Page 55 - Textos de Matemática Vol. 34

SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES 45
Proof.Choose1<p0,p1 <∞,withp0 ̸=p1,and0<θ<1suchthat1 = p
1−θ + θ p0 p1
T :
β ∈Nn0
That this is well-defined comes from the subatomic decomposition for the spaces
HsLpi = Hps = Fps,2, i = 0,1 — cf. [9, 2.6, 2.9, 2.11, 2.12] — together with ii
our Proposition 2.1. Since from these references it also follows that the restric- tion of T to each HsLpi is bounded with values in l∞(Lpi(l2)), using real inter- polation we also get that T takes (HsLp0 , HsLp1 )θ,q linearly and boundedly into (l∞(Lp0 (l2)), l∞(Lp1 (l2)))θ,q. Now just conjugate this with Propositions 3.1, 3.2 and 3.3 in order to get (4.2). As to (4.1), it follows immediately from the corre- sponding result for the elements of HsLpi , i = 0, 1 — cf. [9, 2.6, 2.9, 2.11, 2.12] — and the fact that each f ∈ HsLpq can be written as f = f0 +f1, with f0 ∈ HsLp0, f1 ∈HsLp1.
Proposition4.2. Lets>0,1<p<∞,1≤q≤∞. If λ≡{λβνm}(β,ν,m)∈I ⊂C is such that
and consider the linear operator
HsLp0 + HsLp1 −→ l∞(Lp0 (l2)) + l∞(Lp1 (l2)) .
(4.3)
then
(4.4)
ρ|β| 2νs β 21/2 |λ|ρ,s,p,q := sup 2 2 |λνmχνm(·)|
β ∈Nn0 (ν,m)∈I ′ pq
f:= λβνmψβ(2ν·−m) (β,ν,m)∈I
< ∞,
f → x → {2νs f,Ψβ,ρχ (x)} ′ ν m ν m (ν,m)∈I
(summability in S′) belongs to HsLpq and there is some constant c > 0 (indepen- dent of f and λ) such that
(4.5) ∥f∥HsLpq ≤ c inf |λ|ρ,s,p,q,
where the infimum runs over all λ satisfying (4.3) and giving rise to the same f
according to (4.4).
Proof. Given λ according to the hypotheses and ε > 0, using summability prop- erties, properties of the decreasing rearrangement of functions and the fact that Lpq can be viewed as a Banach space continuously embedded in Lp0 + Lp1 , for some