Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 23–28

The Fourier Transform on Symmetric Spaces and Applications

Sigurdur Helgason*

Key words and phrases: Fourier transform, symmetric spaces, Radon transform, solvability, Huygens’ principle. Mathematics Subject Classification (1991): Primary 22 E 30, 22 E 70, Secondary 43 A 85.

1. The Fourier Transform

The symmetric spaces the title refers to are the spaces X = G/K where G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup. The Fourier transform on X is defined by means of the Iwasawa decomposition G = NAK of G where N is nilpotent and A abelian. Let 𝔤, 𝔫, 𝔞, 𝔨 denote the corresponding Lie algebras. We also need the group M = K^{A}, the centralizer of A in K. To define the Fourier transform we write for g ∈ G

g = n exp A(g)k, n ∈ N, A(g) ∈ 𝔞, k ∈ K

and define A : G/K × K/M ⟶ 𝔞 by

A(gK, kM) = A(k^{−1}g).

If f is a function on X its Fourier transform f̃ is defined in [3a] by

(1)  f̃(λ, b) = ∫_{X} f(x)e^{(−iλ+ρ)(A(x,b))}dx,  b ∈ B,

for all (λ, b) ∈ 𝔞_{c}^{∗}×B for which the integral exists. Here ρ is half the sum of the restricted roots. Definition (1) is the analog of the Euclidean Fourier transform

(2)  F (λ, ω) = ∫_{R^{n}} F (x)e^{−iλ(x,ω)}dx,  |ω| = 1.

With a horocycle in X being an orbit of a group gNg^{−1} the Radon transform f̂ of f is defined by

(3)  f̂(ξ) = ∫_{ξ} f(x) dm(x),  ξ a horocycle,

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*Supported in part by NSF Grant DMS-8805665.

This paper is in final form and no version of it will be submitted for publication elsewhere.

dm being the volume element on ξ. Then (1) can be written

f̃(λ, b) = ∫_{A} e^{(−iλ+ρ)(log a)}( ∫_{A(x,b)=log a} f(x) dm(x))da.

With x = gK, b = kM the inner integral runs over the set

{gK : A(k^{−1}g) = log a} = ka N · o,

where o = {eK}. Writing ξ_{0} for the horocycle N · o this means

(4)  f̃(λ, kM) = ∫_{A} f̂(ka · ξ_{0})e^{(−iλ+ρ)(log a)}da.

This is the analog to the formula

F̃(λω) = ∫_{R} F̂(ω, p)e^{−ipλ}dp

connecting F̃ with the Radon transform

F̂(ω, p) = ∫_{(x,ω)=p} F (x) dm(x)

of F over the hyperplane (x, ω) = p in R^{n}.

The customary Fourier analysis theorems in R^{n} (inversion and Plancherel formula) have their analog for (1) as well; for applications it is more important to have range theorems, i.e. intrinsic descriptions of F(X)^{∼} as F(X) is some naturally defined function space on X.

One basic result of this type is the following characterization of D(X)^{∼} (D(X) = C_{c}^{∞}(X)) from [3b].

Theorem 1. D(X)^{∼} consists of the functions (λ, b) ⟶ ϕ(λ, b) on 𝔞_{c}^{∗} × B which are of exponential type in λ (uniformly in b ∈ B) and satisfy

(5)  ∫_{B} e^{(isλ+ρ)(A(x,b))}ϕ(sλ, b) = ∫_{B} e^{(iλ+ρ)(A(x,b))}ϕ(λ, b)db

for all x ∈ X and all s in the Weyl group W.

2. Applications to Solvability Questions

The following result is a simple consequence of Theorem 1.

Corollary 2. Let G/K have rank one and L the Laplace-Beltrami operator. Let f ∈ D(X). Then the equation

(6)  Lu = f

has a solution u ∈ D(X) if and only if

(7)  ∫_{X} f(x)dx = ∫_{X} f(x)P (x, b)dx = 0,  b ∈ B,

where P is the Poisson kernel.

In fact, if h ∈ D(X) then

(8)  (Lh)^{∼}(λ, b) = −(⟨λ, λ⟩ + ⟨ρ, ρ⟩)h̃(λ, b),

where ⟨ , ⟩ refers to the Killing form, and

P (x, b) = e^{2ρ(A(x,b))}.

Conditions (7) thus amount to f̃(±iρ, b) = 0 which by (8) would be necessary. Also they amount to f̃(λ, b) being divisible by ⟨λ, λ⟩ + ⟨ρ, ρ⟩ so by Theorem 1, they are also sufficient.

Theorem 1 is also instrumental in proving

(9)  DE(X) = E(X)  where E(X) = C^{∞}(X)

for X of any rank, D ≠ 0 being any differential operator on X, invariant under G. Theorem 1 was extended in [2] to the space S(X) of rapidly decreasing functions. From this one can prove ([3b])

(10)  DS′(X) = S′(X)

in analogy with (9), S′(X) being the space of tempered distributions on X.

3. Wave Equations and Huygens’ Principle

We shall now discuss applications of the Fourier transform on X to the question of the validity of Huygens’ principle for wave equations on X.

Let d denote the distance function on X and B_{r}(x) (resp. S_{r}(x)) the open ball (resp. sphere) with radius r and center x. Let L again be the Laplace-Beltrami operator on X and consider the Cauchy problem:

(11)  \frac{∂^{2}}{∂t^{2}} = (L + |ρ|^{2})u,  u(x, 0) = 0, u_{t}(x, 0) = f(x).

Huygens’ principle is said to hold if for each ϵ > 0, the solution u(x, t) only depends on the restriction f|V_{t,ϵ}(x) where V_{t,ϵ}(x) denotes the shell B_{t+ϵ}(x) − B_{t−ϵ}(x).

In the case of the wave equation on R^{n}

(12)  \frac{∂^{2}u}{∂x_{1}^{2}} + ⋯ + \frac{∂^{2}}{∂x_{n}^{2}} = \frac{∂^{2}u}{∂t^{2}},  u(x, 0) = 0, u_{t}(x, 0) = f(x)

the solution is given by the classical Poisson-Tedone formula

(13)  u(x, t) = \frac{1}{(n − 2)!} \frac{∂^{n−2}}{∂t^{n−2}} (∫_{0}^{t}(M^{r}f)(x)r(t^{2} − r^{2})^{\frac{1}{2} (n−3)}dr),

where (M^{r}f)(x) is the average of f on S_{r}(x). If n is odd the integration can be obliterated and we obtain

(14)  u(x, t) = ∑_{k} c_{k} t^{k+1} \frac{d^{k}}{dt^{k}} (M^{t}f(x)),

from which the validity of Huygens’ principle for (12) is obvious.

For (11) we consider in addition to the Radon transform f ⟶ f̂ the dual transform ϕ ⟶ ϕ̌ which to a function ϕ on the space Ξ of horocycles associates the function ϕ̌ on X given by

(15)  ϕ̌(x) = ∫_{ξ∋x} ϕ(ξ)dμ(ξ).

the average over the set of horocycles ξ through x. Then f ⟶ f̂, ϕ ⟶ ϕ̌ are adjoint operators, i.e.

(16)  ∫_{X} f(x)ϕ̌(x)dx = ∫_{Ξ} f̂(ξ)ϕ(ξ)dξ,

where dx and dξ are G-invariant measures. Thus we put for S ∈ E′(X) (space of compactly supported distributions)

(17)  Ŝ(ϕ) = S(ϕ̌)  ϕ ∈ E(X).

Then we have the following support theorem proved in [3d] by the method of Theorem 1.

Theorem 3. For A > 0 let β_{A} = {ξ ∈ Ξ : d(o, ξ) < A}. Suppose T ∈ E′(X) satisfies

supp(T̂) ⊂ β_{A}.

Then

supp(T) ⊂ B_{A}(0).

Combining this with the inversion formula ([3a])

(18)  f = (Λf̂)^{∨},

Λ being a certain pseudo-differential operator, we obtain a certain refinement of Theorem 3.

Theorem 4. For A > 0 let σ_{A} = {ξ ∈ Ξ : d(o, ξ) = A}. Suppose G has all its Cartan subgroups conjugate. Suppose T ∈ E′(X) satisfies

supp(T̂ ⊂ σ_{A}.

Then

supp(T) ⊂ S_{A}(o).

The assumption on G implies that the operator Λ in (18) is a differential operator. These theorems are now used to prove the following result [3c,d]; different proofs are from [4] and [5].

Theorem 5. Let X be odd-dimensional and that G has all its Cartan subgroups conjugate. Then the equation (11) satisfies Huygens’ principle.

To indicate the proof suppose f in (11) belongs to D(X) and take Fourier transforms. Then we find as in [1],

(19)  ũ(t, λ, b) = f̃(λ, b)\frac{sin |λ|t}{|λ|}.

The holomorphic function λ ⟶ |λ|^{−1} sin |λ|t of exponential type is the Euclidean Fourier transform

(20)  \frac{sin |λ|t}{|λ|} = ∫_{A} e^{−iλ(log a)}dT_{t}(a)  T_{t} ∈ E′(A)

and by the extension of Theorem 1 to distributions,

(21)  \frac{sin |λ|t}{|λ|} = ∫_{X} e^{(−iλ+ρ)(A(x,b))}dτ_{t}(x)

where τ_{t} ∈ E′(X).

Lemma 6. In terms of the product representation Ξ = (K/M) × A

(τ_{t})^{∧} = 1 ⊗ e^{ρ}T_{t}.

Now if dim X is odd and rank X > 1 then dim A is odd and > 1. Then it is known that supp(T_{t}) ⊂ S_{t}(e). Then by Lemma 6 and Theorem 4

(22)  supp(τ_{t}) ⊂ S_{t}(o).

Using (21) and the symmetry of τ_{t} we then deduce from (19),

u(g · o, t) = ∫_{X} f(g · x)dτ_{t}(x)

so the support property in (22) implies Huygens’ principle.

References

1. T.P. Branson and G. ’Olafsson, Equipartion of energy for waves in symmetric spaces, J. Funct. Anal. 97 (1991), 403–416.

2. M. Eguchi and K. Okamoto, The Fourier transform of the Schwartz space on a symmetric space, Proc. Japan Acad. 53 (1977), 237–241.

3. S. Helgason, a) Radon–Fourier transforms on symmetric spaces and related group representations.,

Bull. Amer. Math. Soc. 71 (1965), 757–763.

, b) Paley-Wiener theorems and surjectivity of invariant differential operators on symmetric spaces and Lie groups, Bull. Amer. Math. Soc. 79 (1973), 129–132.

, c) Wave equations on homogeneous spaces. in Lie group representations III, Lecture Notes in Math. Springer-Verlag, Berlin 1077 (1984), 254–287.

, d) Huygens’ principle for wave equations on symmetric spaces, J. Funct. Anal. 107 (1992), 279–288.

4. G. ’Olafsson and H. Schlichtkrull, Wave propagation on Riemannian symmetric spaces, J. Funct. Anal. 107 (1992).

5. L.E. Solomatina, Translation representation and Huygens’ principle for an invariant wave equation in a Riemannian symmetric space, Soviet Math. Izv. 30 (1986), 108–111.

Author’s address: Sigurdur Helgason, Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A.

Received 2 November 1992