32.6 Another Approach
The above approach is pretty interesting, but there is a quicker way to do it discussed in
this section. I am also including the case where the operator B is actually a function
of t. I have never had a reason to use this level of generality, but it is here if
it is of any interest. Also, this is presented in the context of complex Banach
spaces. In addition, it is shown that by including i∗ in various formulas, you don’t
need to have V dense in W. Of course, this is typically not of any interest, but
for the sake of generality, it is included. The approach is due to Lions. It is
assumed for convenience that p ≥ 2. This was apparently not needed in the last
section. It may be that this approach can also be generalized to not require
where B′∈ L∞
is a Banach space such that V ⊆ W.
VI ≡ Lp
WI ≡ L2
Now let I = [a,b] and c < a < b < d.Here and in what follows ϕn
0,ϕ ∈ C0∞
The following proposition is known and the
essential features of its proof may be found in [?]
. We give a proof for the convenience of
Proposition 32.6.1 Suppose D
). Suppose also
For u ∈WI and a − n−1 > c,b + n−1 < d, define
where we let u = 0 off I. Then
Proof: First, we show that
is uniformly bounded. Letting
= 0 off
Where C is a positive constant independent of n and u. Thus
Next let u ∈ C0∞
a dense subset of WI.
Then a little computation
Since u is bounded,
By Holder’s inequality, this is no larger than
If t is a Lebesgue point,
so the dominated convergence theorem implies
and so Tnu → 0 for all u in the dense subset, C0∞
We have also the following simple corollary.
Corollary 32.6.2 In the situation of Proposition 32.6.1,
where i is the inclusion map of V into W.
For f ∈ L1
in the sense of V ′
valued distributions as follows. For
ϕ ∈ C0∞
We say f′∈ L1
if there exists
g ∈ L1
necessarily unique, such that for
all ϕ ∈ C0∞
To save on notation, we let V≡V[0,T] and W≡W[0,T]. Define
Note that for u ∈ D
it is automatically the case that i∗Bu ∈V′
Lemma 32.6.3 L is a closed operator.
Then X is isometric to a closed subspace of a product of reflexive Banach spaces and so
X is reflexive by Lemma 15.5.11.
Theorem 32.6.4 Let p ≥ 2 in what follows. For u,v ∈ X, the following hold.
- t → 〈B
〉W′,W equals an absolutely continuous function a.e.,
denoted by 〈Bu,v〉
X for some C > 0 and for all t ∈ [0,T].
- t → B
equals a function in C
a.e., denoted by Bu
}≤ C||u||X for some C > 0.
If K : X → X′ is given by
- K is linear, continuous and weakly continuous.
- Re〈Ku,u〉 = [
- If Bu = 0
, for u ∈ X, there exists un → u in X such that un
is 0 near 0. A
similar conclusion could be deduced at T if Bu = 0
Proof: For h a function defined on [0,T], let h1 be even, 2T periodic, and
= 1 on
t ∈ ℝ
Now let u ∈ X. Then
Thus, if I ⊇ [−T,2T], then
Defining un ≡ũ ∗ ϕn,
then for a.e.
From 32.6.46 and Proposition 32.6.1, the following holds in V
Where the second equality follows from Corollary 32.6.2
, the third follows from the
pointwise a.e. equality of
, while the fourth follows
and standard properties of convolutions.
By choosing a subsequence we can use 32.6.48 to obtain