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

Function Spaces as Dirichlet Spaces 9
Nagel discussed in [19, 20] the rotationally invariant analogue of Lemma 3.2.
Lemma 3.3. Let g : [0, ∞] → [0, ∞] be a monotone increasing concave function
and define µ(t) := ∫ t ∫ ∞
g(s)/s 2 ds dr ∈ [0, ∞]. Then
0 r
( ∫∫
( 1
) ) 1
dx dy
|u(x + y) − u(x)| 2 2
g
R n ×R |y| n 2 |y| n
( ∫
) (19)
1
∼ |û(ξ)| 2 µ(|ξ| 2 2
) dξ
R n
are equivalent (semi-)norms on S(R n ).
Let us now consider the rotationally invariant case. Comparing (16) with
(19) we would like to relate
( 1
) 1
m(t) with g
and f(|ξ| 2 ) with µ(|ξ| 2 ).
t t n/2
Lemma 3.4. Let g(t) = ∫ ∞ t
ρ(ds) be a complete Bernstein function. Then
0+ s+t
the functions r ↦→ g(r −1 ) and r ↦→ r −n/2 g(r −1 ) are completely monotone. In
particular, we have the representations
( 1
)
g =
r
( 1
)
r −n/2 g =
r
∫ ∞
0
∫ ∞
0
( ∫ ∞
e −xr −x/s ρ(ds)
e
s
0+
( 1
e −xr Γ ( )
n
2
∫ ∞
0+
)
dx (20)
s n/2−1 e −x/s ∫ x/s
0
)
y n/2−1 e y dy ρ(ds) dx. (21)
Proof. Since the formulae (20) and (21) prove that r ↦→ r −n/2 g(r −1 ), n 0, is
completely monotone, it is enough to establish these two representations.
From the definition of g we have
Since we can write
0+
( 1
)
g =
r
∫ ∞
0+
1
r
s + 1 r
ρ(ds) =
∫
1
∞
rs + 1 = e −t(rs+1) dt =
0
0
∫ ∞
0+
∫ ∞
0
0
1
rs + 1 ρ(ds).
e −x/s e −xr dx s
we can apply Tonelli’s Theorem to find
( 1
) ∫ ∞ ∫ ∞
g = e −x/s e −xr dx ∫ ∞
( ∫ ∞
)
r s ρ(ds) = e −xr −x/s ρ(ds)
e dx
s
which proves (20). To prove (21), we use the elementary identity
r −n/2 = 1
Γ ( )
n
2
∫ ∞
0
0+
x n/2−1 e −rx dx, n > 0,