Page 97 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 87
The next theorem extends [19, Theorem 3.1]. It also improves [20, Theorem 5.16]
and the result of Edmunds, Gurka and Opic [7, Theorem 4.11]. It refines as well a
result due to Triebel [28, Theorem 14.2 (ii)] (but in the context of Bessel potential
spaces rather than the Triebel-Lizorkin spaces), itself an improvement of Br´ezis
and Wainger super-limiting case result about the “almost Lipschitz continuity” of
1+n/p n
elements of Hp (R ), cf. [4, Corollary 5].
Theorem 5.7. ([11, Theorem 3.2 (i)]) Let σ ∈ (1,n + 1), p = n/(σ − 1), q ∈ (1, +∞), r ∈ [q, +∞] and let b ∈ SV (0, +∞) be such that
(5.26) ∥ t−1/q′ [b(t)]−1∥q′;(0,1) = +∞. Suppose that λr ∈ Lr is defined by
(5.27)
Then
′
λr(t) = t [b(tn)]q /r
2 tn
′ 1/q′+1/r τ−1[b(τ)]−q dτ
, t ∈ (0,1].
HσL
(Rn) → Λλr(.)(Rn). n/(σ−1),q;b ∞,r
(5.28)
Proof. By Lemma 4.1 and by Proposition 5.4, it is enough to prove
∥u|Λλr(.)(Rn)∥ ∥u∥ for all u ∈ S(Rn). ∞,r σ;n/(σ−1),q;b
Let u ∈ S(Rn) ⊂ HσLn/(σ−1),q;b(Rn). Then, by Lemma 4.2, we have ∂u ∈ ∂xi
Hσ−1Ln/(σ−1),q;b(Rn), for i = 1, . . . , n. Now, by Theorem 5.3, with σ − 1 instead of σ, we have
∂ u
∂x , i=1,...,n,
∂ u ∂x
i ∞,r;br;(0,1)
where br ∈ SV (0, +∞) and satisfies (5.10). Again, by Lemma 4.2,
(5.29) n ∂u ∂ x
i=1 i ∞,r;br :(0,1)
Now, from (5.27), the estimate (cf. [14, Proposition 5.12 (i)])
t 0
iσ−1;n/(σ−1),q;b
n ∂u ∥u∥σ;n/(σ−1),q;b. ∂ x
(5.30) ω(u, t)
|∇u|∗(σn) dσ, t > 0,
i=1 i σ−1;n/(σ−1),q;b