Page 52 - Textos de Matemática Vol. 34

42 ANTO´NIO M. CAETANO
is obviously a co-retraction from HsLpi into Lpi , i = 0, 1, with S−1 a correspond- ing retraction from Lpi into HsLpi , i = 0, 1. Therefore, S also establishes a topo- logical isomorphism from (HsLp0 , HsLp1 )θ,q onto the complemented subspace of (Lp0,Lp1)θ,q givenbySS−1(Lp0,Lp1)θ,q,thatis,by(Lp0,Lp1)θ,q itself,which,by the preceding proposition, equals Lpq. That is, f ∈ S′ is in (HsLp0 , HsLp1 )θ,q if, and only if, Sf ∈ Lpq and, moreover, ∥f∥(HsLp0 ,HsLp1 )θ,q is equivalent to ∥Sf∥pq. This finishes our proof.
Let now, for each ν ∈ N0 and m ∈ Zn, Qνm denote the cube in Rn, with sides parallel to the coordinate axes, with centre 2−νm and side length 2−ν. Denote by χν m the characteristic function of Qν m . Let also, from now on, I stand for N n0 × N 0 × Z n .
Definition 3.2. Given 1 < p < ∞, define hLp as the vector space
hLp := λ ≡ {λ(β,ν,m)}(β,ν,m)∈I ⊂ C :
21/2
(3.1) ∥λ∥hLp := endowed with the norm ∥ · ∥hLp .
|λ(β,ν,m)χνm(·)| < ∞ p
(β,ν,m)∈I
Note that the convergence of the sum in (3.1) is considered pointwise (a.e. is enough, of course). Note also that, due to Proposition 2.1, hLp can be equiva- lently described as the space of all λ ≡ {λ(β,ν,m)}(β,ν,m)∈I ⊂ C such that x → {λ(β,ν,m)χνm(x)}(β,ν,m)∈I belongs to Lp(l2). One can take advantage of this char- acterization in order to get an easy proof that hLp is a Banach space.
We would like to remark that, given 1 < p0, p1 < ∞, {hLp0 , hLp1 } is an interpo- lation couple, as both spaces are continuously embedded in (in obvious notation) the Hausdorff topological vector space h(Lp0 + Lp1 ).
AnalogouslytohLp,onecanconsiderhLpq (1<p<∞,1≤q≤∞): inthe definition above one just has to use ∥ · ∥pq instead of ∥ · ∥p and ∥ · ∥hLpq in place of ∥ · ∥hLp , though now ∥ · ∥hLpq is generally only a quasi-norm. However, as for the big H spaces, hLpq can also be considered as a Banach space, for an equivalent norm (see, for example, the proposition below).
Note that, as for hLp, due to Proposition 2.1, hLpq can be equivalently described as the space of all λ ≡ {λ(β,ν,m)}(β,ν,m)∈I ⊂ C such that x → {λ(β,ν,m)χνm(x)}(β,ν,m)∈I belongs to Lpq(l2).
Proposition 3.4. Given numbers 1 < p0,p1 < ∞, with p0 ̸= p1, 1 ≤ q ≤ ∞, 0 < θ < 1 and 1 = 1−θ + θ , the equality
hLpq =(hLp0,hLp1)θ,q holds (with equivalence of (quasi-)norms).
p p0 p1