Page 56 - Textos de Matemática Vol. 34

46 ANTO´NIO M. CAETANO 1<p0,p1 <∞,wehavethat

2(ρ−ε)|β| 2νs β 21/2
(β,ν,m)∈I ≤
21/2
2 2 2 |λν m χν m (·)| pq
2 2 |λν m χν m (·)| pq −ε|β| ρ|β| 2νs β
≤ c
≤ c
(ν,m)∈I ′
2 2 |λν m χν m (·)| pq
(ν,m)∈I ′ 2−ε|β| |λ|ρ,s,p,q < ∞ .
β ∈Nn0
−ε|β| ρ|β| 2νs β 21/2
β ∈Nn0 β∈Nn0
2
Assume, from now on, that ε > 0 has been chosen in such a way that ρ−ε > r. Choose1<p0,p1 <∞,withp0 ̸=p1,and0<θ<1suchthat 1 = 1−θ + θ
and consider the linear operator
U: hLp0+hLp1 −→ HsLp0+HsLp1 ,
{λβνm}(β,ν,m)∈I → 2−(ρ−ε)|β| 2−νsλβνm ψβ(2ν ·−m) (β,ν,m)∈I
where the convergence (summability) of the sum is considered in S′ (or in Lp0 +Lp1 ,
if one wishes some more precise information). That this is well-defined follows easily
from the subatomic representation for the spaces HsLpi = Hps = Fps ,2, i = 0, 1 ii
— cf. [9, 2.6, 2.7, 2.9]. Since these references also guarantee that the restriction of U to each hLpi is bounded with values in HsLpi, using real interpolation one gets that U takes (hLp0 , hLp1 )θ,q linearly and boundedly into (HsLp0 , HsLp1 )θ,q. Conjugating this with Propositions 3.3 and 3.4 and the first part of the present proof, it follows that f given by (4.4) is well-defined (with summability meant in S′, or even in Lp0 + Lp1 ) and (4.5) holds true.
Remark 4.1. We would like to stress that the summability implied by (4.4) in S′ is not an assumption, but rather a consequence of (4.3). Also that the summability can even be taken in Lp0 + Lp1 , for suitable 1 < p0, p1 < ∞ according to the proof given above.
Now the theorem on the subatomic representation for spaces HsLpq which follows is a simple corollary of the two preceding propositions:
Theorem4.3. Lets>0,1<p<∞,1≤q≤∞. Thenf∈S′ belongstoHsLpq if, and only if, it can be represented by
(4.6) f= λβνmψβ(2ν·−m) (β,ν,m)∈I
p p0 p1