Page 70 - Textos de Matemática Vol. 34

60 SUSANA D. MOURA
4.1. Lp(Γ) as a space of distributions. It is possible to identify Lp(Γ), for p ≥ 1,
to a subspace of S′(Rn). The identification operator idΓ : Lp(Γ) → S′(Rn) is
defined as follows: if f ∈ Lp(Γ), then the functional idΓf acts on S(Rn) as
f(γ)φ|Γ(γ) dμ, φ ∈ S(Rn). (s,Ψ) n
(4.2) ⟨idΓf,φ⟩ :=
One can check that idΓ is linear and injective. The distributions in idΓLp(Γ) belong
Γ
to a suitable class of subspaces of some spaces Bp,q (R ) defined below. Definition4.1. LetΓbea(d,Ψ)-setinRn,0<p,q≤∞,s∈Randψbeaslowly
varying function. Then
(4.3) B(s,ψ),Γ(Rn) = g ∈ B(s,ψ)(Rn) : ⟨g, φ⟩ = 0 if φ ∈ S(Rn) and φ|Γ = 0.
p,q p,q
One can easily see that the distributions in Bp,q (R ) have support in Γ,
but in general the condition suppf ⊂ Γ for f ∈ Bp,q (R ) is not sufficient for (s,ψ),Γ n
f ∈ Bp,q (R ). We refer to [4, p. 91] for examples and details.
Theorem 4.1. Let Γ be a (d,Ψ)-set in Rn, 1≤p≤∞ and denote by p′ the con- jugate exponent of p.
(i) Then
(ii) Supposep>1andeither(I)d<nor(II)d=nand∞j=0Ψ(2−j)−1/p′ <∞.
Then
id L (Γ) = B(−(n−d)/p′,Ψ−1/p′ ),Γ(Rn). Γp p,∞
id L (Γ) ⊂ B(−(n−d)/p′,Ψ−1/p′ ),Γ(Rn). Γp p,∞
Remark 4.1. The theorem above is covered by [7, Theorem 5.4 & Corollary 5.8], which is stated for general h-sets. In the case of Ψ being an admissible function, this had already been obtained by Edmunds and Triebel in [17, Theorem 2.16], extending previous results of Triebel in [28, Theorem 18.2] from d-sets to (d,Ψ)- -sets.
4.2. Lp(Γ) as a trace space. First we recall what is meant by traces. Suppose (s,ψ) n
that for some space Bp,q (R ), there exists a constant c > 0 such that ∥tr φ|L (Γ)∥≤c∥φ|B(s,Ψ)(Rn)∥,
Γp p,q
n n (s,ψ) n
forallφ∈S(R ). Ifmax(p,q)<∞,asS(R )isdenseinBp,q (R ),thedefinition (s,ψ) n
of trΓ on the whole space Bp,q (R ) is a matter of completion. In this way, any (s,ψ) n
g ∈ Bp,q (R ) has a (uniquely determined) trace trΓg ∈ Lp(Γ), and the bounded operator
tr : B(s,ψ)(Rn) → L (Γ) Γp,q p
is denoted as trace operator.
(s,ψ),Γ n (s,ψ) n