Huygens ’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian

In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace–Beltrami operator on hyperbolic spaces or Damek–Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish–Chandra c–function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.


Introduction
In Euclidean space X = R n of odd dimension, any solution u(x, t) to the wave equation ) is determined by the value of its initial data in an arbitrarily thin shell around the sphere S(x, |t|).This is Huygens' principle.Moreover the total energy splits eventually equally into its kinetic and potential components if the initial data are compactly supported.See Duffin [7] and Branson [3].Similar results were established by Branson, Helgason, Olafsson, Schlichtkrull in [4], [8], [12] for the modified wave equation on Riemannian symmetric spaces of noncompact type X = G/K, under the assumptions that dim X is odd and G has only one conjugacy class of Cartan subgroups.Otherwise these phenomena may not hold strictly speaking, but they do asymptotically, as shown by Branson, Olafsson and Schlichtkrull in [5].
This paper is devoted to another setting, which is known to share features with the previous one.Specifically we consider the modified wave equation with initial data associated to certain second order differential operators on (0, +∞).Following Chébli and Trimèche, we assume that the function A(x) behaves as follows : • A(x) ∼ x 2α+1 as x 0, where α > − 1 2 .More precisely where B : R → (0, +∞) is a smooth even function with B(0) = 1.
Our paper is organized as follows.In Section 2, we recall some basic harmonic analysis associated to L. This theory was developed initially by Chébli [6] and Trimèche [13] (see also Trimèche [14] and Yacoub [15]) and was resumed by Bloom and Xu in the framework of hypergroups (see for instance [2]).We apply it next to solve the Cauchy problem (1.2).
In Section 3, we resume the analysis carried out in the symmetric space case by Branson, Olafsson and/or Schlichtkrull ( [4], [12]) and we establish the following two properties of the wave equation (1.2), under the assumption that the inverse c(λ) −1 of the Harish-Chandra c-function is a polynomial : On one hand, Huygens' principle holds and, on the other hand, the potential and kinetic energies contribute eventually equally to the total energy.
In Section 4, we show that these properties hold in general asymptotically, resuming again the analysis carried out in the symmetric space case by Branson, Olafsson and Schlichtkrull, this time in [5] .
For every λ ∈ C, the equation has a unique solution on [ 0, +∞) such that ϕ(0) = 1 and ϕ (0) = 0.It is denoted by ϕ λ .If λ = 0, the equation (2.1) has two other linearly independent solutions Φ ±λ on (0, ∞) with the following behaviour at infinity : Moreover there exists a function c(λ) (the so-called Harish-Chandra cfunction) such that In the Jacobi setting (1.7), everything can be expressed in terms of classical special functions : . (2.2) is a smooth even function in x ∈ R and an analytic even function in λ ∈ C.
• Integral representation of Mehler type : where K(x, .) is an even nonnegative function on R, which is supported in [−x, x] and which is smooth in (−x, x).
Hence there are positive constants C 1 , C 2 such that
• For every 0 < γ < δ, there exist positive constants C 1 , C 2 such that Hence c(λ) −1 and its derivatives have polynomial growth on R.More precisely
Recall some classical function spaces : D(R) denotes the space of smooth functions on R with compact support, S(R) the space of Schwartz functions on R, and H(C) the space of entire functions h on C, which are of exponential type and rapidly decreasing.This means that there exists R ≥ 0 such that A less familiar function space is the L 2 Schwartz space S 2 * (R) = ϕ 0 (x) S * (R).The subscript * means that we restrict our attention to even functions.
The generalized Fourier transform and the generalized Weyl transform are defined, let say for f ∈ S 2 * (R), by the converging integrals and where K(x, y) is the kernel occurring in (2.3).The terminology comes from Jacobi function theory, where W is expressed in terms of Weyl fractional transforms.In particular, let us recall that (2.7) is the Abel transform of radial functions on hyperbolic spaces and more generally on Damek-Ricci spaces.These transforms are related by means of the classical Fourier transform e iλy g(y) dy = +∞ 0 cos λy g(y) dy .

Properties of W :
• W is a topological isomorphism between S 2 * (R) and S * (R).

Properties of F :
• F is a topological isomorphism between S 2 * (R) and S * (R).• F is a topological isomorphism between D * (R) and H * (C).More precisely, f is supported in [−R, +R ] if and only if h = Ff is of exponential type R in the sense of (2.5).

Modified Weyl transform
Recall that λ −→ c(λ) −1 is a smooth function on R, which is tempered.Thus c −1 is a pointwise mutiplier of S(R) and we may consider the corresponding convolution operator on S(R).Similarly, let us denote by J the convolution operator corresponding to the multiplier c(λ) −1 = c(−λ) −1 .One may modify the Weyl transform W by composing it with J : We shall mostly do so when c −1 is a polynomial i.e. when J is a differential operator.Such modified transforms were considered by Lax and Phillips ([10], chap.IV; [11]), both for (odd-dimensional) Euclidean and for (3-dimensional) hyperbolic spaces, and more generally by Olafsson and Schlichtkrull [12] for the class of symmetric spaces X = G/K where G has only one conjugacy class of Cartan subgroups.

Modified wave equation
Consider the Cauchy problem with initial data f 0 , f 1 ∈ D * (R).
• Conservation ot the total energy : By applying the Weyl transform W or W to (2.12), one gets the classical wave equation on R : whose solution is well known :

Asymptotic Huygens' principle and equipartition of energy
In this section we drop the assumption that c(λ) −1 is a polynomial and we show that the properties investigated in Section 3 hold asymptotically, for lack of holding strictly speaking.This will be achieved by resuming the analysis carried out in [5] and by using the generalized Fourier transform F instead of the modified Weyl transform W. Specifically, (2.12) is transformed by F into the ordinary differential equation Transforming backwards by F −1 , one gets where Notice that h j (x, λ) is an even function both in x and in λ, which is smooth in x ∈ R and analytic in the strip |Imλ| < δ.Moreover there is no actual singularity in (4.1), since c(λ) −1 hence h 1 (x, λ) vanish at λ = 0.
Proof: We may assume that x and t are nonnegative.Since

Remarks 4.2:
• The condition on γ can be improved, provided c(λ) −1 extends holomorphically to a larger strip.This is the case for hyperbolic spaces and more generally for Damek-Ricci spaces, where δ can be replaced by ρ.
• If c(λ) −1 is a polynomial, the estimate (2. with a constant C ≥ 0 independent of γ > 0. Letting γ +∞, we obtain this way a new proof of Theorem 3.2.

Asymptotic equipartition of energy
Our aim in this subsection is to estimate in general the gap between the kinetic and the potential energies.