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Functional Spaces 89
(5.6.8) u H s w (R) := u H [s]
w (R) +
R×R
|v(x) − v(y)| 2
|x − y| 1+2σ 1
2
dxdy , ∀u ∈ H s w(R),
where v := d[s]
dx [s] (u √ w) and σ := s − [s].
Depending on the context, we shall make use either of the representations (5.6.3) and
(5.6.8) or of the expression (5.6.6). In all three cases, H s w(R) turns out to be a complete
space.
We can easily check that any function of the form pe−x2, where p is a polynomial,
belongs to H s w(R), ∀s ≥ 0. At the same time, if u ∈ C k (R), k ∈ N, and u and its
derivatives of order less or equal to k decay faster than e −x2 /2 at infinity, then we obtain
u ∈ H k w(R). The converse of this statement leads to the Sobolev embedding theorem (see
triebel(1978), p.206), which can be stated as follows. For any k ∈ N, if u ∈ H s w(R),
with s > k + 1
2 , then u ∈ Ck (R). Furthermore, a constant K > 0 exists such that
(5.6.9) u √ w C k (R) ≤ K u H s w (R), ∀u ∈ H s w(R), s > k + 1
2 .
This result is significant since it relates abstract Sobolev spaces with the classical spaces
of continuous and differentiable functions. To be precise, the inclusion H s w(R) ⊂ C 0 (R),
for s > 1
2 , means that any class Cu ∈ H s w(R) contains a continuous representative u.
We also have (see funaro and kavian(1988))
(5.6.10) ρ µ u √ wC0 (R) ≤ K uHs w (R), ∀u ∈ H s w(R), s > 1
2 + µ and µ ≥ 0.
An interesting property is that any function in H s w(R), s ≥ 0, can be approximated in
the norm of H s w(R) by a sequence of functions in C ∞ 0 (R).
Finally, we mention the following inequality. Let σ,τ ≥ 0 and u ∈ H σ w(R) ∩ H τ w(R),
then we have u ∈ H s w(R), where s = (1−θ)σ+θτ, θ ∈ [0,1], and we can find a constant
K > 0 such that
(5.6.11) u H s w (R) ≤ K u 1−θ
H σ w (R) uθ H τ w (R), ∀u ∈ Hσ w(R) ∩ H τ w(R).