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is finite. That is,
d
n
(G)
H
n
i
(I ),
H
=
(13.3)
i
=
1
s
(G)
by interpolation. For later
purpose, we additionally introduce anisotropic Sobolev spaces, i.e. spaces which
consist of functions with different smoothness in different coordinate directions.
Recall the definition of
H
s
(
d
≥
where
I
:= [
0
,
1
]
. For arbitrary
s
∈ R
0
, we define
H
2
H
s
(
R
)
for
s
≥
0 via the Fourier transform
u
)
:=
R
2
d
ξ
. Naturally, this extends to
isotropic
Sobolev spaces of mul-
tivariate functions via
)
2
s
(
1
+|
ξ
|
|
u(ξ)
|
R
1
+|
ξ
|
2
s
2
H
s
(
R
2
d
ξ,
u
d
)
:=
|
u(ξ )
|
R
d
(
j
=
1
x
j
)
1
/
2
. Similarly, for a multi-index
s
d
≥
where
|
ξ
|=
∈ R
0
, we can define
anisotropic
Sobolev spaces
H
s
(
R
d
)
with norm
d
2
H
s
(
R
+
ξ
j
)
s
j
2
d
ξ.
u
d
)
:=
(
1
|
u(ξ )
|
R
d
j
=
1
It is useful to notice that by [129, Sect. 9.2] the spaces
H
s
(
d
)
admit an intersection
R
structure, and we have
d
d
H
s
j
j
H
s
(
R
d
)
=
d
)
2
H
s
(
R
+
ξ
j
)
s
j
/
2
2
L
2
(
R
(
R
and
u
d
)
∼
1
(
1
u
d
)
.
(13.4)
j
=
1
j
=
Similarly to the one dimensional case, we finally define the space
H
s
(G)
:=
u
0
.
H
s
(
d
), u
|
G
:
u
∈
R
|
R
G
=
(13.5)
d
\
1
(G)
is different from the space
H
1
(G)
:
u
Note that, for example, the space
H
∈
1
(G)
implies
∂
x
1
···
L
2
(G)
,but
u
H
1
(G)
is equivalent to
u, ∂
x
1
u,...,
H
∂
x
d
u
∈
∈
L
2
(G)
. The following holds for
s
∂
x
d
u
∈
≥
0:
H
s
(G)
H
s
(I )
L
2
(I )
L
2
(I )
L
2
(I )
L
2
(I )
H
s
(I ).
=
⊗
⊗···⊗
∩···∩
⊗
⊗···⊗
L
2
(G)
, we have as a consequence of (12.1) and (
13.1
) the series
For a function
u
∈
representation
∞
u
=
u
,
k
ψ
,
k
.
(13.6)
i
=
0
k
i
∈∇
i
Using the norm equivalences (12.3) and the underlying tensor product structure
(
13.3
), we obtain
∞
2
H
(
2
2
s
1
1
+···+
2
s
d
d
)
2
2
H
s
(G)
|
u
,
k
|
u
u
s
(G)
,
(13.7)
i
=
0
k
i
∈∇
i
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