Page 80 - Textos de Matemática Vol. 34

70 SUSANA D. MOURA
Proof. Note that the quadratic form belonging to the self-adjoint operator Gβ in W21(Rn) is given by

Γ
and, hence, Gβ is a lower semi-bounded self-adjoint operator.
Since Γ is a compact (d,Ψ)-set, for each j ∈ N0 there exist Mj disjoint balls
Bj,l, l = 1,...,Mj, centred at xj,l ∈ Γ and of radius 2−j, with Mj ∼ 2jd Ψ(2−j)−1, j ∈ N0.
Let φ be a non-negative C∞ function with
suppφ⊂ξ∈Rn :|ξ|≤1/4 and φ(x)>0 for |x|≤δ
for some 0 < δ < 1/4, and define
φj,l(x):=φ2j(x−xj,l), j∈N0, l=1,...,Mj.
Then suppφj,l ⊂ Bj,l and so, for fixed j ∈ N0, the functions φj,l, l = 1,...,Mj, have disjoint supports. We may assume that {ψj,l : l = 1,...,Mj}, with ψj,l := 2−j(1−n/2) φj,l, is an orthonormal system in W21(Rn). Using the definition of (d, Ψ)- -set, we can see that there are positive constants c1 and c2 such that

(6.15) Qβ[f]:=(Gβf,f)W21(Rn) =∥f|W21(Rn)∥2 −β Then,
Qβ[f] ≥ (1 − cβ)∥f |W21(Rn)∥2
|(trΓf)(γ)|2dμ(γ).
(6.16) c1 2−j(2−n+d) Ψ(2−j) ≤
for all j ∈ N0 and l ∈ {1,...,Mj}. Let βj := c3 2j(2−n+d) Ψ(2−j)−1 for c3 a fixed
|ψj,l(γ)|2 μ(dγ) ≤ c2 2−j(2−n+d) Ψ(2−j) positive constant such that c > c−1. Then, by (6.15) and (6.16), it follows
31
(6.17) Qβj[ψj,l]≤1−βjc12−j(2−n+d)Ψ(2−j)=1−c3c1 <0
Γ
for all j ∈ N0 and l = 1,...,Mj. Let Vj be the span of the Mj functions ψj,l, l = 1,...,Mj, and denote by Pj the orthonormal projection of W21(Rn) onto Vj. It is not hard to verify that Pj Gβj Pj|Vj has eigenvalues (Gβj ψj,l,ψj,l)W21(Rn), l = 1, . . . , Mj . Following the notation in [14, p. 489] with respect to the Max-Min principle for semi-bounded self-adjoint operators, in view of (6.17) and by Lemma 1.4 (Rayleigh-Ritz) in [14, p. 492], we get
μMj(Gβj):= sup (Gβjψ,ψ)W1(Rn) :ψ∈SM⊥ −1,∥ψ|W21(Rn)∥=1
2j
SMj −1
(6.18) ≤ sup(Gβj ψj,l,ψj,l)W21(Rn) : l = 1,...,Mj < 0,
where the first supremum is taken with respect to all subspaces SMj−1, with di- mension less or equal to Mj − 1. Suppose that Gβj has k negative eigenvalues