1.2. DERIVATIVE OPERATOR: DEFINITION AND PROPERTIES 5
We denote by
Dk,p
the closure of S with respect to the seminorm · k,p. For any
k 1 and p q we have
Dk,p

Dk,q.
We set
D∞
=
∩k,pDk,p.
For p = 2, the space
D1,2
is a Hilbert space with the scalar product
F, G
1,2
= E(F G) + E( DF, DG
H
).
By Proposition 1.8 the operator D is well defined in the space D1,p. Also, by the
same arguments as in Proposition 1.8 we can show that the iterated derivative Dk
is closable from Lp(Ω) to Lp(Ω; H⊗k) for any p 1 and k 1. As a consequence,
Dk
is well defined in
Dk,p.
Remark 1.9. The above definitions can be exended to Hilbert-space-valued
random variables. That is, if V is a separable Hilbert space, then Dk,p(V ) is the
completion of the set SV of V -valued smooth and cylindrical random variables of
the form F =
∑m
j=1
Fj vj , Fj S, vj V , with respect to the seminorm
F
k,p,V
=

⎣E
( F
p
V
) +
k
j=1
E
Dj
F
p
H⊗j ⊗V
⎤1/p

.
The following result is the chain rule for the derivative operator.
Proposition 1.10. Let ϕ : Rm R be a continuously differentiable function
with bounded partial derivatives, and fix p 1. Suppose that F = (F 1, . . . , F m) is
a random vector whose components belong to the space D1,p. Then ϕ(F ) D1,p,
and
D(ϕ(F )) =
m
i=1
∂ϕ
∂xi
(F )DF
i.
Proof. Approximate the components of the random vector F by smooth and
cylindrical random variables in the norm of
Dk,p,
and the function ϕ by ϕ ψn,
where ψn is an approximation of the identity.
The following older inequality implies that
D∞
is closed under multiplication.
Proposition 1.11. Let F
Dk,p,
G
Dk,q
for k 1, 1 p, q and let r
be such that
p−1
+
q−1
=
r−1.
Then, F G
Dk,r
and
F G
k,r
cp,q,k F
k,p
G
k,q
.
Proof. If F, G S, then D(F G) = F DG + GDF . Hence,
D(F G)
H
|F|DG
H
+ |G|DF
H
.
Similarly
D2(F
G) = F
D2G
+ 2DF DG +
GD2F,
and
D2(F
G)
H⊗2
|F|
D2G
H⊗2
+ 2 DF DG
H⊗2
+ |G|
D2F
H⊗2
|F|
D2G
H⊗2
+ 2 DF
H
DG
H
+ |G|
D2F
H⊗2
.
In this way we obtain
k
j=0
Dj
(F G)
H⊗j
ck


k
j=0
Dj
F
H⊗j
⎞⎛
⎠⎝
k
j=0
Dj
G
H⊗j


.
Then, it suffices to use older’s inequality and an approximation argument.
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