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

8 N. Jacob and R. L. Schilling
Proof. Denote by T −y the translation operator T −y u(x) := u(x + y). Since
|e it − 1| 2 = 2(1 − cos t), t ∈ R, we find using Plancherel’s Theorem
∫∫
1
|u(x + y) − u(x)| 2 dx ν(dy)
2 R n ×(R n \{0})
= 1 ∫
‖T −y u − u‖ 2 L
2
ν(dy)
2 R n \{0}
= 1 ∫
∥ (T−y u)̂ − û ∥ 2 ν(dy)
2 R n L 2 \{0}
= 1 ∫ ( ∫
)
|û(ξ)e −iyξ − û(ξ)| 2 dξ ν(dy)
2 R n \{0} R n
= 1 ∫
)
(|û(ξ)|
2
∫R 2 |e −iyξ − 1| 2 ν(dy) dξ
n R n \{0}
∫
)
=
(|û(ξ)|
∫R 2 (1 − cos yξ) ν(dy) dξ
n R n \{0}
∫
= |û(ξ)| 2 ψ(ξ) dξ.
R n
If ψ(ξ) = f(|ξ| 2 ), we see from Lemma 2.1 that ν(dy) = m(|y| 2 ) dy with a
completely monotone density m(r) and
∫
|û(ξ)| 2 f(|ξ| 2 ) dξ = 1 ∫∫
|u(x + y) − u(x)| 2 m(|y| 2 ) dy dx. (16)
R 2
n R n ×R n
Up to notational changes, the following result is proved in Maz’ya and
Nagel [18].
Lemma 3.2. Let g j
functions such that
: [0, ∞] → [0, ∞], 1 j n, be monotone increasing
∫ 1
0
g j (s)
s
ds +
∫ ∞
1
g j (s)
s 2 ds < ∞. (17)
Then µ j (|t|) := ∫ |t|
0
n∑
( ∫∫
j=1
∫ ∞
g
r j (s)/s ds dr is well-defined and
( 1
|u(x + t j e j ) − u(x)| 2 g j
R 1 ×R t 2 n j
( ∫
n∑
∼ |û(ξ)| 2 µ j (|ξ j | 2 ) dξ
R n
j=1
) ) 1
dx dtj
2
|t j |
) 1
2
(18)
are equivalent seminorms on S(R n ).