Page 50 - Textos de Matemática Vol. 34

40 ANTO´NIO M. CAETANO
Proposition 2.1. Let I be a countable set and lr := lr(I,C) be the Banach space of complex-valued r-summable families indexed by I, with 1 ≤ r < ∞ (see end of sectionforaprecisedefinition). Let1<p<∞,1≤q≤∞. If
f:Rn −→lr
x → {fi (x)}i∈I
(possibly defined only a.e.) is such that |f(·)|r ∈ Lpq(Rn,C) and fi ∈ M(Rn,C) for every i ∈ I, then f ∈ Lpq(Rn, lr).
Proof. All that remains to prove is that f ∈ M(Rn,lr), because then we will have ∥f∥pq = ∥ |f(·)|r ∥pq < ∞.
By hypothesis, each fi is strongly measurable, so the same happens with the composition with the natural embedding in lr and, afterwards, with any finite sum of such compositions. Since, also by hypothesis, (i∈I |fi(·)|r)1/r ∈ M(Rn,C), so in particular the infinite sum i∈I |fi(·)|r is a.e. pointwise convergent, then f is a.e. the pointwise limit of the above considered finite sums, and therefore is also strongly measurable.
It is clear that the above result also holds with R in place of C. However, from now on, we shall assume that all vector spaces are complex ones.
Since our function spaces will always be of “functions” defined on Rn, from now on we shall omit Rn from the notation. We shall also omit the Banach space E when E = C. Therefore, Lpq(E) will stand for Lpq(Rn,E) and Lpq(Rn,C) will be simply written as Lpq.
We shall also need to deal with the Banach “sequence” spaces lr(I,E) of E- valued r-summable (if 1 ≤ r < ∞) or bounded (if r = ∞) families indexed by the countable set I and equipped with the norm | · |r :
r1/r
|ai|E <∞ ,
the notation if it is clear from the context. We shall also omit E when E = C.
lr(I,E):= a≡{ai}i∈I ⊂E :|a|r :=
where |a|∞ must be interpreted as supi∈I |ai|E. Usually the I will be omitted from
3. Interpolation spaces
Recall that two Banach spaces A and B are said to form an interpolation couple {A,B} if they are both continuously embedded in the same Hausdorff topological vector space.
As far as methods of interpolation are concerned, we deal only with the real method (·,·)θ,q, where 0 < θ < 1 and 1 ≤ q ≤ ∞ (see [7], for example).
We begin with a somewhat abstract result, dealing with the spaces of type l∞(I, E) introduced before.
i∈I