Page 86 - Textos de Matemática Vol. 34

76 JU´LIO S. NEVES
for every q < +∞. However, when k ≥ 2, Strichartz [27] noted that Trudinger’s result could be improved with the appearance of the larger power λ = p′ = n .
n−k Note that in 1966, Peetre [23, Theorem 9.1] proved a limiting embedding concerning
Besov spaces from which Trudinger’s and Strichartz’s limiting embeddings follow for p = 2.
In 1979 Hansson [13, pp. 96–101] and, independently, in 1980 Br´ezis and Wainger [4, Theorem 2, p. 781] proved that
Wpk(Ω) → L∞,p(log L)−1(Ω),
where k p = n, Ω is a bounded domain with smooth boundary and L∞,p(log L)−1(Ω) is a Lorentz-Zygmund space. As pointed out in both [13] and [4], the space L∞,p(logL)−1(Ω) is strictly smaller than the various versions of the space LΦ(Ω) which appear in (1.2).
In the super-limiting case, i.e. p > n/k, Sobolev’s classical embedding theorem
asserts that if and Ω ⊂ Rn is a domain with a sufficiently smooth boundary, then
Wpk(Ω) → CB(Ω), where CB(Ω) is the space of scalar-valued bounded continuous
functions on Ω. However, in this case more can be said, as embeddings in H¨older
spaces C0,α(Ω) are possible. For example, Wpk(Ω) → C0,α(Ω), for all α ∈ (0,1),
when k = 1+n/p ∈ N and Ω ⊂ Rn is a domain with a sufficient smooth boundary,
cf. e.g. [1, Lemma 5.17]. In the particular case k = 1 + n and p = 1, the previous
embedding can be improved as the Sobolev space is embedded into the Lipschitz
space. Therefore, we may ask if an embedding into a Lipschitz space for p > 1
and k = 1 + n/p ∈ N would be possible. The answer is negative; see [8, Theorem
3.3] where a more general result is established, which deals with the sharpness of
embeddings of logarithmic Bessel potential spaces into general H¨older spaces. This
result for Sobolev spaces is common knowledge, but as remarked in [8] it is hard
to find a precise argument in the literature: [1, Example 5.28] does not settle the
question as there is a slip in the calculations, cf. [8, Remark 5.2]. Nevertheless,
Br´ezis and Wainger (cf. [4, Corollary 5]) proved that elements of the Bessel potential
1+n/p n
space Hp (R ), with 1 < p < +∞, are “almost Lipschitz continuous”, in the
sense that there is a positive constant c for which
|f(x)−f(y)| ≤ c∥f|H1+n/p∥ |x−y| |log|x−y||p′
1
In more recent times, Sobolev type embeddings have attracted some attention. A systematic research of embeddings of Bessel potential spaces modelled upon gener- alised Lorentz-Zygmund (GLZ) spaces was carried out in [6]-[8]. We refer to [21] for a survey of these results. The embeddings mentioned above were extended in [18]- -[20] to the case when Bessel-potential spaces are modelled upon LK-spaces, where in several cases improvements of embeddings from [6]-[8] were obtained. In [10] and
1+n/p n n
forallf∈Hp (R )andx,y∈R suchthat0<|x−y|<2.
p
1