Page 53 - Textos de Matemática Vol. 34

SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES 43
Proof. We use again the method of retraction and co-retraction (cf. [7, p. 22]). We show that
S: hLp0 +hLp1 −→ Lp0(l2)+Lp1(l2) {λ(β,ν,m)}(β,ν,m)∈I → x → {λ(β,ν,m)χνm(x)}(β,ν,m)∈I
is a co-retraction from hLpi into Lpi (l2 ), i = 0, 1, with R: Lp0(l2)+Lp1(l2) −→ hLp0 +hLp1
x → {a(β,ν,m)(x)}(β,ν,m)∈I → 2νn

χνm
a(β,ν,m)(y) dy
(β,ν,m)∈I
a corresponding retraction from Lpi (l2) into hLpi , i = 0, 1.
Using the equivalent characterization of each hLpi , given immediately after its
definition, it is straightforward to see that S is well-defined, it is linear, its restriction to each hLpi is a bounded linear operator into Lpi(l2) and, assuming R is well- -defined, that the restriction of RS to each hLpi is the identity operator.
As to R, it is clear that it will be a linear well-defined operator if one can prove that its restriction to each Lpi (l2) is well-defined taking this space into the corresponding hLpi . We show this next:
Given any x → {a(β,ν,m)(x)}(β,ν,m)∈I ∈ Lpi (l2), it is clear, since pi is assumed
greater than 1, that, for each (β,ν,m) ∈ I, a(β,ν,m) is a (strongly) measurable lo-
cally integrable function, so that the expression 2νn a(β,ν,m)(y) dy makes sense. χνm
Furthermore, it is also clear that x → 2νn a(β,ν,m)(y)dy χνm(x) χνm (β,ν,m)∈I 2
is measurable in the sense of a function with values in the extended real number system. Therefore
Rx → {a(β,ν,m)}(β,ν,m)∈I hLpi
=
≤

(β,ν,m)∈I
νn
21/2 a(β,ν,m)(y) dy χνm(·) pi
≤
≤ c
(β,ν,m)∈I
χνm
(β,ν,m)∈I
2
νn
χνm
21/2 |a(β,ν,m)(·)| pi
(β,ν,m)∈I
= cx → {a(β,ν,m)(x)}(β,ν,m)∈ILpi(l2)
21/2 pi
|a(β,ν,m)(y)| dy χνm(·) 21/2
2
|(M a(β ,ν,m) )(·)| pi
< ∞,