Page 95 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 85
Proof. With no loss of generality we shall assume that |Ω|n = 1. Let u ∈ HσLn,q;b(Rn). Then u = gσ ∗f, where f ∈ Ln,q;b(Rn) and ∥f∥n,q;b = ∥u∥σ;n,q;b.
σσσσ
Hence by O’Neil’s inequality we have (5.4), which together with the estimate (5.2) yields, for every t ∈ (0, 1),
nt1 +∞ (5.12) u∗(t) ≤ σAtσ/n−1 f∗(τ)dτ + gσ∗(τ)f∗(τ)dτ +
0t1 By Lemma 3.5, there is a positive constant c such that
∗ ∗∗ t−σ/n
(5.13) f (t)≤f (t)≤c b(t) ∥f∥n,q;b forall t>0.
gσ∗(τ)f∗(τ)dτ.
τ−σ/n b(τ) dτ
(5.14)
σ
+∞ +∞ σ
gσ∗(τ)f∗(τ)dτ ≤ C∥f∥n ,q;b τ n −1 exp(−Bτ1/n) σ
11 = C1∥f∥n,q;b.
σ
Using (5.1) and (5.13) we obtain
The estimates (5.12) and (5.14) yield
(5.15) where
∥u∗∥∞,r;br;(0,1) N1 + N2 + N3∥f∥n ,q;b, σ
σ1t
N1= tn−1−r br(t) f∗(τ)dτ
0 r;(0,1)
11
N2= t− r br (t) gσ∗ (τ )f ∗ (τ ) dτ
;
;
t r;(0,1)
−1 N3=trbr(t) .
r;(0,1) By(5.9),itiseasytoverifythatN3 <+∞. Sincebr bon(0,1)andσ<n,
Lemma 3.2 gives
(5.16) N1
σ1
tn −1+q′ b(t)f∗(t)q;(0,1)
σ1 ≤ tn−1+q′ b(t)f∗(t)
Finally, Lemma 3.3, with ˜b = br, and the estimate (5.1) yield 1σ
(5.17) N2 tq′ b(t)tn −1 f∗(t) ∥f∥n ,q;b.
=∥f∥n,q;b. q;(0,+∞) σ
q;(0,1) Now the result follows from inequalities (5.15)-(5.17).
σ