Page 71 - Textos de Matemática Vol. 34

ON THE PROBLEM OF THE NEGATIVE SPECTRUM 61
Theorem4.2. LetΓbea(d,Ψ)-setinRn,0<p<∞and0<q≤min(1,p). (i) Then
(4.4) tr B((n−d)/p,Ψ1/p)(Rn) ⊂ L (Γ). Γ p,q p
(ii) Supposep>1andeither(I)d<nor(II)d=nand∞j=0Ψ(2−j)−1/p <∞.
Then
tr B((n−d)/p,Ψ1/p)(Rn) = L (Γ). Γ p,q p
Remark 4.2. The last theorem is covered by [7, Theorem 5.9 & Corollary 5.10], which is stated for general h-sets. For Ψ an admissible function this type of issue had already been studied by Edmunds and Triebel in [17, Theorem 2.19] and by Bricchi in [3, Proposition 4.2]; see also [23, 24].
Note that, for 0 < q ≤ 1,
B(0,Ψ0)(Rn) ⊂ B0 (Rn) ⊂ C(Rn),
∞,q ∞,1
where C(Rn) stands for the set of all complex-valued bounded and uniformly contin- uous functions on Rn (cf. [27, 2.5.7, p. 89]), and hence (4.4) can be complemented for p = ∞. On the other hand, by the embedding assertion in Proposition 3.1(i), we have
B(s+ε,Ψ)(Rn) ⊂ B(s,Ψ1/p) (Rn) p,q p,min(1,p)
for any s ∈ R, 0 < p,q ≤ ∞, Ψ, Ψ slowly varying functions and ε > 0. Therefore, in (σ,Ψ) n
view of (4.4), it makes sense to speak about traces on Γ for all spaces Bp,q (R ) with 0 < p, q ≤ ∞ and σ > (n − d)/p; justifying the consideration of special subspaces of Lp(Γ) defined below.
Definition4.2. LetΓbea(d,Ψ)-setinRn,0<p,q≤∞,s>0anda∈R. Then (4.5) B(s,Ψa)(Γ) := tr B(s+(n−d)/p,Ψ1/p+a)(Rn),
p,q Γ p,q equipped with the quasi-norm
∥f | B(s,Ψa)(Γ)∥ := inf ∥g | B(s+(n−d)/p,Ψ1/p+a)(Rn)∥, p,q p,q
(s+(n−d)/p,Ψ1/p+a ) n
where the infimum is taken over all g ∈ Bp,q (R ) with trΓg = f.
Remark 4.3. In some cases where Γ is a particularly regular d-set (with d ∈ N) – as, for instance, the boundary of a smooth domain Ω in Rn –, one already has a definition of Besov spaces on these sets, usually denoted by the letter B. That definition and the one above do not generally coincide, that is why we preferred to use the letter B instead of B. However, under the restriction 1 < p < ∞ there is no danger of confusion by using B. We refer to [28, 20.3, pp. 160-161] for a discussion of this point.