(\partial K^{+}) + \operatorname {area}(\partial K^{-})\) and, analogously as in [19], we obtain the estimate

$$\operatorname {area}\bigl(\partial K^{\pm}\bigr) \ge\int_U e^{f^\pm(x) H} \omega_0(x). $$

This yields the desired estimate

$$\frac{\operatorname {area}(\partial K)}{\operatorname {vol}(K)} \ge\frac{\operatorname {area}(\partial K^+) + \operatorname {area}(\partial K^-)}{\operatorname {vol}(K)} \ge H. $$

Let \(f(r) = \log \operatorname {vol}(B_{r}(x))\). Then, for all r>0,

$$f'(r) = \frac{\operatorname {area}(S_r(x))}{\operatorname {vol}(B_r(x))} \ge h(X). $$

It was shown in [16] that \(A(r) := \operatorname {area}(S_{r}(x))\) is strictly increasing in r and that \(\frac{A'}{A}\) is monotone decreasing with limit H≥0. Applying l’Hôpital’s rule twice, we conclude that

$$\lim_{r \to\infty} \frac{f(r)}{r} = \lim_{r \to\infty} f'(r) = \lim_{r \to\infty} \frac{A(r)}{\operatorname {vol}(B_r(x))} = \lim _{r \to \infty} \frac{A'(r)}{A(r)} = H. $$

Thus we have

$$h(X) \le\mu(X) = H = \lim\frac{\log \operatorname {vol}(B_r(x))}{r}. $$

Both estimates together prove the theorem. □

Remark 2

C. Connell proved in [5] that the Cheeger constant and the exponential volume growth of simply connected strictly negatively curved homogeneous spaces agree. This fails without the curvature condition: horospheres \(\mathcal {H}\) with barycentric normal directions in higher rank symmetric spaces of non-compact type are unimodular solvable groups with \(h(\mathcal {H}) = 0\) and \(\mu(\mathcal {H}) > 0\) (see [18]). The simplest example of this type is \({\rm Solv}(3)\), the diagonal horosphere in the product of two hyperbolic planes. Note that our Theorem 5.1 does not contain any curvature condition.

Applying Cheeger’s inequality and Brooks’s result \(\lambda_{0}^{\rm ess} \le\mu(X)^{2}/4\) (see [3]), we obtain

Corollary 5.2

Let (X,g) be a non-compact, simply connected harmonic space and H≥0 be the mean curvature of its horospheres. Then the bottom of the spectrum and of the essential spectrum agree, and

$$\lambda_0(X) = \lambda_0^{\rm ess}(X) = \frac{H^2}{4}. $$

Applying [14, Prop. 2.4], we obtain

Corollary 5.3

Let (X,g) be a non-compact, simply connected harmonic space of dimension n. If X has vanishing Cheeger constant, then X is isometric to the flat Euclidean space \(\mathbb {R}^{n}\).

Finally, we consider the Abel transform of the heat kernel on non-compact harmonic manifolds. We first state a useful lemma.

Lemma 5.4

Let \(\mathcal {H}\subset X\) be a horosphere and \(x\in \mathcal {H}\). Then there is C>0 so that

$$\operatorname {vol}_\mathcal {H}\bigl(B_r(x)\cap \mathcal {H}\bigr)\leq C e^{Cr} $$

for all r≥0.

Proof

Without loss of generality, we can assume that x=x 0 and \(\mathcal {H}_{0} = \mathcal {H}\). We use the diffeomorphisms \(\varPsi_{s}: \mathcal {H}\to \mathcal {H}_{s}\), introduced earlier.

Let R>0 be fixed. Let \(A_{r}=B_{r}(x_{0})\cap \mathcal {H}\) and

$$G_{R,r}=\bigcup_{s\in[-R,R]} \varPsi_s(A_r). $$

We compute

By the triangle inequality, G R,r ⊂B r+R (x 0), and, since

$$\operatorname {vol}\bigl(B_{r+R}(x_0)\bigr)\leq C'e^{C'(r+R)} $$

with some constant C′>0, by Bishop’s volume comparison theorem, we have

$$\operatorname {vol}_\mathcal {H}(A_r)=\frac{H\operatorname {vol}(G_{R,r})}{2\sinh(RH)}\leq\frac {HC'e^{C'(r+R)}}{2\sinh(RH)}= \frac{HC'e^{C'R}}{2\sinh(RH)}e^{C'r}. $$

 □

It is a well-known fact that a general complete Riemannian manifold (X,g) with Ricci curvature bounded from below has a unique heat kernel \(p_{t}^{X}(x,y)\) (see, e.g., [4, Thm. VIII.3]). In the case that (X,g) is harmonic, the heat kernel is a radial kernel function (see, e.g., [22, Thm. 1.1]), and is therefore uniquely determined by the function \(k_{t}^{X}(x) := p_{t}^{X}(x_{0},x)\), where x 0∈X is a fixed reference point. Our main result states that the Abel transform of the heat kernel on a non-compact harmonic space agrees, up to the factor \(e^{-H^{2}t/4}\), with the Euclidean heat kernel \(k_{t}^{\mathbb {R}}(s) = p_{t}^{\mathbb {R}}(0,s) = \frac{1}{\sqrt{4 \pi t}} e^{-s^{2}/(4t)}\). Since the heat kernel of a non-compact harmonic manifold does not have compact support, one has to guarantee that its Abel transform (centered at x 0), evaluated via the integral (14) over the horospheres \(\mathcal {H}_{s} = \varPsi_{s}(\mathcal {H}_{0})\), is well defined. This follows from the following result.

Lemma 5.5

Let t>0 be fixed, \(x_{0} \in \mathcal {H}_{0}\), and \(\varPsi_{s}: \mathcal {H}_{0} \to \mathcal {H}_{s}\) be the diffeomorphisms introduced earlier. Let \(x_{s} = \varPsi_{s}(x_{0}) \in \mathcal {H}_{s}\). For all ϵ>0, there exists an r 0>0 such that we have for all \(s \in \mathbb {R}\):

$$ 0 \le\int_{\mathcal {H}_s \backslash B_{|s|+r_0}(x_s)} k_t^X(x) d\omega_s(x) \le \epsilon. $$
(26)

Proof

Since the Ricci curvature of the non-compact harmonic manifold (X,g) is bounded below, there exist constants C t ,α t >0 such that

$$ 0 \le k_t^X(x) \le C_t e^{-\alpha_t r(x)^2} \qquad \text{for all $x \in X$}, $$
(27)

by a classical result of Li and Yau [15]. Using Lemma 5.4, we derive for arbitrary r=r 0+|s|>0:

Since α t is positive, the sum over j converges. By choosing r 0 sufficiently large, we can make the rightmost factor and thus the whole expression as small as we wish. □

Theorem 5.6

Let (X,g) be a non-compact, simply connected harmonic space and H≥0 be the mean curvature of its horospheres. Then the Abel transform \(\mathcal {A}k_{t}^{X}\) of the heat kernel \(k_{t}^{X}(x) = p_{t}^{X}(x_{0},x)\) is

$$\bigl(\mathcal {A}k_t^X\bigr) (s) = e^{-H^2t/4} \frac{1}{\sqrt{4\pi t}} e^{-s^2/4t}. $$

Proof

In the case H=0, (X,g) is the Euclidean space and there is nothing to prove. So we can assume that H>0.

Since \(\mathcal {A}k_{t}^{X}: \mathbb {R}\to \mathbb {R}\) is an even function, we have

by the heat conservation property \(\int_{X} k_{t}^{X}(x) dx = 1\) for all t>0 (see, e.g., [4, Thm. 8.5]). This shows that \(\mathcal {A}k_{t}^{X} \in L^{1}(\mathbb {R})\). Next, we show that \(\mathcal {A}k_{t}^{X}\) is continuous. Let \(s_{0} \in \mathbb {R}\) and ϵ>0 be given. We conclude from Lemma 5.5 that there is an r 0>0 such that

$$e^{-\frac{H}{2}s} \int_{\mathcal {H}_s \backslash B_{2s+r_0}(x_s)} k_t^X(x)d \omega_s(x) \le\epsilon/3, $$

for all s∈(s 0−1,s 0+1). Since the map

$$s \mapsto F(s) := e^{-\frac{H}{2}s} \int_{\mathcal {H}_s \cap B_{r_0+2|s|}(x_s)} k_t^X(x)d\omega_s(x) $$

is obviously continuous, we can find 0<δ<1 such that

$$\bigl| F(s) - F(s_0) \bigr| \le\epsilon/3, $$

for all s∈(s 0−δ,s 0+δ). This implies that

for all s∈(s 0−δ,s 0+δ). This shows that \(\mathcal {A}k_{t}^{X} \in C(\mathbb {R}) \cap L^{1}(\mathbb {R})\).

For the proof of the theorem, it only remains to show that the Fourier transforms of the L 1-functions \(\mathcal {A}k_{t}^{X}\) and \(e^{-H^{2}t/4} k_{t}^{\mathbb {R}}\) agree. To show this, we need some growth information of \(k_{t}^{X}\) and φ λ and their derivatives.

Let us first consider φ λ for \(\lambda\in \mathbb {R}\). An immediate consequence of φ λ =aψ λ is

$$\bigl| \varphi_\lambda(r) \bigr| \le C e^{\frac{H}{2} r}, $$

with a suitable constant C>0. Moreover, \(\varphi'' + \frac{\theta'}{\theta} \varphi' = L \varphi\) with L=−(λ 2+H 2/4) implies that

$$\bigl\Vert\nabla\varphi_\lambda(r) \bigr\Vert\le|L| \int_0^r \underbrace{\biggl \vert \frac{\theta(t)}{\theta(r)} \biggr \vert }_{\le1} \bigl| \varphi_\lambda(t) \bigr| dt, $$

which shows that ∥∇φ λ ∥ grows also at most exponentially in the radius.

Next, we derive superexponential decay of the derivatives \(\frac{\partial}{\partial t} k_{t}^{X}\) and \(\Vert\nabla k_{t}^{X} \Vert\). Since the Ricci curvature of (X,g) is bounded from below and all balls of the same radius have the same volume, we conclude from [10, Prop. 1.1] that

$$p(x,x,t) \le C \begin{cases} e^{-\lambda_0(X) t},&\text{if $t \ge1$}, \\ t^{-n/2}, &\text{if $t \le1$}, \end{cases} $$

with a suitable constant C>0. Since we have λ 0(X)=H 2/4>0, we can find another constant C′>0 such that

$$p(x,x,t) \le\frac{C'}{t^{n/2}}, $$

for all x∈X and t>0. Then we are in Case 1 of [7], and Theorems 2 and 6 in [7] imply that, for any fixed time t>0, the above heat kernel derivatives decay at the rate \(q(r) e^{-r^{2}/4t}\), with a suitable polynomial q.

To finish the proof, let \(\lambda\in \mathbb {R}\). We need to show that

$$f(t) := \widehat{\mathcal {A}k_t^X}(\lambda) = \bigl\langle \mathcal {A}k_t^X, \psi_\lambda \bigr\rangle= \bigl\langle k_t^X, a \psi_\lambda\bigr\rangle= \int _X p_t^X(x_0,x) \varphi_\lambda(x) dx $$

and

$$g(t) := e^{-\frac{H^2}{4}t} \widehat{k_t^\mathbb {R}}(\lambda) = e^{-\frac {H^2}{4}t} e^{-\lambda^2 t} $$

agree.

Obviously, both functions satisfy lim t→0 f(t)=lim t→0 g(t)=1. So it only remains to show that we have f′(t)=g′(t) for all t>0. Now,

All steps in this calculation are justified by the growth properties derived above. □

The result corresponding to Theorem 5.6 in the special case of Damek–Ricci spaces can be found, e.g., in [1, (5.6)].