1.2. ‘FOURIER TRANSFORM’ FC ON THE ISOTROPIC CONE C 7

2) Continuous operators A on L2(C) satisfying (1.2.4) are unique up to scalar. In

particular, any such operator A is a scalar multiple of the unitary operator FC,

and A2 is a scalar multiple of the identity operator.

Joint eigendistributions on C

Next, we consider the system of differential equations on C:

(1.2.5) Pjψ = 4ξjψ (1 ≤ j ≤ n).

The coeﬃcient 4 in the right-hand side is just for simplifying later notation.

We shall deal with solutions in an appropriate class of distributions on C (the

dual space

L2(C)−∞

of smooth vectors

L2(C)∞,

see (2.5.9)) and prove the follow-

ing:

Theorem 1.2.4 (see Theorem 2.5.5). Fix ξ = (ξ1,...,ξn) ∈

Rn

\ {0}.

1) If Q(ξ) = 0, then any distribution ψ on C satisfying (1.2.5) is zero.

2) If ξ ∈ C, then the solution space of (1.2.5) in L2(C)−∞ is one-dimensional.

The first statement is an immediate consequence of P4. By the explicit formula

given in Theorem 1.3.1, we shall see that the unique solution in Theorem 1.2.4 (2)

is not real analytic if n1,n2 1.

Abstract properties of FC

We will prove in this book that the distribution solution ψ(x) in Theorem 1.2.4

(2) can be normalized depending on ξ ∈ C, which we denote by K(x, ξ) for now, in

such a way that the following key properties are fulfilled:

K1 For each fixed ξ ∈ C, K(·,ξ) is a distribution solution on C to (1.2.5).

K2 K(x, ξ) = K(ξ, x) as a distribution on C × C.

K3 We define

(1.2.6) (FC f)(ξ) :=

C

K(x, ξ)f(x)dμ(x) for f ∈ C0

∞(C).

Then, (1.2.6) is well-defined, and we have a linear map FC : C0 ∞(C) →

C∞(C) ∩ L2(C).

K4 FC extends to a unitary operator on L2(C).

K5 FC ◦ 4xj = Pj ◦ FC,

FC ◦ Pj = 4ξj ◦ FC.

K6 FC 2 = id.

K7 FCu(ξ) =

R

Ψ(t)Rf(ξ, t)dt.

K8 FC gives the automorphism of each of the following topological vector spaces:

L2(C)∞

⊂

L2(C)

⊂

L2(C)−∞.

These properties K1–K8 are stated in parallel to the Euclidean case F1–F8. In

K7, R is the (singular) Radon transform on the isotropic cone C which will be

defined in (1.2.7), and Ψ(t) is a distribution on R which will be defined in Theorem

1.3.1. We note that the transform in K7 by Ψ(t) collapses to the Hankel if n2 = 1.

In K8, we have the following inclusive relation

C0

∞(C)

⊂

L2(C)∞

⊂

L2(C)

⊂

L2(C)−∞

⊂ D (C)

as in the Euclidean case (see F8):

C0

∞(Rn)

⊂

S(Rn)

⊂

L2(Rn)

⊂ S

(Rn)

⊂ D

(Rn).