Page 76 - Textos de Matemática Vol. 34

66 SUSANA D. MOURA
with
tr : W1(Rn) → B(1−(n−d)/2,Ψ−1/2)(Γ) Γ 2 2,2
id : B(1−(n−d)/2,Ψ−1/2)(Γ) → L (Γ) 12,2 q
(6.9) b : Lq(Γ)→Lt(Γ) with 1:=1+1 tqr
id : L (Γ) → B(−(n−d)/q,Ψ−1/q )(Rn) Γ t t,∞
(id − ∆)−1 : B(−(n−d)/q,Ψ−1/q)(Rn) → B(2−(n−d)/q,Ψ−1/q)(Rn) t,∞ t,∞
id : B(2−(n−d)/q,Ψ−1/q )(Rn) → W 1(Rn). 2t,∞ 2
The boundedness of trΓ above is a consequence of Definition 4.2. The embedding id1 is just the compact embedding in (6.5), while id2 is justified by Proposition 3.1(i),(ii), thanks to
t≤2 and 2−n−d−n=2−n+d− d >1−n, qt 22r2
where we used (6.2). As for b it is due to H¨older’s inequality, idΓ comes from Propo- sition 4.1 with q = t′ and (id − ∆)−1 is a lift assertion (cf. [4, Proposition 2.2.19]). Let f ∈ W21(Rn) be an eigenfunction of B with eigenvalue μ ̸= 0. Then, by the
construction in (6.9), we infer that f belongs to the space B(2−(n−d)/q,Ψ−1/q)(Rn) t,∞
and so it is an eigenfunction of the operator B restricted to this space. Obvi- ously the converse is also true. Hence, the root systems of B in W21(Rn) and in
B(2−(n−d)/q,Ψ−1/q)(Rn) coincide. Then the eigenvalues of B considered in these t,∞
spaces also coincide, inclusively their multiplicities. We factorise the operator B in B(2−(n−d)/q,Ψ−1/q)(Rn) by
t,∞
where now
B = (id − ∆)−1 ◦ idΓ ◦ b ◦ id ◦ trΓ, B(2−(n−d)/q,Ψ−1/q)(Rn) → B(2−n+d,Ψ−1)(Γ)
Γ t,∞ t,∞
tr :
B(2−n+d,Ψ−1 ) (Γ) → L (Γ) t,∞ q
id : (6.10) b :
id
(id − ∆)−1 : B(−(n−d)/q,Ψ−1/q )(Rn) → B(2−(n−d)/q,Ψ−1/q )(Rn). t,∞ t,∞
Lq (Γ) → Lt (Γ)
: L (Γ) → B(−(n−d)/q,Ψ−1/q )(Rn)
Γ t t,∞