Page 89 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 79
3. Slowly varying functions and Lorentz-Karamata spaces
Following [12], we say that a positive and Lebesgue-measurable function b is slowly varying on (0, +∞), and we write b ∈ SV (0, +∞), if, for each ε > 0, tεb(t) is equivalent to a non-decreasing function on (0,+∞) and t−εb(t) is equivalent to a non-increasing function on (0, +∞). If the role of the interval (0, +∞) is replaced by the interval (0, 1), we obtain the class SV (0, 1).
Properties and examples of slowly varying functions can be found in [32, p. 186], [3], [9], [16], [18] and [12]. The following functions belong to SV (0, +∞):
(i) b(t) = lα(t), α ∈ Rm;
(ii) b(t) = lα(t)χ(0,1)(t) + lβ(t)χ[1,+∞)(t), α, β ∈ Rm; (iii) b(t) = exp(|logt|α), 0 < α < 1;
(iv) bm(t)=exp(lαm(t)),0<α<1andm∈N.
It can be shown (cf. [12]) that any b ∈ SV(0,+∞) is equivalent to a b ∈ SV (0, +∞) which is continuous in (0, +∞).
Let p, q ∈ (0, +∞] and b ∈ SV (0, +∞). The Lorentz-Karamata (LK) space Lp,q;b(Ω) is defined to be the set of all functions f ∈ M(Ω) such that
(3.1) ∥f∥p,q;b;Ω := ∥t1/p−1/q b(t) f∗(t)∥q;(0,+∞)
is finite. Here ∥.∥q;(0,+∞) stands for the usual Lq (quasi-)norm over the interval (0, +∞).
When 0 < p < +∞, the Lorentz-Karamata space Lp,q;b(Ω) contains the charac- teristic function of every measurable subset of Ω with finite measure and hence, by linearity, every μn-simple function. When p = +∞, the Lorentz-Karamata space Lp,q;b(Ω) is different from the trivial space if, and only if,
∥t1/p−1/qb(t)∥q;(0,1) <+∞.
If m ∈ N, α = (α1,...,αm) ∈ Rm and b = lα, then Lp,q;b(Ω) is precisely the generalised Lorentz-Zygmund (GLZ) space Lp,q;α(Ω) introduced in [7] and endowed with the (quasi-)norm ∥f∥p,q;α;Ω. When α = (0,...,0), we obtain the Lorentz space Lp,q(Ω) endowed with the (quasi-)norm ∥.∥p,q;Ω, which is just the Lebesgue space Lp(Ω) equipped with the (quasi-)norm ∥.∥p;Ω when p = q; if p = q and m = 1, we obtain the Zygmund space Lp(log L)α1 (Ω) endowed with the (quasi-) norm ∥.∥p;α1;Ω.
We shall need weighted Hardy inequalities, where the weights are slowly varying functions. The proofs simply involve checking well-known criteria (cf. e. g. [22, Theorems 5.9 & 5.10 & 6.2. & 6.3.]).