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class
S
m
(x)
ρ,δ
C
∞
(
d
d
)
such that for all multi-indices
as the set of all
ψ(x,ξ)
∈
R
× R
d
0
α, β
∈ N
there exists a constant
C
α,β
>
0 with
D
x
D
ξ
ψ(x,ξ)
≤
C
α,β
d
d
+
ξ
i
)
(m
i
(x)
−
ρα
i
+
δ
|
β
|
)/
2
.
∀
x,ξ
∈ R
:
(
1
i
=
1
(16.12)
An anisotropic Sobolev space of variable order can now be defined using the
variable order Riesz potential
Λ
m
(x)
:=
i
=
1
(
1
m
(x)
with symbol
ψ(x,ξ)
=
ξ
+
ξ
i
)
2
m
i
(x)
,
m
i
(x
i
)
1
,...,d
. Clearly,
ψ(x,ξ)
is an element of
S
m
(x)
1
,δ
≥
0,
i
=
for
(
0
,
1
)
. The norm on
H
m
(x)
δ
∈
is given by
2
H
m
(x)
Λ
2
m
(x)
u
2
L
2
(
2
L
2
(
d
)
.
There is an alternative representation of the above space, when
m
(x)
admits the
following representation
m
(x)
u
:=
d
)
+
u
R
R
(m
1
(x
1
),...,m
d
(x
d
))
. This is very useful for the
proof of norm equivalences, which play a crucial role in wavelet discretization the-
ory, we refer to [137] for details. We consider the anisotropic Sobolev spaces
H
m
i
(x
i
)
i
=
of variable order
m
i
(x
i
)
in direction
x
i
, equipped with the following norms:
u
2
H
m(x)
i
:=
Λ
2
m
i
(x
i
)
i
2
L
2
(
R
2
L
2
(
R
u
d
)
+
u
d
)
,
where
Λ
m
i
(x
i
)
i
)
m
i
(x
i
)
. It then
is a pseudodifferential operator with symbol
(
1
+ |
ξ
i
|
follows by the elementary inequality
2
2
d
d
d
a
i
C
1
a
i
≤
≤
C
2
a
i
,
i
=
1
i
=
1
i
=
1
with
a
i
>
0 and
C
1
,
C
2
only dependent on
d
, that
d
2
H
m(x)
(
R
2
H
m
j
(x
j
)
j
u
d
)
∼
1
u
,
(16.13)
d
)
(
R
j
=
and therefore,
d
H
m
j
(x
j
)
j
H
m
(x)
(
R
d
)
=
d
).
(
R
(16.14)
j
=
1
=
i
=
1
(a
i
,b
i
)
d
, we define for a variable order
On the bounded set
G
=
(
a
,
b
)
⊂ R
m
(x)
,
a
≤
x
≤
b
the space
H
m
(x)
(G)
u
|
G
0
.
H
m
(x)
(
d
), u
:=
∈
R
|
R
G
=
u
d
\
This space coincides with the closure of
C
0
(G)
(the space of smooth functions
with support compactly contained in
G
) with respect to the norm
u
H
m
(x)
(G)
:=
u
H
m
(x)
(
R
d
)
,
(16.15)
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