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there holds
D
α
u
2
L
w
(
[
d
|
D
α
u(x)
|
2
w
2
(x)
d
x
d
)
:=
0
,
1
]
[
0
,
1
]
d
2
2
,
α
w
i
(
2
−
i
k
i
)
2
.
|
u
,
k
|
i
≥
0
k
i
∈∇
i
i
=
1
Proof
Let
α
=
(
0
,...,
0
)
.Asin[19], for 1
≤
i
≤
d
,let
1
0
w
i
(x
i
)ψ
i
,k
i
(x
i
)ψ
i
,k
i
(x
i
)
d
x
i
w
i
(
2
−
i
k
i
)w
i
(
2
−
i
k
i
)
M
i
(
i
,k
i
),(
i
,k
i
)
=
(
i
,k
i
),(
i
,k
i
)
ψ
i
,k
i
,ψ
i
,k
i
w
i
:=
.
(
i
,k
i
),(
i
,k
i
)
We estimate by Cauchy-Schwarz
2
L
w
(
d
)
=
u
u
,
k
u
,
k
[
0
,
1
]
1
,...,
d
1
,...,
d
k
i
∈∇
i
k
i
∈∇
i
d
w
i
(
2
−
i
k
i
)w
i
(
2
−
i
k
i
)
ψ
i
,k
i
,ψ
i
,k
i
w
i
×
i
=
1
d
M
d
2
1
,...,
d
w
i
(
2
−
i
k
i
)
2
.
≤
M
1
⊗···⊗
|
u
,
k
|
k
i
≤∇
i
i
=
1
M
d
2
≤
i
=
1
M
1
⊗···⊗
M
i
2
and
M
i
2
≤
Since
c
i
by [19, Theorem 3.2], this
shows the upper estimate. The lower estimate follows by a duality argument (see the
proof of [19, Theorem 3.3] for
d
=
1). Now let
α
be such that
|
α
|
∞
=
1. Then the
claim follows by the same arguments as for the case
α
=
(
0
,...,
0
)
and by (
15.25
)
with
j
=
1.
For most stochastic volatility models under consideration, the function spaces
V
for which the corresponding bilinear form is continuous and satisfies a Gårding
inequality are weighted Sobolev spaces. In particular, these spaces are equipped
with a norm of the form
2
D
α
v
2
v
V
:=
1
L
w
α
(G)
,
(15.26)
|
α
|
≤
1
:=
i
=
1
w
i
(x
i
)
where the weight
w
α
d
depending on
α
is given by
w
α
(x)
: R
→ R
+
for univariate weights
w
i
1
)
weights
w
i
: R → R
+
. We assume that all the
d(d
+
satisfy Assumption
15.4.1
(iii).
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