Page 96 - Textos de Matemática Vol. 34

86 JU´LIO S. NEVES
The next result extends [7, Corollary 4.6 & Remark 4.7], concerning the loga- rithmic Bessel potential space, and [20, Proposition 5.6].
Proposition 5.4. Let q ∈ [1, +∞) and let b ∈ SV (0, +∞). Suppose that either
(5.18)
or
(5.19)
Then
Proof. It follows the proof of [20, Proposition 5.6]. The following theorem, which is a refinement of [15, Theorem 5.7.8 (i)], improves
q′
σ∈(0,n), p=n and ∥t−1 [b(t)]−1∥′ <+∞
σ q ;(0,1)
σ ∈ (0,+∞) and max{1, n} < p < +∞.
σ HσLp,q;b(Rn) → CB(Rn).
[7, Theorem 4.9] and [20, Theorem 5.10].
Theorem 5.5. ([11, Theorem 3.1 (i)]) Let σ ∈ [1,n + 1), max{1, n} < p < n ,
σ σ−1 q ∈ (1,+∞), r ∈ [q,+∞] and let b ∈ SV(0,+∞). Let λ : (0,1] → (0,+∞) be
defined by
(5.20) λ(t) = tσ−n/p[b(tn)]−1, t ∈ (0, 1].
(Note that λ ∈ Lr for any r ∈ [1, +∞].) Then
(5.21) HσL (Rn) → Λλ(.) (Rn). p,q;b ∞,r
The next result is a refinement of [15, Theorem 5.7.8 (iii)] and extends [7, The- orem 4.13] and [20, Theorem 5.12].
Theorem 5.6. Let 1 ≤ q < +∞ and let b ∈ SV (0, +∞). Suppose that either
n − 1 ′ −1
<+∞
σ∈(1,n+1), p=
σ∈(1,+∞) and max{1, n }<p<+∞.
(5.22)
or
(5.23)
and t q [b(t)] σ−1 q′;(0,1)
σ−1 Then for any u ∈ HσLp,q;b(Rn) and all x,y ∈ Rn,
(5.24) |u(x) − u(y)| ∥u∥σ;p,q;b |x − y|. Moreover,
(5.25) HσLp,q;b(Rn) → Lip(Rn). Proof. It follows the proof of [20, Theorem 5.10].