Function Spaces as Dirichlet Spaces (About a Paper by Maz'ya and ...

Function Spaces as Dirichlet Spaces 23
The following functions are Bernstein functions. Although they are also Stieltjes
transforms, it is not clear whether they are complete Bernstein functions, since
the representing measure ρ (shown in the table below) might become negative.
No. f(x) ρ(dt) Comments
27.
28.
29.
xK √ ν(α x)
K √ , J √ ν(β t)Y √ ν(α t)−J √ ν(α t)Y √ ν(β t)
ν(β x) Jν 2 (β √ t)+Yν 2 (β √ t)
dt
α > β > 0
π
ν 0; [13]
xI √ ν(β x)
I √ , ∑
2 ∞ j ν,nJ ν(βj ν,n/α)
ν(α x) α 2 n=1 J ν+1 (j ν,n)
δ j 2 ν,n /α2(dt) α > β > 0,
ν > −1; [13]
K √ (
ν(α x)
K √ − ν(β x)
) ν
β J √ ν(α t)Y √ ν(β t)−J √ ν(β t)Y √ ν(α t)
α
Jν 2(β√ t)+Yν 2(β√
t)
dt
α > β > 0
πt
ν 0; [12]
Appendix 2: ‡
Proofs of Lemma 2.1 and Theorem 2.2
We begin with the proof of Lemma 2.1.
Proof of Lemma 2.1. Assume first that ψ(ξ) = g(|ξ| 2 ) with some Bernstein
function g,
Then
g(x) = a + bx +
∫ ∞
∫ ∞
( )
g(|ξ| 2 ) = a + b |ξ| 2 + 1 − e
−s |ξ| 2 τ(ds)
0+
∫ ∞
( ∫
= a + b |ξ| 2 + (1 − e −iyξ )
0+ R
∫
n ( ∫ ∞
= a + b |ξ| 2 + (1 − e −iyξ )
R n 0+
Switching to polar coordinates we see
∫R n |y| 2
1 + |y| 2 ( ∫ ∞
= 2 πn/2
Γ ( )
n
2
0+
∫ ∞
0+
1
(
(4πs) exp n/2
(
r 2 ∫ ∞
1 + r 2
0+
0+
(1 − e −xs ) τ(ds).
)
− |y|2 τ(ds)
4s
1
(
(4πs) exp n/2
1
(
(4πs) exp n/2
1
(
(4πs) exp n/2
)
dy
− |y|2
4s
− |y|2
4s
) )
dy
)
τ(ds)
) )
− r2
τ(ds) r n−1 dr
4s
‡ Based on an unpublished manuscript of the second author, see also [21].
τ(ds)
)
dy.