Page 68 - Textos de Matemática Vol. 34

58 SUSANA D. MOURA
where B(γ,r) denotes the closed ball centred at γ with radius r.
(ii) d = n, Ψ is non-increasing with limr→0 Ψ(r) = ∞ and (2.1) and (2.2) hold
true (with d = n) for some Radon measure μ.
Remark 2.1. This definition generalises the notion of d-sets, introduced by Jons- son and Wallin in [20], and the definition of (d, Ψ)-sets (for Ψ an admissible function) considered by Edmunds and Triebel in [16, 17] (see also [23, 24]). One could also define an (0, Ψ)-set, but this is of no interest for us in what follows. The definition above is subsumed in the larger class of h-sets, extensively studied by Bricchi in [4, 5, 6], which consist in replacing the right-hand side of (2.2) by h(r) where h is a general gauge function. We refer to the latter above-mentioned works for details in this connection.
Remark 2.2. The measure μ in (2.2) is unique, up to equivalence, and we can always think of μ as being Hh|Γ, the restriction to Γ of the generalised Hausdorff measure Hh related to the gauge function h(r) = rdΨ(r) – cf. Proposition 3.5(i) of [7]. As Ψ is slowly varying, limr→0 log Ψ(r)/ log r = 0 (cf. [1, Proposition 1.3.6, p. 16]), so that, in view of Proposition 3.5(ii) of [7], any (d,Ψ)-set has (local) Hausdorff and packing dimension d. Furthermore, it has Lebesgue measure zero (even in the case d = n, due to [4, Lemma 1.8.4]).
3. Besov spaces in Rn
Before introducing the function spaces under consideration we need to recall some notation. By S(Rn) we denote the Schwartz space of all complex-valued, infinitely differentiable and rapidly decreasing functions on Rn and by S′(Rn) the dual space of all tempered distributions on Rn. Furthermore, Lp(Rn) with 0 < p ≤ ∞, stands for the usual quasi-Banach space with respect to the Lebesgue measure, quasi-normed by
n p 1/p
∥f|Lp(R)∥:=
|f(x)| dx ,
Rn
with the obvious modification if p = ∞. Let φ0 ∈ S(Rn) with
(3.1) φ0(x)=1 if |x|≤1 and suppφ0 ⊂{x∈Rn :|x|≤2},
and for each j ∈ N let φj (x) := φ0(2−j x) − φ0(2−j+1x), x ∈ Rn. Then (φj )j∈N0 form a smooth dyadic resolution of unity. Given any f ∈ S′(Rn), we denote by Ff and F−1f its Fourier transform and its inverse Fourier transform, respectively.
Definition 3.1. Let 0 < p,q ≤ ∞, s ∈ R and Ψ be a slowly varying function. (s,Ψ) n ′ n
Then Bpq (R ) is the collection of all f ∈S (R ) such that
∞ 1/q 2jsq Ψ(2−j)q ∥F−1[φ Ff] | L (Rn)∥q
(3.2) ∥f | B(s,Ψ)(Rn)∥ :=
p,q jp
j=0