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ON THE PROBLEM OF THE NEGATIVE SPECTRUM 59
(with the usual modification if q = ∞) is finite.
Remark 3.1. The above spaces were introduced by Edmunds and Triebel in [16, 17]
and also considered by Moura in [23, 24] when Ψ is an admissible function. Spaces
of generalised smoothness have already been considered in some generality: see, for
instance, [13, 21, 22]. We refer to the paper [18], by Farkas and Leopold, which
represents a unified and general approach on this topic. Note that if Ψ ≡ 1 then the (s,Ψ) n s n
spaces Bp,q (R ) coincide with the usual Besov spaces Bp,q(R ) and the following elementary embeddings hold
(3.3) Bs+ε(Rn) → B(s,Ψ)(Rn) → Bs−ε(Rn), p,q p,q p,q
for all ε > 0 (cf. Propositon 3.1(i)).
Taking advantage of the properties of slowly varying functions, in particular Proposition 2.1(ii), one can easily prove the embedding assertions below. We refer also to [2, 4, 24] in this connection.
Proposition3.1. (i) Let0<p,q1,q2 ≤∞,s∈R,ε>0,Ψ1 andΨ2 beslowly varying functions. Then
B(s+ε,Ψ1)(Rn) → B(s,Ψ2)(Rn). p,q1 p,q2
(ii) Let0<p1 ≤p2 ≤∞,0<q1,q2 ≤∞,s1,s2 ∈Rwiths1−n/p1 =s2−n/p2,
and Ψ1, Ψ2 be slowly varying functions such that
∞, if q1≤q2
4. Function spaces on (d,Ψ)-sets
Let Γ be a (d,Ψ)-set in Rn according to Definition 2.2. Recall that the cor- responding Radon measure μ is unique up to equivalence – cf. Remark 2.2. If 0 < p ≤ ∞, then Lp(Γ) is the usual complex quasi-Banach space (Banach space if p ≥ 1) with respect to μ, quasi-normed by
Ψ (2−j) ∞
2 ∈lq∗ −j
q1q2 , if q2 <q1 <∞. q1 −q2
Ψ1(2)j=0 Then
where
q∗ := B(s1,Ψ1)(Rn) → B(s2,Ψ2)(Rn).
q2, ifq1=∞ p1 ,q1 p2 ,q2
p 1/p |f(γ)| dμ(γ)
Γ
(with the usual modification if p = ∞).
In this section we recall the relationships between the spaces Lp(Γ) and some
Besov spaces with generalised smoothness.
(4.1) ∥f |Lp(Γ)∥ :=
