Page 51 - Textos de Matemática Vol. 34

SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES 41
Proposition 3.1. Given an interpolation couple {A, B} and 0 < θ < 1, 1 ≤ q ≤ ∞, {l∞(A),l∞(B)} is also an interpolation couple and the continuous embedding
(l∞(A),l∞(B))θ,q →l∞((A,B)θ,q)
holds.
The proof is straightforward, so we shall omit it.
The Lorentz spaces introduced in the preceding section can be seen as interpo- lation spaces of Lebesgue spaces:
Proposition 3.2. Given a Banach space E and numbers 1 < p0,p1 < ∞, with p0 ̸= p1, 1 ≤ q ≤ ∞, 0 < θ < 1 and 1 = 1−θ + θ , the equality
p p0 p1 Lpq(E) = (Lp0 (E), Lp1 (E))θ,q
holds (with equivalence of (quasi-)norms).
A proof of this result in a somewhat different framework can be seen in [7, pp. 133-134]; see also [1, pp. 109-110].
Recall that HsLp stands for Hps, the Bessel potential space of the tempered 2 s/2 ˆ
distributions f such that ∥f∥HsLp := ∥((1+|x| ) f)ˇ∥p < ∞, s ∈ R, 1 < p < ∞. With the norm ∥ · ∥HsLp , it becomes a Banach space.
Definition 3.1. We define HsLpq, the Bessel potential space modelled on a Lorentz space, as
s ′ 2 s/2 ˆ
H Lpq :={f ∈S :∥f∥HsLpq :=∥((1+|x| ) f)ˇ∥pq <∞},
s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞, quasi-normed by ∥ · ∥HsLpq .
Though the quasi-norm above is not, in general, a norm, HsLpq can still be considered as a Banach space, for an equivalent norm. This can, for example, be seen as a consequence of the next result.
Proposition3.3. Givennumberss∈R,1<p0,p1 <∞,withp0 ̸=p1,1≤q≤∞, 0 < θ < 1 and 1 = 1−θ + θ , the equality
p p0 p1
HsLpq =(HsLp0,HsLp1)θ,q
holds (with equivalence of (quasi-)norms).
Proof. We use the method of retraction and co-retraction (cf. [7, p. 22]). The operator
S:S′ −→S′
f → ((1+|x|) f)ˇ
2 s/2 ˆ