The Shifted Wave Equation on Non-flat Harmonic Manifolds

We solve the shifted wave equation ∂2∂t2φ(x,t)=(Δx+ρ2)φ(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial ^2}{\partial t^2}\varphi (x,t)=(\Delta _x+\rho ^2)\varphi (x,t) \end{aligned}$$\end{document}on a non-compact simply connected harmonic manifold with mean curvature of the horospheres 2ρ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\rho >0$$\end{document}. We give an explicit representation of the solution as the inverse dual Abel transform of the spherical means of their initial conditions using the local injectivity of the Abel transform and symmetry properties of the spherical mean value operator. Furthermore, we investigate the shifted wave equation using the Fourier transform on harmonic manifolds of rank one. Additionally, we obtain a result analogous to the classical Paley–Wiener theorem and use it to show an asymptotic Huygens principle as well as asymptotic equidistribution of the energy of a solution of the shifted wave equation under assumptions on the Harish–Chandra type c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{c}$$\end{document}-function.


Introduction
In their paper [AMPS13] the authors solved the shifted wave equation on Damek-Ricci spaces explicitly.These spaces together with Euclidean and hyperbolic spaces, provide all known examples of non compact simply connected harmonic manifolds.A harmonic manifold is a complete Riemannian manifold (X, g) such that for all p ∈ X the volume density function in geodesic polar coordinates g ij (p) = θ q (p) only depends on the geodesic distance.The Euclidean and non flat symmetric spaces of rank one are harmonic.It was a long standing conjecture that all harmonic manifolds are of this type, referred to as the Lichnerowicz conjecture [Lic44].The conjecture was proven for compact simply connected spaces by Szabo [Sza90] but shortly after this, in 1992 Damek and Ricci [DR92a] provided for dimension 7 and higher a class of homogeneous harmonic spaces that are non symmetric.These manifolds are called Damek-Ricci spaces.In 2006 Heber [Heb06] showed that all homogeneous non compact simply connected harmonic Date: February 16, 2023.The author would like to thank Gerhard Knieper and Norbert Peyerimhoff for their support, helpful comments and advise.The author is partially supported by the German Research Foundation (DFG), CRC TRR 191, Symplectic structures in geometry, algebra and dynamics.
spaces are of the type mentioned above.Since these spaces have a rich algebraic structure one obtains tools from harmonic analysis, see [Hel94] and [RS09].In [BKP21] the authors showed that one can obtain these tools without the assumption of homogeneity by assuming purely exponential volume growth or equivalently rank one.Furthermore in [PS15] the authors showed that tools like the Abel transform and its dual are accessible without the assumption of rank one.We now use their methods to generalise the results from [AMPS13].The idea of the proof is identical: We use the symmetries of the mean value operator to express the solution of the shifted wave equation via the inverse dual Abel transform of spherical means of its initial conditions.In section 2 we provide all the generalities on harmonic manifolds needed for this discussion.In section 3 we recall important properties of the Abel transform and its dual form [PS15], and in section 4 we show the symmetry of the spherical mean operator before solving the wave equation with smooth compactly supported initial conditions explicitly in section 5.In section 6 we investigate the wave equation under the additional assumption that X is of rank one and thereby obtain a similar results as in [ADB10].To conduct this investigation we will use the Fourier transform on X.For this purpose we give a brief overview over the Fourier transform on harmonic manifolds of rank one and look at the action of the Laplacian under Fourier transform.Then in section 7 we in particular generalise the Paley-Wiener type theorem from [ADB10] and use it to obtains bounds on the energy of a solution of the shifted wave equation on X under assumptions on the initial conditions.In section 8 we improve the result form the previous section by showing an analogous of the classical Paley-Wiener theorem on harmonic manifolds of rank one, generalising the results from [Hel94] and [ACB97] for symmetric and non symmetric Damek-Ricci spaces respectively.The main idea of the proof of this theorem is to use the Radon transform from [Rou21] to translate the problem to the real line.We then use this to obtain an asymptotic Huygens principle (section 9) and asymptotic equidistribution of energy (section 10).Under the assumption that the c-function of X has a polynomial holomorphic extension into a strip on the upper half plane in C with the first pole of multiplicity one.This generalises the results of symmetric spaces ( [BO91], [Hel92], [OS92], [BOS95], [BOP05]), non symmetric Damek-Ricci spaces ( [ADB10]) and gives a non radial version of the results in [EKY05].

Preliminaries
In this section we give a brief introduction into non compact simply connected harmonic manifolds.For more information we refer the reader to the surveys [Kre10] and [Kni16].Let (X, g) be a non compact simply connected Riemannian manifold without conjugate points.Denote by C k (X) the space of k-times differentiable functions on X and by C k c (X) ⊂ C k (X) those with compact support.With the usual conventions for continuous, smooth and analytic functions.Furthermore for x ∈ X denote by C k (X, x) the functions in C k c (X) radial around x i.e f ∈ C k (X, x) if there exists a even function u ∈ C k even (R) on R such that f = u • d(x, •) where d : X × X → R ≥0 is the distance induced by g.Furthermore for p ≥ 1 L p (X) refers to the L p -space of X with regards to the measure induced by the metric and integration over a manifold is always be interpreted as integration with respect to the canonical measure on this manifold unless stated otherwise.For p ∈ X and v ∈ S p M denote by c v : R → M the unique unit speed geodesic with c(0) = p and ċ(0) = v.Define A v to be the Jacobi tensor along c v with initial conditions A v (0) = 0 and A ′ (0) = id.For details on Jacobi tensors see [Kni02].Then using the transformation formula and the Gauss lemma the volume of the sphere of radius r around p is given by: vol S(p, r) = SpM det A v (r) dv. (1) The second fundamental form of S(p, r) is given by A ′ v (r)A −1 v (r) and the mean curvature by Definition 2.1.Let (X, g) be a complete non compact simply connected manifold without conjugate points and SX its unit tangent bundle.For v ∈ SX let A v (t) be the Jacobi tensor with initial conditions A v (0) = 0 and A ′ v (0) = id.Then X is said to be harmonic if and only if A(r) = det(A v (r)) ∀v ∈ SX.
Hence the volume growth of a geodesic ball centred at π(v) only depends on its radius.
From (2) one easily concludes that the definition above is equivalent to the mean curvature of geodesic spheres only depending on the radius.More precise the mean curvature of a geodesic sphere S(x, r) of radius r around a point x ∈ X is given by A ′ (r) A(r) .Using A v one can construct the Jacobi tensor S v,r along c v with S v,r (0) = id, S v,r (r) = 0, and U v,r = S v,−r .
Then the stable respectively unstable Jacobi tensor is obtained via the limiting process: Note that these limits exist [Kni02].
Let v ∈ S p X and c v the unit speed geodesic with initial direction v. Now define for x ∈ X the Busemann function b v (x) = lim t→∞ b v,t (x), where b t,v (x) = d(c v (t), x)−t.This limit exists and is a C 1,1 function on X, see for instance [Kni86].The level sets of the Busemann functions, H s v := b −1 v (s) are called horospheres and in the case that b v ∈ C 2 (X) their second fundamental form in π(v) = p is given by U ′ v (0) =: U(v).Hence their mean curvature is given by the trace of U(v).In the case of a harmonic manifold v → trace U(v) is independent of v ∈ SX, hence the mean curvature of horospheres is constant.Using this notion of stable and unstable Jacobi tensors Knieper in [Kni12] generalised the well known notion of rank for general spaces of nonpositive curvature introduced by Ballmann, Brin and Eberlein [BBE85] to manifolds without conjugated points.
Define for v ∈ SX S(v) Furthermore Knieper showed that for a non compact harmonic manifold rank(X) = 1 is equivalent to other important notions in geometry these are stated in section 6.
For f ∈ C 2 (X) the Laplace-Beltrami operator is defined by ∆f := div grad f and for local coordinates {x i } is given by where g = {g ij } is the matrix which defines the metric tensor g : T X × T X → [0, ∞) and {g ij } its inverse.∆ is by definition linear on C ∞ c (X) and we have where • g is the norm induced by g.Hence −∆ is a non negative symmetric operator.Furthermore −∆ is formally self adjoint hence by the density of C ∞ c (X) in L 2 (X) we can extent ∆ to a self adjoint operator on L 2 (X) which in abuse of notation we will again denote by ∆.The above also implies that the spectrum of ∆ is contained in the negative half line.From now on assume that (X, g) is a non compact simply connected harmonic manifold with mean curvature of the horosphere h = 2ρ.In this case the authors showed in [RS03] that ∆b v = h and hence the Busemann functions as well as all eigenfunctions of ∆ are analytic by elliptic regularity since harmonic manifolds are Einstein, see for instance [Wil96,Sec. 6.8], and therefore analytic by the Kazdan-De Truck theorem [DK81].Furthermore the authors in [PS15, Corollary 5.2] showed that the top of the spectrum of ∆ is given by −ρ 2 .Lemma 2.2 ([BKP21], Lemma 3.1).Let f be a C 2 function on (X, g) and u a C ∞ function on R. Then we have: With Lemma 2.2 we can calculate the spherical and horospherical part of the Laplacian, by choosing f = d x for some x ∈ X.We obtain with ∆d x (r) = A ′ (r) A(r) • d x (r) using spherical coordinates around x From this we have that the radial part of the Laplacian, does only depend on the radius and not on specific points.Therefore we obtain: X) function and x ∈ X then for the mean value operators we have Especially we have for Proof.We can decompose the Laplacian ∆f (y) = ∆ S(x,d(x,y)) f (y) + ∆ radial f (y).
Where ∆ S(x,d(x,y)) denotes the Laplacian of S(x, d(x, y)) and ∆ radial is defined by: where for v ∈ SX, c v is the geodesic corresponding to the initial conditions c v (0) = π(v) and ċv (0) = v.Since S(x, d(x, y)) is closed Greens first identity implies: Now the radial part of the Laplacian only depends on radial derivatives and the mean curvature of the geodesic sphere which since X is harmonic also only depends on the radius hence: The second part of the Lemma follows now from (3).
Remark 2.4.Note that the fact that the Laplace operator commutes with the mean value operator is equivalent to X being harmonic.See Proof.Let f, g ∈ C ∞ c (X) and x 0 ∈ X.We integrate in geodesic polar coordinates using equation (1) and the fact that X is harmonic: where ω n−1 = vol S n−1 .

The Abel Transform and Its Dual
Peyerimhoff and Samion discussed the Abel transform and its dual for radial functions as well as its connection to the radial Fourier transform in [PS15].We will use these to construct a solution to the shifted wave equation.Therefore we recall the definition and some imported facts that we will need in the prove of our main theorems.For this purpose we need the following version of the Co-area formula.
Theorem 3.1 ([Cha06, p.160]).Let M be a connected Riemannian manifold.Given a C 1 -function f : M → R such the gradient grad f never vanishes on M, let S t denote the hypersurface defined by is a diffeomorphism and is an orientation preserving diffeomorphism.Furthermore the Jacobian of Ψ v,s is given by e hs (see [PS15, Proposition 3.1]).Hence, for a measurable function f : X → R we get : The dual with respect to the L 2 -inner product of R and X is called the Abel transform and is denoted by A. This means that for every g Furthermore the authors in [PS15] showed in Proposition 3.5 that: Furthermore A(f ) is smooth, has compact support and is even.
Then bottom equality follows immediately from (6).Therefore we only need to show that: and that g(s) is even, since the smoothness follows after showing the equality from the smoothness of Ψ s,v in s.Now we prove (7) Let for λ ∈ C, ϕ λ,x 0 be a eigenfunction of the Laplacian with eigenvalue −(λ 2 + ρ 2 ) radial around x 0 with ϕ λ,x 0 (x 0 ) = 1.Now evenness follows similar to (7) if we observe that since the Laplacian commutes with R x 0 and by (4) e (iλ−ρ)bv (x) is for all λ ∈ C a eigenfunction of ∆ with eigenvalue −(λ 2 + ρ 2 ) we have Then using this and integration in horospherical coordinates yields: R g(s)e iλs ds = R e iλs e −ρs Now we have that ϕ λ,x 0 = ϕ −λ,x 0 , hence: By taking λ ∈ R this implies that g is even.
Furthermore the authors showed in [PS15, Proposition 3.10] that the Euclidean Fourier transform of the Abel transform is equal to the radial Fourier transform, given for a function radial around x 0 with compact support by where ϕ λ,x 0 is the radial eigenfunction of the Laplacian around x 0 with eigenvalue −(λ 2 + ρ 2 ) and ϕ λ,x 0 (x 0 ) = 1.This means that where F (u)(λ) = R e iλs u(s) ds for u : R → R sufficiently regular is the Euclidian Fourier transform.
Remark 3.4.Applying F −1 to both sides in equation (9) yields that the Abel transform and thereby its dual are independent of the choice of v ∈ S x 0 X.See also Lemma 8.5.
Theorem 3.5 ([PS15], Theorem 3.8).The dual Abel transform is a topological isomorphism between the spaces of smooth even functions on R and smooth radial functions around x 0 .This fact is going to be exploited to characterise solutions of the wave equation on X with smooth initial conditions with compact support.

Symmetry of the Mean Value Operator
From here one out we will consider complex valued functions u : X → C, where the Laplacian of u is given via the decomposition of u in real and imaginary part u = u 1 + iu 2 by ∆u = ∆u 1 + i∆u 2 .The proof of the following lemma follows the lines of the proof of Theorem 17 in [Hel59] which in turn follows the proof in [ Á37, p.334].
Lemma 4.1.Let (X, g) be a non compact simply connected harmonic manifold, and u : X ×X → C a twice continuous differentiable function with where ∆ i denotes to Laplacian with respect to the i-th variable.Then for each (x 0 , y 0 ) ∈ X × X we have 1 vol(S(x 0 , r)) for all r, s ≥ 0.
Proof.Let (x 0 , y 0 ) ∈ X × X be arbitrary points define U(x, y) := 1 vol(S(x 0 , r)) 1 vol(S(y 0 , s)) S(x 0 ,r) S(y 0 ,s) u(z 1 , z 2 ) dz 2 dz 1 with r = d(x 0 , x) and s = d(y 0 , y).Then U can both be viewed as a function on X × X and R + × R + .Since the Laplacian ∆ commutes with the mean value operator (see Lemma 2.3) and u is twice continuous differentiable we have: Then with the representation of the Laplacian in radial coordinates (see(3)) we have: If we set F (r, s) = U(r, s) − U(s, r) we obtain: Our goal is it now to show that F ≡ 0. Since F (r, r) = 0 is sufficient to show that all partial derivatives of F vanish.We have: and Therefore multiplying (10) by 2A(r) ∂F ∂s we obtain: Let C > 0 be arbitrary and consider the line r + s = C.We want to integrate the formula (12) over the triangle D with oriented boundary ∂D = OMN (see Figure 4.1), where O = (0, 0), M = ( C 2 , C 2 ) and N = (0, C), using Stokes theorem.With this we then show F vanishes on D. For this we first need the check that the expressions in (12) have no singularities in D. The critical term is 2 A ′ (s)A(r) A(s) .To rule out such a singularity let r ≤ s then since A is monotonous increasing we have A ′ (s)A(r)

A(s)
≤ A ′ (s) and A ′ (0) = 1 hence we have no singularity at O. Using Stokes theorem and equation ( 12) we get: We have to break the path along the boundary into the three lines.First consider the line r = s parameterised by the curve γ 1 (t) = (t, t) ending at M denoted by OM.Then we have γ1 = (1, 1) and therefore: From this we conclude that the integral (14) vanishes.
Next we consider the line ON.We have that A(r) = 0 therefore Lastly we consider the curve jointing N and M given by γ 2 (t) = (t, C − t).Then we have γ2 (t) = (1, −1) and obtain: Now we have using ( 13) dr ds = 0.
Since A ′ (s) ≥ 0 both integrals are non negative.This implies that Now since C > 0 is arbitrary (15) together with ( 16) implies that all partial derivatives of F vanish and therefore that F is constant on the left side of the line (t, t).Since F (r, r) = 0 we conclude F (s, r) = 0 on the left side of the line (t, t).Since F is antisymmetric, see equation (11), the same holds true for the the rest of R 2 + hence the claim follows.
Corollary 4.2.Under the conditions and with the notations of the proof of Lemma 4.1 we have that U(r, 0) = U(0, r) for all r ≥ 0 hence we obtain: With a classical Lemma by Willmore [Wil96, p.249] one can deduce a near equivalence in Corollary 4.2.
Corollary 4.3.Let u : X × X → R be a smooth function such that equation (17) holds for a small neighbourhood of (x 0 , y 0 ) ∈ X × X and all small r > 0 then: Proof.We have by [Wil96,p.249]for f ∈ C ∞ (X), x ∈ X and r > 0: where n = dim X. Applying this to u yields: Since the terms on the left hand side coincide, we obtain the claim.

The Shifted Wave Equation
In this section we solve the shifted wave equation: via the inverse Abel transform.This is analogous to Ásgeirsson characterisation of the solutions of the wave equation on R n [ Á37] and generalises work on non compact symmetric spaces and Damek-Ricci spaces by [Hel59], [Nog02] and [AMPS13] respectively.The methods used are to a large part identical and rely heavily on [PS15, Theorem 3.8] and Corollary 4.2.Where our approach differs is in that we do not have an explicit formula for the inverse dual Abel transform and hence need to rely on the local infectivity of the dual Abel transform shown in [PS15, Theorem 3.8] to obtain the existence of solutions and that they posses finite speed of propagation.
the Laplacian ∆ 2 of U with respect to the second variable is given by Proof.Define h : X × R → C by h(x, t) = e −ρt u(x, t), then by the representation of the Laplacian in horospherical coordinates (4) the Laplacian with respect to the second variable can be expressed by Now plugging (19) into (18) yields the claim.
where a is the dual Abel transform on X based at a point x 0 ∈ X.
Proof.Let x 0 ∈ X and v ∈ S x 0 X.And denote by ∆ i the Laplacian with respect to the i-th variable.First consider a solution to the wave equation ϕ 1 (x, t) with initial conditions ϕ 1 (x, 0) = f (x) and ∂ ∂t ϕ 1 (x, 0) = 0 for all x ∈ X.Because of this we can assume that ϕ 1 is even in t.

Define the function Φ
).Then since ϕ 1 (x, t) is a solution of the wave equation we have: Furthermore by Lemma 5.1 we have that: Therefore: Now we can apply Corollary 4.2 above and obtain that for every pair where a : C ∞ even (R) → C ∞ (X, x 0 ) denotes the dual Abel transform with the choice of v ∈ S x 0 X as above.Hence by Theorem 3.8 in [PS15] we get for every t ∈ R and x ∈ X: Now let ϕ 2 be a solution of the wave equation on X with ϕ 2 (x, 0) = 0 and ∂ ∂t ϕ 2 (x, 0) = g(x) for all x ∈ X.Then the initial conditions imply: Since ϕ 2 is a solution of the wave equation and by Lemma 5.1 Now we can again apply Corollary 4.2 and obtain that for every pair Now by Theorem 3.8 in [PS15] and integrating with respect to time we have for t ∈ R Since the shifted wave equation is linear we obtain a solution to the shifted wave equation with ϕ(x, 0) = f (x) and ∂ ∂t ϕ(x, t) = g(x) by ϕ = ϕ 1 + ϕ 2 .This yields the claim.
Corollary 5.3.From the characterisation in the Theorem 5.2 it follows now that ϕ is a unique solution to the initial data f, g as above.
Next we are going to show that a solution of the shifted wave equation has finite speed of propagation.
Corollary 5.4.Under the assumption of the Theorem 5.2 assume that f, g have support in a geodesic ball of radius R around x 0 ∈ X then By the local injectivity of the dual Abel transform [PS15, proof of Theorem 3.8] we have that for u : R → R smooth and even Furthermore we have R ′ = d(x 0 , x) − R > |t| hence since ǫ > 0 is arbitrary we obtain from (21) and (22):   Next we provide an intrinsic prove of the existence of solution to the shifted wave equation without using general existence results mentioned in Remark 5.8.Theorem 5.6.Let f, g ∈ C ∞ c (X) then the functions: and are solutions of the shifted wave equation with initial condition respectively.Consequently ϕ = ϕ 1 + ϕ 2 is a solution of the shifted wave equation with initial conditions ϕ(x, 0) = f (x) and ∂ ∂t t=0 ϕ(x, t) = g(x).
Proof.Because f and g have compact support there exists an R > 0 such that the support of f and of g is contained in the closed ball B(x 0 , R).We choose an orthonormal basis of eigenfunctions of the Dirichlet Laplacian on B(x 0 , 2R), with respect to the L 2 norm on B(x 0 , 2R), First we observe that by Lemma 2.3 for x ∈ B(x 0 , R) where ϕ λ k is a eigenfunction of the operator L A (see Lemma 2.3 for the definition) with Using (23) we obtain for all r ≤ R and x ∈ B(x 0 , R) Applying the inverse dual Abel transform a −1 yields, using that [PS15,Proposition 3.4]) and that a −1 is linear, that: Therefore if we can show that (24) converges uniformly in x and t we get: Hence ϕ 1 solves the shifted wave equation and satisfies the initial conditions ϕ 1 (x, 0) = f and ∂ ∂t t=0 ϕ 1 (x, t) = 0 as one sees by (24).Now suppose that (25) converges uniformly in x and s then by integration we obtain: where we interpret sin(λ j t) • 1 λ j = t if λ j = 0. Now applying the Laplacian yields: and we also get: Therefore ϕ 2 satisfies the shifted wave equation, with the required initial conditions, as one can see by (25).Hence the proof would be complete if we show that (24) and (25) converge uniformly in both variables.This will follow from Lemma 5.7.Under theses assumptions we have shown that ϕ 1 and ϕ 2 satisfy the theorem locally on the ball B(x 0 , R).If we now take R ′ > R and repeat the construction above, we have by the local infectivity of the dual Abel transform [PS15, proof of Theorem 3.8] that the series above coincide on B(x 0 , R). Therefore using the finite speed of propagation of the solution we can repeat the argument for a series R n → ∞ to obtain the theorem.
The lemma that finishes the proof of the theorem above is already contained in the proof of Theorem 3.8 in [PS15].
Lemma 5.7.Let x 0 ∈ X, R > 0 and f ∈ C ∞ c (X) such that the support of f is contained in the closed ball B(x 0 , R) and {φ k } k∈N an orthonormal basis of eigenfunctions of the Dirichlet Laplacian on B(x 0 , R), with respect to the L 2 norm on B(x 0 , r) with converges uniformly in x ∈ B(x 0 , R).And hence all series in the proof of the Theorem 5.6 converge uniformly.
Proof.First we observe that by the Sobolev embedding theorem (see for instance [Heb96, Chapter 3]) there exists a constant C 0 > 0, such that for every function u in the Sobolev space H 2 2n (B(x 0 , R)) we have: where • sup is the sup norm on C 0 (B(x 0 , R)) and n = dim X.Now since φ k is an orthonormal basis with respect to the L 2 norm on B(x 0 , R) we have . By Weyl's law (see for instance [CRD84, p.155]) we obtain that k ∼ µ n/2 k , meaning that for k > 0 there is a constant C ≥ 1 such that X) for every j ∈ N and has support in B(x 0 , R). Therefore: With this we obtain for l ∈ N arbitrarily and any x ∈ B(x 0 , R): Now using Weyl's law and µ k = λ 2 k + ρ 2 we conclude: and using Weyl's law there is a constant C 2 such that: This yields the claim.In their context one would consider the product manifold R × X with the canonical space time structure where the shifted wave equation corresponds to a lower order perturbation of the ordinary wave equation.

The rank one case
A non compact simply connected harmonic manifold X is said to be of purely exponential volume growth if there exists some constants C ≥ 1 and ρ > 0 such that: . This property is by [Kni12] equivalent to • The Geodesic Flow in SX is Anosov with respect to the Sasaki metric • Gromov Hyperbolicity • Rank one.Note that non positive curvature implies purely exponential volume growth.
From now on let (X, g) to be a non compact simply connected harmonic manifold of rank one.The geometric boundary ∂X is defined by equivalence classes of geodesic rays.Where two rays are equivalent if their distance is bounded.The topology on ∂X is the cone topology with the property that for X = X ∪ ∂X and Furthermore we obtain a cocycle property: By the above if v ∈ S σ X defines the unique geodesic ray such that For a proof see [BKP21, Lemma 2.2].With this we have ∆B ξ,σ = 2ρ where 2ρ is the mean curvature of the horospheres.And obtain: g(y) = e (iλ−ρ)B ξ,x (y) is a eigenfunction of the Laplacian with g(x) = 1 and ∆g = −(λ 2 + ρ 2 )g for λ ∈ C. Furthermore, by pushing forward the probability measure induced by the metric θ x on S x X under pr x we obtain a probability measure µ x on ∂X.Hence, we have a family of probability measures {µ x } x∈X , that are pairwise absolutely continuous with Radon-Nikodym derivative dµ x dµ y (ξ) = e −2ρB ξ,x (y) .(30) For a detailed proof see [KP16, Theorem 1.4].

Fourier Transform and Plancherel Theorem on Rank One
Harmonic Manifolds.The main tool in defining the Fourier transform on rank one harmonic manifolds is the theory of hypergroups.This was first presented for harmonic manifolds with pinched negative curvature in [Bis18] and then extended in [BKP21] to rank one harmonic manifold.Since we refrain ourselves from details, we refer the reader to [BH11] for a thorough discussion of the topic and the definition.In [BKP21, Section 4.2] the authors showed that the density function A(r) of a harmonic manifold of rank one satisfies the following conditions (C1) A is increasing and A(r) > 0. (C3) For r > 0, A(r) = r 2α+1 B(r) for some α > − 1 2 and some even And therefore A(r) defines a Chébli-Triméche hypergoup.The structure is of the so defined hypergroup is related to the second order differential operator given by the radial part of the Laplacian: and which admits a smooth extension to zero with ϕ λ (0) = 1.Under conditions (C1)-(C4) it was shown in [BX95] that there is a complex function c on C\{0}.Such that for the two linear independent solutions of L A u = −(λ 2 + ρ 2 )u Φ λ and Φ −λ which are asymptotic to exponential functions i.e.
Φ ±λ (r) = e (±iλ−ρ)r (1 + o(1)) as r → ∞ (34) we have Imposing the additional condition that |α| > 1 2 the authors in [BX95] showed that c-function dose not have zeros on the closed lower half plane.Hence this would exclude the case dim X = 3 (see [BKP21]) but the Lichnerowicz conjecture is affirmed in the case dim X < 6 and therefore the Jacobin analysis applies, and we can use the c-function obtained in this context.We then can define the radial Fourier transform by: Definition 6.1.Let f : X → C be, i.e. f = u • d σ for some σ ∈ X, where u : [0, ∞) → C and d σ : X → R is the distance function.The radial Fourier transform of f is given by: Note that in the following we will omit to mention the base point σ unless there is the possibility of confusion.For f radial around σ ∈ X, we will use σ as base point for the radial Fourier transform unless stated otherwise.Now observe that we obtain the radial eigenfunctions of the Laplace operator with eigenvalue −(λ 2 + ρ 2 ) by: Using the results from [BX95] the authors in [BKP21] showed that there is a constant Moreover the radial Fourier transform extends to an isometry between the L 2 -radial functions denoted by L 2 (X, σ) and See [BKP21, Theorem 4.7].In the same fashion as in the case of the Helgason Fourier transform on symmetric spaces we can extend the Fourier transform to non radial functions.By using radial symmetry of the Poisson kernel.Again the main reference for this is [BKP21].Definition 6.2.Let σ ∈ X for f : X → C measurable, the Fourier transform of f based at σ is given by for λ ∈ C, ξ ∈ ∂X for which the integral above converges.

We can immediately note that because of the cocycle property of the Busemann function (29)
we obtain: X) and x, σ ∈ X then we have: Proof.Let x, σ ∈ X and f ∈ C ∞ c (X) then we have for λ ∈ C and ξ ∈ ∂X that: Furthermore the Fourier transform coincides with the radial Fourier transform on radial functions.For details see [BKP21, Lemma 5.2].The inversion formula follows now from the representation of the radial eigenfunctions via convex combination of non radial eigenfunctions, [BKP21, Theorem 5.6],: This is analogous to the well known formula on a rank one symmetric space G/K and harmonic NA groups.See for the symmetric case [Hel94, Chapter III, Section 11] and for the harmonic NA group [DR92b] and [RS09].Using equation (39) the authors obtain: where C 0 is the same constant given in (37).Additionally the authors obtain a Plancherel theorem: and the Fourier transform extends to an isometry between

Wave Equation Under Fourier Transform and conservation of Energy.
Using the Fourier transform we can obtain the conservation of energy for solutions of the wave equation similar to the result in [ADB10] for Damek-Ricci spaces.For this we first need to study the action of the Laplacian under Fourier transform.
Lemma 6.5.Let f ∈ L 2 (X) such that ∆f ∈ L 2 (X), where ∆f is meant in the sense of distributions i.e. ∆f is defined by and σ ∈ X then: and by using the Plancherel theorem it is sufficient to prove the assertion for f ∈ C ∞ c (X).To be more precise: If f, ∆f ∈ L 2 (X) then there is a sequence For this see [Str83, Corollary 2.5].Let σ ∈ X then the above implies by the the Plancherel theorem that fn σ → f σ and ∆f Therefore we find subsequences such that both converge point wise almost everywhere.
Then since the Laplacian is essentially self adjoint and we have almost every where: Therefore we have after if necessary passing to a subsequences that almost everywhere.
Theorem 6.6.Suppose (X, g) is a harmonic manifold of rank one.Let σ ∈ X then the Fourier transform of a C ∞ solution to the shifted wave equation ϕ : is given by Proof.Since by Remark 5.5 ϕ(•, t) and all its derivatives in t have compact support for every t ∈ R we obtain: Now the wave equation becomes: therefore applying the inverse Fourier transform yields the claim.
Remark 6.7.While the representation of the solutions of the shifted wave equation from Theorem 5.2 corresponds to the classical representation of the solutions of the wave equation on R n by Ásgeirsson [ Á37] the representation obtained in Theorem 6.6 corresponds to the operator expression for the operator In turn this again corresponds to the expression of the solution as a power series in the proof Theorem 5.6.Definition 6.8.Let ϕ : X × R → C be a solution of the shifted wave equation, we define its kinetic energy K(ϕ) by: and its potential energy P(ϕ)(t) by The total energy is defined by Lemma 6.9.Suppose (X, g) is a harmonic manifold of rank one.Let σ ∈ X and ϕ : X × R → C be a solution to the shifted wave equation with initial conditions then we have Proof.Using the Plancherel theorem for the Fourier transform and Theorem 6.6 we obtain for the kinetic energy For the potential energy we are using the Plancherel theorem for the Fourier transform, Theorem 6.6 and Lemma 6.5: Theorem 6.10.Suppose (X, g) is a harmonic manifold of rank one.Let σ ∈ X and ϕ : X × R → C a solution to the shifted wave equation with initial conditions f, g ∈ C ∞ c (X) then the total energy E(ϕ)(t) is independent of t.In particular If we look at the terms under the integrals in Lemma 6.9 separately we obtain: Hence we obtain: Therefore the total energy is given by and is independent of the time.
Note that using a different method one can proof the conservation of energy of solutions of the shifted wave equation on an arbitrary oriented Riemannian manifolds (see [Hel94]CH.V Lemma 5.12).But via this proof one does not obtain the explicit expression for the total energy above.Using Theorem 6.10, Greens identity and the fact that f has compact support we obtain that: . Hence comparing the above with the expression for the energy from Theorem 6.10 we obtain using the Plancherel theorem and Lemma 6.5 In the next section we are going to investigate the term on the right hand side to obtain bounds on the energy just using the L 2 norms of the initial conditions.

A Paley-Wiener Type Theorem on Harmonic Manifolds of Rank One
The classical Paley-Wiener theorem (see for instance [Yos74, p.161]) gives shape bounds on the decay of the Fourier transform of a compactly supported function on R n : Theorem 7.1.A holomorphic function F : C n → C is the Fourier transform of a smooth function with support in the ball {x ∈ R n | x ≤ R} if and only if for every N ∈ N >0 there exists a constant In this section we want to show a weaker statement (Theorem 7.4) namely that a sufficient decay of the derivatives of a function forces there Fourier transform to have support within a bounded set.Using mainly Lemma 6.5 and the Plancherel theorem this is an extension of a Paley-Wiener type theorem from [ADB10] to harmonic manifolds of rank one.The proof follows the lines in [ADB10] closely with the addition of some details, but the statement of the Paley-Wiener type theorem is weaker then the one in [ADB10] since it is still not known if the Fourier transform on harmonic manifolds is surjective.Furthermore we use this result to show that the total energy of a solution to the shifted wave equation with specific initial conditions is bounded by bounds only depending on the L 2 norm of the initial conditions and bounds on the support of the Fourier transform of the initial conditions.Let g : R + × ∂X → C be a measurable function with respect to the measure C 0 |c(λ)| −2 dµ σ (ξ) dλ then we define Note that this might be infinite.
Lemma 7.2.Let g be a function on Proof.First we assume R g < ∞ then let 0 < ǫ < R g and we get for some δ > 0 that: Hence we have: On the other hand: Then for every M > 0 we have: Definition 7.3.Let R > 0. We define: Theorem 7.4.Let R > 0 then, if it exists, the inverse Fourier transform of a function in ) and denote its inverse Fourier transformed with respect to σ ∈ X by f .f is smooth by the Lebesgue's dominant convergence theorem and f satisfies condition (1) since by Lemma 6.5 we have: Using the Plancherel theorem, Lemma 6.5 and Lemma 7.2 we have: , then by the Plancherel theorem and Lemma 6.5 we have: and by Lemma 7.2 we have R g = R.
Corollary 7.5.Let σ ∈ X and R > 0 then for a smooth solution of the shifted wave equation ϕ : we have . Furthermore we obtain: . Proof.We have by Theorem 6.10 that and since f ∈ P W 2 R (X) we obtain: . Therefore applying the Plancherel theorem yields: . Now using equation (43), equation ( 44) and the Plancherel theorem we conclude: 8. The Paley Wiener Theorem for Harmonic Manifolds of Rank One Theorem 8.1.Let f : X → C be a smooth function with compact support in the ball B(σ, R) for some σ ∈ X and R > 0 then the Fourier transform of f based at σ is a holomorphic function in λ and we have: The above is a generalisation of theorem 4.5 in [ACB97] but our method differs from theirs which relies on the homogeneity of Damek-Ricci spaces.Furthermore the boundary structure of the Damek-Ricci space NA used consist of the non compact group N wheres we use the geometric boundary which is equivalent to using the one point compactification of N, for an explanation of this correspondence see for example [ADB08, Section 3].The idea of the proof: We first show that for f ∈ C ∞ c (X) the Radon transform R σ (f )(s, ξ), a modification of the one introduced in [Rou21], is smooth in s.Then we argue that it vanishes for s > R and all ξ ∈ ∂X.Using the connection of the Radon transform and the Fourier transform via the Euclidean Fourier transform we apply the classical Paley-Wiener theorem to show the claim.This approach is also used by Helgason to show the Paley Wiener theorem for non compact symmetric space (see [Hel94,p.278]).We begin by introducing the Radon transform, a generalisation of the Abel transform to non radial functions.
8.1.The Radon transform.We define the Radon transform R σ (f ) : f (z) dz for all s ∈ R and ξ ∈ ∂X.Note that this definition differs from the one given in [Rou21] by the factor e −ρs , furthermore all signs are swapped compared to his work since he chooses the Busemann function to be defined with the opposite sign to ours.We choose this factor deliberately to have a direct correspondence to the Fourier transform via the Euclidean Fourier transform in Lemma 8.5 and obtain the Abel transform on radial functions.
In coordinates given by the diffeomorphisem (5) and by (6) the regularity of R σ (f )(s, ξ) in s is given by the minimum of the regularity of f and Ψ s .But since the Busemann functions and the metric are analytic Ψ s is analytic in s.Hence R σ (f )(s, ξ) is smooth in s.
The lemma is a version of the projection slice theorem for harmonic manifolds.
The proof follows from the following lemma with the relation Proof.Recall the relations (29), ( 30), ( 38) and (39).Then we obtain for x, σ ∈ X: The interchange of integrals is justified by the Fubini-Tonelli theorem and the facts that f has compact support and ∂X has finite measure (dµ σ (ξ) is a probability measure).
Corollary 8.9.Let R > 0 and denote by P W 0 R all functions

Huyghens' principle
In this section we want to prove an asymptotic Huyghens' principle along the lines of the proof of [BOS95].For this we need to make assumptions on the c-function, namely we need that the function η defined by η(λ) −1 := c(λ)c(λ) on the lower have plane of C has a holomorphic extension up to Im(λ) = ǫ max > 0 where it has a singular pole and is a polynomial with real coefficients up to this point such that η(λ) = λ n−1 η 0 (λ) where all poles of η are also poles of η 0 with the same multiplicity.This condition is satisfied in the case of symmetric spaces of rank one and Damek-Ricci spaces whose nilpotent part has a centre of even dimension as well as on the hyperbolic spaces of odd dimension.For this see [EKY05].For more detail on the c-function of Damek-Ricci space see [Var06], especially proposition 4.7.13-4.7.15 and theorem 6.3.4.Remark 9.1.Note that η(λ) = |c(λ)| −2 and that by [BX95, Lemma 3.4 and Proposition 3.17] (alternatively one can observe this from (33) combined with (34) and ( 35)) we have:

From this we get that for all
Theorem 9.2.Let (X, g) be a non compact simply connected harmonic manifold of rank one of dimension bigger then one, such that the cfunction satisfies the condition above.And let ϕ be a solution of the shifted wave equation with initial conditions f, g supported in a ball of radius R around σ ∈ X.Let ǫ max be as above and 0 < ǫ < ǫ max < ∞ then there is a constant C > 0 such that The proof of this statement will be conducted via a series of lemma occupying the remainder of the section.We will always require the assumptions of the theorem.
Lemma 9.3.Let h : C → C be a function holomorphic on the stripe Proof.Consider the contour in Figure 9.1.Let γ 1 : [0, 1] → C be given by γ 1 (s) = r + isǫ and γ 2 : [0, 1] → C be given by γ 2 (s) = −r + i(1 − s)ǫ then by the bounds on h on the stripe P there are constants C 1 , C 2 > 0 such that: Therefore since both integrals tend to zero for r → ±∞ and we get the assertion.
Lemma 9.4.Let f, g ∈ C ∞ c (X) then the functions are even in λ and Proof.Since η ,by Remark 9.1, is even in λ and by Proposition 8.7 F (λ, x) and G(λ, x) are even in λ.Now using this and 2 cos(λt) = e iλt + e −iλt Re(z) we get: Since 2i sin(λt) = e iλt − e −iλt and G(λ, x) is even in λ we obtain: By [Tri18, Prop.6.1.1 and Prop.6.1.4]and (36) we have the following bounds for the radial eigenfunctions of the Laplacian: Lemma 9.5.For all x, σ ∈ X and λ ∈ C we have: ( Furthermore, we have: for some positive constant k > 0. Lemma 9.6.Assume the assumptions of the Theorem 9.2.Let f, g ∈ C ∞ c (X) with support in the ball of radius R > 0 around σ ∈ X then F and G admit holomorphic extensions in λ up to ǫ max and for every N ∈ N we can find a constant C N such that for all λ ∈ C with 0 ≤ Im λ ≤ ǫ < ǫ max and x ∈ X Furthermore if dim X > 1 we have that for every N ∈ N there is a constant D N such that ≤ e |Im λ|d(x,σ) .Now using Theorem 8.1, the assumption that η has a singular pole at ǫ max and is a polynomial and since ∂X is compact we can conclude that for every N ∈ N there is a constant C N such that for all 0 ≤ Im λ ≤ ǫ < ǫ max |F (λ, x)| ≤ C N (ǫ max − ǫ) −1 (1 + |λ|) −N e ǫd(x,σ)+R|Im λ| ≤ C N (ǫ max − ǫ) −1 (1 + |λ|) −N e ǫd(x,σ)+Rǫ .
For the last estimate on |λ −1 G(λ, x)| one only need to consider that η(λ) = λ n−1 η 0 (λ) where all poles of η are also poles of η 0 with the same multiplicity.Hence one only need to exclude the case where dim X = 1.Then we get using the same lines as above: Hence we obtain using the same arguments as above that for every N ∈ N there is a constant D N such that for 0 ≤ Im λ ≤ ǫ < ǫ max Since the last integral is bounded we obtain the claim.For the case that the c-function is an entire function and a polynomial one notice that we can ignore the therm (ǫ max − ǫ) −1 in all the estimates which yields the assertion in this case.

Equidistribution of Energy
Under the same assumptions on the c-function as in the last section we now want to proof an asymptotic equidistribution of the energy between the kinetic and potential energy of a wave on X.
For us to be able to use the same arguments as in section 9 the following lemma is essential.

Figure 5
Figure 5.1.A sketch for the proof of Corollary 5.4.

Figure 5 . 2 .
Figure 5.2.Finite propagation speed of a solution of the shifted wave equation with initial conditions supported in B(x 0 , R).
5. The finite speed of propagation also follows from the general theory in [Fri75, Chapter 5] or [Tay11, Chapter 2, Proposition 8.1] by choosing the canonical space time structure on R × X. See also [BO91, Lemma 1.1].
turns out that since the geodesic flow is Anosov the Busemann function only depends on the direction of the ray.Hence for x ∈ X and ξ ∈ ∂X being the point at infinity of the geodesic γ we can alternatively define the Busemann function B ξ,x : X → R by B ξ,x (y) = lim t→∞ (d(y, γ(t)) − d(x, γ(t)).