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Functional Spaces 87
The use of ρ(t) in place of t will be understood later on. Now, due to (5.5.8) and a
proposition proven in escobedo and kavian(1987), if w(x) = ex2, x ∈ R, one can
deduce that
(5.5.9) u ∈ L 2 w(R), u ′ ∈ L 2 w(R) ⇐⇒ ρF(u √ w) ∈ L 2 (R;C).
More generally, for any integer k ≥ 1, when u is differentiable k times, we have
(5.5.10)
d m u
dx m ∈ L2 w(R), 0 ≤ m ≤ k ⇐⇒ ρ k F(u √ w) ∈ L 2 (R;C).
5.6 Sobolev spaces in R
Let k ≥ 1 be an integer. We introduce the weighted Sobolev space of order k in R as
follows (see adams(1975), bergh and löfström(1976), triebel(1978)):
(5.6.1) H k
w(R) :=
u differentiable k times and
Cu
dmu dxm ∈ L2
w(R), for 0 ≤ m ≤ k .
The derivative is intended in the weak sense and dm u
dx m = u when m = 0. We also
set H 0 w(R) := L 2 w(R). Recalling (5.2.3) and (5.2.4), an inner product and a norm are
introduced in H k w(R), k ∈ N, namely
(5.6.2) (u,v) H k w (R) :=
(5.6.3) u H k w (R) :=
k
m=0
k
dmu dxm
m=0
m d u
dxm , dmv dxm
L2 w (R)
, ∀u,v ∈ H k w(R),
2
L2 w (R)
1
2
, ∀u ∈ H k w(R).
With this norm, H k w(R), k ∈ N, is a Hilbert space. The higher is the index k and the
smoother the functions belonging to H k w(R) will be.