Page 73 - Textos de Matemática Vol. 34

R+0 be such that Then the embedding
δ+:=s1−s2−d1−1 >0. p1 p2 +
id : B(s1,Ψa1 )(Γ) → B(s2,Ψa2 )(Γ) p1 q1 p2 q2
ON THE PROBLEM OF THE NEGATIVE SPECTRUM 63 (s,Ψa )
Given a (d, Ψ)-set Γ, the spaces Bp,q (Γ) with s > 0 have been introduced in Definition 4.2. In the following, to avoid awkward formulations, we agree on
B(0,Ψa)(Γ):=L (Γ) for 0<p,q≤∞ and a∈R, p,q p
simply as a notation.
Theorem 5.1. Let Γ be a (d,Ψ)-set in Rn. Let 0 < p1,p2,q1,q2 ≤ ∞ and s1,s2 ∈
is compact and the related entropy numbers satisfy
ek(id) ∼ k Ψ♯d(k−1))−(s1−s2)/d Ψ♯d(k−1a2−a1 ,
where Ψ♯d is the Bruijn conjugate of Ψ(r1/d)−1, according to Proposition 2.1(iv).
Remark 5.1. The above theorem (excluding the case d = n) is given in [8, Propo- sition 7.16] and it is a special case of an assertion for general h-sets – cf. [8, Theorem 7.12]. For Ψ an admissible function we refer to the estimates obtained in [16, 17, 24].
6. The problem of the negative spectrum
In this section, which is the heart of this note, we apply the results collected in the previous sections to the so called problem of the negative spectrum related with Schr¨odinger operators with fractal potentials. The motivation for this kind of problems can be found in [28, 31.1] and it comes from quantum mechanics in connection with the typical hydrogen-like operator
2
H = −2m∆ + V,
where V is the potencial, is the Planck’s constant and m is the mass of the electron. We obtain an extension of Triebel’s results in [28, Theorems 31.3 & 31.5] from d-sets to (d, Ψ)-sets.
As before, let Γ be a (d,Ψ)-set in Rn, according to Definition 2.2. Let b ∈ Lr(Γ) be real with r ∈ [1, ∞] and
0 ≤ 1 ≤ 1 + 1 = 1 ≤ 1. pprt
Based on Theorems 4.2 and 4.1, and by Ho¨lder’s inequality we have
trΓ : B((n−d)/p,Ψ1/p)(Rn) → B(−(n−d)/t′,Ψ−1/t′ )(Rn), b p,1 t,∞
k ∈ N,