Page 94 - Textos de Matemática Vol. 34

84 JU´LIO S. NEVES
We have by O’Neil’s inequality (cf. e.g. [31, Lemma 1.8.8]) that
(5.4) u∗(t)≤u∗∗(t)≤tg∗∗(t)f∗∗(t)+
The estimates (5.2) and (5.4) yield, for every t > 0, t +∞
(5.5) u∗(t) tσ/n−1 f∗(τ)dτ + 0t
Now, from (5.5) we obtain (5.6)
where
N1 = N2 =
∥u∥pσ,r;b N1 +N2, 11t
t p −1− r b(t) f ∗ (τ ) dτ
0 r;(0,+∞)
;
+∞ σσ
t
g∗(τ)f∗(τ)dτ foreach t>0.
τσ/n−1f∗(τ)dτ.
1σ1+∞ tp −n −r b(t) τσ/n−1f∗(τ)dτ
.
t Applying Lemma 3.1 (i), we have
11 (5.7) N1 tp −q b(t)f∗(t)
q;(0,+∞)
Finally, Lemma 3.1 (ii) gives
r;(0,+∞)
= ∥f∥p,q;b.
11 (5.8) N2 tp −q b(t)f∗(t)
= ∥f∥p,q;b. The result now follows from inequalities (5.6)-(5.8).
The next result concerns the limiting case p = n/σ and extends [6, Theorem 3.2 and Theorem 6.6], [7, Theorem 4.3] and [18, Theorem 5.2].
Theorem 5.3. ([10, Theorem 3.2 (i)]) Let σ ∈ (0,n), p = n/σ, q ∈ (1,+∞], r ∈ [q, +∞] and let b ∈ SV (0, +∞) be such that
(5.9) ∥ t−1/q′ [b(t)]−1∥q′;(0,1) = +∞.
Suppose that Ω ⊂ Rn is a nonempty domain with |Ω|n < +∞ and that br ∈
SV (0, +∞) satisfies
(5.10) br(t) ≈ [b(t)]−q /r τ−1[b(τ)]−q dτ
′ 2 ′ −1/q′−1/r t
for all
t ∈ (0,1].
Then
(5.11) HσLn/σ,q;b(Rn) → L∞,r;br (Ω).
q;(0,+∞)