The Stieltjes convolution and a functional calculus for non-negative ...

ounded on H(R n +). Representation (13) shows that S(ϕ) does not depend on
the choice of the measures µ ⃗k .
From now on, whenever we write S(ϕ) for a distribution S ∈ D S ∗(R n +) ′ and a
function ϕ ∈ H(R n +), we identify S with its extension to H(R n +) from Definition
2.
Alternatively, one could have used the isomorphism Π from Theorem 1 to
define the extension via S(ϕ) := (S ◦ Π)(Π −1 ϕ) for ∀ϕ ∈ H(R n +), where
S ◦Π ∈ B(R ˙ n ) ′ is identified with its extension to B(R n ). Π can also be used to
prove Lemma 1 (ii). The approach we chose in this section shows nicely the
similarities between the extension of integrable distributions to B(R n ) and of
strongly Stieltjes-transformable distributions to H(R n +).
4 Stieltjes-transformable Distributions
In this section let n = 1. We want to enlarge the class D S ∗(R + ) ′ in a way that
also measures with ∫ ∞ 1
0 d|µ|(λ) < ∞ for ∀a > 0 but with ∫ ∞
λ+a 0 λ −1 d|µ|(λ) =
∞ are included.
Note that we have to restrict ourselves to the one-dimensional case since for
n > 1 we do not have any information about the support of the functions f ⃗k .
This also explains why we could not start with these weaker assumptions right
away because we want to develop a multidimensional functional calculus.
Definition 3 A distribution S ∈ D(R) ′ is called Stieltjes-transformable if
(i)
suppS ⊆ R + and
(ii) S ◦ τ ∈ D S ∗(R + ) ′ ,
where τ denotes the translation (τϕ)(x) = ϕ(x + 1). We denote the class of
all Stieltjes-transformable distributions by D S (R + ) ′ .
Theorem 2 (Characterization of D S (R + ) ′ ) A distribution S ∈ D(R) ′ is
Stieltjes-transformable if and only if S has the form
for some measures µ k on R + , 0 ≤ k ≤ K, with
∫
K∑
S = D k µ k (15)
k=0
R +
(x + 1) −k−1 d|µ k |(x) < ∞. (16)
In this case we can assume that the measures µ k are given by measurable
11