Page 49 - Textos de Matemática Vol. 34

SUBATOMIC REPRESENTATION OF BESSEL POTENTIAL SPACES 39
elements of M(Rn,E) as functions (in corresponding equivalence classes), and even functions which are only defined a.e. from Rn into E (and strongly measurable in their domains of definition) can be thought of as elements of M(Rn,E); the important thing is that the equivalence classes they point to in M(Rn,E) are clearly identified.
For 1 ≤ p < ∞, the Lebesgue space Lp(Rn, E) is defined as the linear subspace of M (Rn , E ) of equivalence classes containing at least a (strongly measurable) function f : Rn → E such that |f(·)|pE is (Lebesgue) integrable. Endowed with
p 1/p
|f(x)|E dx
it is a Banach space continuously embedded in M(Rn,E). The space L∞(Rn,E) is defined as the linear subspace of M(Rn,E) of equivalence classes containing at least a (strongly measurable) essentially bounded function f. Endowed with ∥f∥∞ := esssupx∈Rn|f(x)|E, it is also a Banach space continuously embedded in M(Rn,E).
For any f ∈ M(Rn,E), the decreasing rearrangement of f is defined (with possibly infinite values) by
where
Note that f∗ = |f(·)|∗E.
∥f∥p :=
Rn
f∗(t):=inf{σ>0:ρ(f,σ)≤t}, t>0, ρ(f,σ):=|{x∈Rn :|f(x)|E >σ}|, σ>0.
Definition 2.1. Let E be a Banach space, 1 < p < ∞, 1 ≤ q < ∞. The Lorentz space Lpq(Rn,E) is defined as
∞ dt1/q Lpq(Rn, E) := {f ∈ M(Rn, E) : ∥f∥pq := (t1/pf∗(t))q
t>0
With the expressions ∥·∥pq, these are in general only quasi-Banach spaces, but since there are norms equivalent to ∥·∥pq (see, for example, next section on interpolation), one can say that these Lorentz spaces are also Banach spaces. Moreover, it follows from properties of the decreasing rearrangement that Lpp(Rn,E) = Lp(Rn,E), 1<p<∞,with∥·∥pp =∥·∥p.
It is clear that if f ∈ Lpq(Rn, E) then also |f(·)|E ∈ Lpq(Rn, C) (or Lpq(Rn, R)). In the sequel we shall also need a partial converse of these result:
< ∞};
0t Lp∞(Rn, E) := {f ∈ M(Rn, E) : ∥f∥p∞ := sup t1/pf∗(t) < ∞}.
the Lorentz space Lp∞(Rn,E) is defined by