Page 93 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 83
[7, Theorem 4.2] also proved that an analogous assertion holds when the role of the Lebesgue spaces is played by the generalised Lorentz-Zygmund space Lp,q;α(Rn), with α ∈ Rm and p, q ∈ (1, +∞). The next result extends [7, Theorem 4.2] and [20, Theorem 5.3], and establishes the relation between HσLp,q;b(Rn), when σ is a natural number k, and the space
WkLp,q;b(Rn) = {u : Dαu ∈ Lp,q;b(Rn), if |α| ≤ k} endowed with the (quasi-)norm |α|≤k ∥Dαu∥p,q;b.
Theorem 4.3. Let k ∈ N, p ∈ (1,+∞), q ∈ (1,+∞) and let b ∈ SV(0,+∞). Then
HkLp,q;b(Rn) = WkLp,q;b(Rn) and the corresponding (quasi-)norms are equivalent.
Proof. Similar to that of [26, Chapter. V, Theorem 3]. 5. Embeddings
In this section we present some embedding results of Bessel-potential spaces modelled upon Lorentz-Karamata spaces either into Lorentz-Karamata spaces or Lipschitz-H¨older-type spaces.
The next lemma, due to Edmunds, Gurka and Opic [6, Lemma 3.5], provides us the important estimate (5.1) for the non-increasing rearrangement of the Bessel kernel.
Lemma 5.1. Let 0 < σ < n. Then there exist constants A, B ∈ (0, +∞) such that (5.1) gσ∗(t) ≤ Atσ/n−1 exp(−Bt1/n) for all t > 0,
and
(5.2) g∗∗(t) ≤ nAtσ/n−1 for all t > 0.
σσ
The following result, which considers the sub-limiting case, extends [7, Theorem
4.8] and [18, Theorem 5.1] and refines [15, Theorem 5.7.7 (i)].
Theorem 5.2. ([10, Theorem 3.1 (i)]) Let σ ∈ (0, n), 1 < p < n/σ, q ∈ [1, +∞], r∈[q,+∞],1/pσ =1/p−σ/nandletb∈SV(0,+∞). LetΩ⊂Rn beanonempty domain. Then
(5.3) HσLp,q;b(Rn) → Lpσ,r;b(Rn).
Proof. Let u ∈ HσLp,q;b(Rn). Then u = gσ ∗ f, where f ∈ Lp,q;b(Rn) and ∥f∥p,q;b = ∥u∥σ;p,q;b.