Page 79 - Textos de Matemática Vol. 34

ON THE PROBLEM OF THE NEGATIVE SPECTRUM 69
Hence, f(k) ≤ c1 ∥b|Lr(Γ)∥β implies k ≤ c2 gc1 ∥b|Lr(Γ)∥β
d dn−2 ≤ c3 ∥b | Lr(Γ)∥ β 2−n+d (1/Ψ♯d)♯ (∥b | Lr(Γ)∥ β)− n−2 2−n+d
for any k and β sufficiently large, completing the proof, in view of (6.4).
Remark 6.2. When Ψ is of the form k
Ψ(r) = logj(cr−1)αj j=1
forsomek∈N,αj ∈R,j=1,···,k,onehasΨ♯ ∼1/Ψ–cf. [1,Appendix5, p. 433]. Since then (6.14) is fulfilled, we can replace (1/Ψ♯d)♯ by Ψ in the above upper estimate for Nβ. In the special case of Ψk,b, given by (2.1), this had already been obtained in [24, Corollary 4.2.4].
In the next theorem we obtain an estimate from below for Nβ in the particular case of b ≡ 1 ∈ L∞(Γ), extending previous estimates of Triebel for d-sets – cf. [28, Theorem 31.5, p. 249].
Theorem6.3.LetΓbea(d,Ψ)-setinRn with(n−2)+ <d≤nandΨbea slowly varying function. Let trΓ be the operator trΓ := idΓ ◦ trΓ. Then B, given by
B := (id − ∆)−1 ◦ trΓ
is a compact non-negative self-adjoint operator in W21(Rn) with null space
N(B) = {f ∈ W21(Rn) : trΓf = 0}. Moreover, B is generated by the sesquilinear form

(trΓf)(γ) (trΓg)(γ) μ(dγ) = (Bf, g)W21(Rn). Γ
Let
and let Nβ be the number of non-positive eigenvalues of Gβ (counting multiplicities).
Gβ :=id+βB with β>0, Then, there are positive numbers c and β0 such that
Nβ ≥ c βd/(2−n+d) Ψ♯d,n(β−1)(n−2)/(2−n+d),
for all β > β0, where Ψ♯d,n denotes the Bruijn conjugate of Ψ(r1/(2−n+d))−1 accord- ing to Proposition 2.1(iv).