Page 92 - Textos de Matemática Vol. 34

82 JU´LIO S. NEVES
4. Bessel-potential-spaces
In this section we collect some properties of the Bessel-potential spaces modelled upon Lorentz-Karamata spaces.
The Bessel kernel gσ, σ > 0, is defined as that function on Rn whose Fourier transform is g(ξ) = (2π)−n/2(1+|ξ|2)−σ/2, ξ ∈ Rn, where the Fourier transform fˆ
−iξ·x
σ
ˆ −n/2 ofafunctionf isgivenbyf(ξ)=(2π) Rn e
f(x)dx.Itisknownthatgσ is a positive, integrable function which is analytic except at the origin.
Let σ > 0, p ∈ (1, +∞), q ∈ [1, +∞] and b ∈ SV (0, +∞). The Bessel-potential- type space HσLp,q;b(Rn) is defined to be
{u:u=gσ ∗f, f ∈Lp,q;b(Rn)} and is equipped with the (quasi-)norm ∥u∥σ;p,q;b := ∥f∥p,q;b.
For σ = 0, we put
HσLp,q;b(Rn) = Lp,q;b(Rn).
If m ∈ N, α = (α1,...,αm) ∈ Rm and b = lα, we obtain the logarithmic Bessel potential space HσLp,q;α(Rn), endowed with the (quasi-)norm ∥u∥σ;p,q;b and considered in [7]. Note that if α = (0,...,0), HσLp,p;α(Rn) is simply the (fractional) Sobolev space Hpσ(Rn) of the order σ.
We shall need some density results, which generalise [7, Lemma 3.12] and [20, Lemma 5.1].
Lemma4.1. Let1<p<+∞,1≤q<+∞andletb∈SV(0,+∞). Then
(i) the space C0∞(Rn), of all scalar-valued infinitely continuously differentiable functions on Rn and having compact support, is dense in Lp,q;b(Rn);
(ii) C0∞(Rn) is dense in W1Lp,q;b(Rn);
(iii) the Schwartz space S(Rn), of all scalar-valued rapidly decreasing infinitely
differentiable functions on Rn, is dense in HσLp,q;b(Rn) for all σ ≥ 0. Proof. It follows the proof of [20, Lemma 5.1].
The next lemma extends [26, Chapter V, Lemma 3], [7, Lemma 4.1] and [20, Lemma 5.2].
Lemma 4.2. ([11, Lemma 4.5]) Let σ ∈ [1, +∞), p ∈ (1, +∞), q ∈ (1, +∞) and
b ∈ SV(0,+∞). Then f ∈ HσLp,q;b(Rn) if, and only if, f ∈ Hσ−1Lp,q;b(Rn)
and the distributional derivatives ∂f belong to Hσ−1Lp,q;b(Rn) (j = 1, . . . , n). ∂xj
Moreover, the (quasi-)norms ∥f∥σ;p,q;b and ∥f∥σ−1;p,q;b + n ∥ ∂f ∥σ−1;p,q;b are j=1 ∂xj
equivalent on HσLp,q;b(Rn).
It is well known that if 1 < p < +∞, then the Bessel potential space HkLp(Rn), when k ∈ N, coincides with the Sobolev space Wpk(Rn). Edmunds, Gurka and Opic