Page 67 - Textos de Matemática Vol. 34

ON THE PROBLEM OF THE NEGATIVE SPECTRUM 57
The proposition below gives some properties of slowly varying functions that will be useful in the sequel. We refer to the monograph [1] for details and further prop- erties. See also [26, 29]. We just remark that those references deal with functions defined in a neighbourhood of infinity rather than the origin; to relate a statement in this paper to a corresponding statement in this literature one should replace: (i) every occurrence of r (as r ↓ 0) with 1/x (as x → ∞); (ii) every occurrence of Ψ(·), where Ψ is slowly varying at the origin, with ψ(1/·), where ψ is slowly varying at the infinity.
Proposition 2.1. Let Ψ be a slowly varying function.
(i) There exists a C∞ slowly varying function Ψ0 such that Ψ(r) ≈ Ψ0(r) (r ↓ 0).
(ii) For any chosen δ > 0 there exists a constant c = c(δ) > 0 such that
Ψ(r) ≤ c max(r/s)δ , (r/s)−δ , r, s ∈ (0, 1]. Ψ(s)
(iii) For each α > 0 there is a C∞ non-increasing (strictly decreasing in a neigh- bourhood of zero) function φ and a C∞ non-decreasing (strictly increasing in a neighbourhood of zero) function φ with
r−αΨ(r)≈φ(r) and rαΨ(r)≈φ(r) (r↓0).
(iv) There exists a uniquely determined (up to equivalence) slowly varying function
Ψ♯ with
Ψ(r) Ψ♯rΨ(r)−1 ≈ 1 and Ψ♯(r) ΨrΨ♯(r)−1 ≈ 1 (r ↓ 0);
then Ψ♯♯ ≈ Ψ. The function Ψ♯ is called Bruijn conjugate of Ψ.
(v) Let α > 0 and f(x) := xα Ψ(x−1), x ∈ [1,∞). Then there exists a uniquely determined (up to equivalence) slowly varying function l such that for g(x) :=
x1/α l(x−1) it holds
fg(x) ≈ gf(x) ≈ x (x → ∞).
The function g is called asymptotic inverse of f. In particular, if f(x) = xab Ψ(x−b)a
for some a, b > 0, then
g(x) ≈ x1/ab Ψ♯(x−1/a)1/b (x → ∞).
Definition 2.2. Let Γ be a non-empty compact set in Rn, Ψ be a function equivalent to a slowly varying function and d ∈ (0, n]. Then Γ is a (d, Ψ)-set if either:
(i) d ∈ (0, n) and there exists a Radon measure μ with
(2.1) supp μ = Γ,
(2.2) μ(B(γ,r)) ∼ rd Ψ(r), r ∈ (0,1], γ ∈ Γ;