Page 90 - Textos de Matemática Vol. 34

80 JU´LIO S. NEVES
Lemma 3.1. ([10, Lemma 4.1]) Let 1 ≤ q ≤ r ≤ +∞, ν ̸= 0 and let b,˜b ∈ SV(0,+∞).
(i) The inequality t′
(3.2) tν−1/r˜b(t) g(u)du tν+1/q b(t)g(t)
0 r;(0,+∞) holds for all g ∈ M+(0, +∞) if, and only if,
(3.3) ν<0 and ˜bb on (0,+∞).
q;(0,+∞)
(ii) The inequality +∞′
(3.4) tν−1/r ˜b(t) g(u)du tν+1/q b(t)g(t)
t r;(0,+∞) holds for all g ∈ M+(0, +∞) if, and only if,
(3.5) ν>0 and ˜bb on (0,+∞).
q;(0,+∞)
Lemma3.2. Let1≤q≤r≤+∞,ν̸=0andletb,˜b∈SV(0,1). Theinequality t′
(3.6) tν−1/r˜b(t) g(u)du tν+1/q b(t)g(t)
0 r;(0,1) holds for all g ∈ M+(0, 1) if, and only if,
(3.7) ν<0 and ˜bb on (0,1).
q;(0,1)
Lemma3.3. ([18,Lemma3.3])Let1≤q≤r≤+∞andletb,˜b∈SV(0,1). Then the following inequality
111 (3.8) t−r ˜b(t) g(u)du tq′ b(t)g(t)
q;(0,1)
holds for all g ∈ M+(0, 1) if, and only if, there is a positive constant c such that
t r;(0,1) 1 1 −1
(3.9) t−r ˜b(t) tq′ b(t) ≤c forall x∈(0,1).
r;(0,x)
Let p, q ∈ (0, +∞] and let ∥.∥(p,q;b);Ω defined by
q′ ;(x,1)
b ∈ SV (0, +∞). Let us introduce the functional
(3.10) ∥f∥
(p,q;b);Ω
:= ∥t1 −1 b(t) f∗∗(t)∥
p q q;(0,+∞)
;