Page 88 - Textos de Matemática Vol. 34

78 JU´LIO S. NEVES
by (∆hf)(x) = f(x + h) − f(x) for all x ∈ Rn. The modulus of smoothness of a function f in CB(Rn) is defined by
If
ω(f,t) := sup ∥∆hf|L∞(Rn)∥ for all t ≥ 0. |h|≤t
ω(f,t):=ω(f,t)/t foreach t>0,
then ω(f,.) is equivalent to a non-increasing function on (0,+∞). We refer to [2, pp. 331–333] and to [5, pp. 40–50] for more details.
Let q ∈ (0, +∞] and let Lq be the class of all continuous functions λ : (0, 1] →
(0, +∞) which are increasing on some interval (0, δ), with δ satisfy limt→0+ λ(t) = 0 and
−1/q t
(2.2) t λ(t) < +∞.
q;(0,δ) When q = +∞, we simply write L instead of Lq.
= δλ ∈
(0, 1], and
λ(.) n
Let q ∈ (0,+∞] and let λ ∈ Lq. The space Λ∞,q(R ) consists of all those
functions f ∈ CB(Rn) for which the (quasi-)norm
λ(.) n n −1/q ω(f,t)
∥f|Λ (R )∥:=∥f|L (R )∥+ t
∞,q λ(.)
∞
λ(t) q;(0,1)
is finite. The space Λ∞,∞(R ) coincides (cf. [17, Proposition 3.5]) with the space
n
C0,λ(.)(Rn) of λ(.)-H¨older continuous functions defined by
∥f|C0,λ(.)(Rn)∥ := sup |f(x)| + sup x∈Rn x,y∈Rn
|f(x) − f(y)| < +∞. λ(|x − y|)
0<|x−y|≤1
If λ(t) = t, t ∈ (0,1], then Λ∞,∞(R ) coincides with the space Lip(R ) of the
λ(.) n
Lipschitz functions. Note also that if (2.2) does not hold, then Λ∞,q(R ) consists
only of constant functions on Rn. Nowletm∈Nandα=(α1,...,αm)∈Rm. Wedenotebylαthereal
function defined by lα(t) = m lαi(t) for all t ∈ (0,+∞), where l1,...,lm are i=1 i
positive functions defined on (0, +∞) by l1(t) = 1 + | log t|, and, if m ≥ 2, li(t) = 1+logli−1(t), i ∈ {2,...,m}.
For two non-negative expressions (i.e. functions or functionals) A, B, the symbol A B means that A ≤ c B, for some positive constant c independent of the variables intheexpressionsAandB. IfABandBA,wewriteA≈BandsaythatA and B are equivalent. We adopt the convention that a/+∞ = 0 and a/0 = +∞ for alla>0. Ifp∈[1,+∞],theconjugatenumberp′ isgivenby1/p+1/p′ =1.
λ(.) n
n