Page 54 - Textos de Matemática Vol. 34

44 ANTO´NIO M. CAETANO
where M is the Hardy-Littlewood maximal function and we have used the maximal inequality of Fefferman-Stein (cf. [6, pp. 14-15]).
It is also now clear that the restriction of R to each Lpi (l2) is a bounded linear operator into hLpi .
Therefore S establishes a topological isomorphism from (hLp0 , hLp1 )θ,q onto the complemented subspace of (Lp0 (l2), Lp1 (l2))θ,q given by SR(Lp0 (l2), Lp1 (l2))θ,q, i.e., due to Proposition 3.2, onto the complemented subspace of Lpq(l2) given by SRLpq(l2).
From this it immediately follows that x→{λ(β,ν,m)χνm(x)}(β,ν,m)∈ILpq(l2) ≤c{λ(β,ν,m)}(β,ν,m)∈I(hLp0,hLp1)θ,q,
which proves the continuous embedding (hLp0,hLp1)θ,q →hLpq.
On the other hand, given {λ(β,ν,m)}(β,ν,m)∈I ∈ hLpq, one knows that x → {λ(β,ν,m)χνm(x)}(β,ν,m)∈I belongs to Lpq(l2), so that applying R one gets {λ(β,ν,m)}(β,ν,m)∈I ∈ (hLp0 , hLp1 )θ,q and the continuous embedding
hLpq →(hLp0,hLp1)θ,q.
4. Subatomic representations

Letr≥0andψ≥0beaC∞ functioninRn withsuppψ⊂{y∈Rn :|y|<2r} and m∈Zn ψ(x−m) = 1 for all x ∈ Rn. Given any β ∈ Nn0, define ψβ(x) := xβ ψ(x).
Consider, in what follows, that the real number ρ has been chosen greater than r and that Ψβ,ρ ∈ S has the same meaning as in the Introduction.
νm
Let also I stand for Nn0 × N0 × Zn and I′ := N0 × Zn.
Proposition 4.1. Let s > 0, 1 < p < ∞, 1 ≤ q ≤ ∞. If f ∈ HsLpq, then (4.1) f= λβνm(f)ψβ(2ν·−m),
≤ c ∥f∥HsLpq ,
(β,ν,m)∈I
summability in S′, where the λβ (f) := 2−ρ|β| f,Ψβ,ρ satisfy the relation
νm
ρ|β| 2νs β
νm
(4.2)
|λ(f)|ρ,s,p,q := sup 2 2 |λνm(f)χνm(·)|
21/2
β ∈Nn0 (ν,m)∈I ′ pq
for some c > 0 independent of the f considered.