Page 91 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 81
this is identical with that defined in (3.1) except f∗ has been replaced by f∗∗. The following result extends [7, Lemma 3.2] and [18, Lemma 3.5].
Lemma 3.4. ([12, (6.9)]) Suppose 1 < p ≤ +∞, 1 ≤ q ≤ +∞ and let b ∈ SV(0,+∞). Then
(3.11) ∥f ∥p,q;b;R ≤ ∥f ∥(p,q;b);Ω ∥f ∥p,q;b;R for all f ∈ M(Ω).
In particular, the Lorentz-Karamata (LK) space Lp,q;b(Ω) consists of all those func-
tions f for which ∥f∥(p,q;b);Ω is finite.
Proof. The first inequality follows immediately since f∗ ≤ f∗∗ for all f ∈ M(Ω).
Asforthesecond,sincep>1,weseefromLemma3.1(i),withν=1 −1,that p
∥f∥(p,q;b);Ω
11t = tp −1−q b(t) f∗(s) ds
0 q;(0,+∞) ∥t1−1+1 b(t)f∗(t)∥ =∥f∥
p q′ q;(0,+∞) p,q;b;Ω
.
The next lemma extends [7, Lemma 3.3] and [18, Lemma 3.6]. We refer as well to ([12, (2.6) & (5.7)]) for a more general result.
Lemma 3.5. Let p ∈ (1,+∞), q ∈ [1,+∞] or p = q = +∞, and let b ∈ SV (0, +∞). Then there exists a positive constant c = c(p, q, b) such that, for every f ∈ Lp,q;b(Ω) and all t > 0,
t−1 p
f∗(t) ≤ f∗∗(t) ≤ c b(t) ∥f∥p,q;b;Ω.
Proof. The first inequality is obvious. To prove the second inequality, we use the
fact
(3.12) ∥f∥ ≈ ∥t1 −1 b(t) f∗∗(t)∥ ,

p,q;b;Ω p q q;(0,+∞)
according to Lemma 3.4. If q = +∞ the result follows immediately. If q ∈ [1, +∞),
then (3.12) and [12, Proposition 2.2 (iv)] give
∥f∥p,q;b;Ω f
∗∗
(t)
t 1 − 1 q sp q b(s)
0
1/q ds
≈ f∗∗(t)t1 b(t) for all t > 0, and the result now follows.
p