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

Function Spaces as Dirichlet Spaces 21
No. f(x) τ(ds) ρ(dt) Comments
1. 1 − e −γx δγ(ds) ̸∃ γ 0; [4, p. 71]
2. log(1 + x) e
3. log x+γ
e
γ
(
4. log α x+α
β x+β
5. (x + γ) log(x + γ)
− x log x − γ log γ
−s ds
1(1,∞)(t) dt
[4, p. 71]
s t
−γs ds
) ( ) e
−αs − e
−βs ds
1(α,β)(t) dt
s
6. (x + β) log(x + β) − β log β
− (x + α) log(x + α) + α log α
1(γ,∞)(t) dt
γ > 0; [21, p. 35]
s t
( ) 1 − e
−γs ds
(t ∧ γ) + dt
s 2
7. x α α ds
, s−α
Γ(1−α) s
√
8. x √ 1 −1/2 ds
4π
s
s
√
9. x + m
2 − m √ 1
4π
e −m2 s s
−1/2 ds
t
α, β > 0; [21, p. 35]
t
γ > 0; [21, p. 35]
( ) [ ]
e
−αs − e
−βs ds
+
s (t − α) ∧ (β − α) dt
β > α > 0; [21, p. 36]
2 t
sin απ
1
π
t α dt
π t
√
0 < α < 1; [4, p. 71]
t
dt
1
s π (m 2 ,∞)(t) √ t − m 2 dt
t
m 0;
x
10. γe −γs ds x+γ
δγ(dt) γ > 0; [4, p. 71]
11.
x
12.
13.
14.
15.
= 1 − 1
e −γs 1
ds δ γ(γ+x) γ γ+x γ γ(dt) γ > 0;
√ ∫
x arctan √x
γ γ
0 u2 e −su2 du ds 1 (0,γ 2 ) (t) dt
2 √ t
= ∫ sγ 2
ue −u du ds
0 2s 2
√ √ x log(1 + x) anon log(1 + t)
dt
2π √ t
√ (
x 1 − e
−4 √ ) x
anon (sin 2 √ t) 2 2 dt
π √ t
√ √ √ √
x log(1 + coth x) anon t log(1 + coth t)
dt
[7]
2π
t
γ > 0; [21, p. 36]
[7]
[7]
More complicated complete Bernstein functions can be obtained by composition of two complete Bernstein functions,
e.g., f(x α ) and f α (x), 0 < α < 1, or by setting f 1/α (x α ), 0 < |α| 1, cf. [22].