Page 85 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES
JU´LIO S. NEVES
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. Embedding theorems for Bessel-potential spaces modelled upon appropriate Lorentz-Karamata spaces are presented. Target spaces are either Lorentz-Karamata spaces or Lipschitz-H¨older-type spaces.
1. Introduction
Sobolev’s classical embedding theorem [25] states that if Ω ⊂ Rn is a domain
with a sufficient smooth boundary, then Wpk(Ω) → Lq(Ω) continuously, whenever
p< n andp≤q≤ np ,seealsoforinstance[1,Lemmas5.12and5.14]. Inthe k n−pk
limiting case, i.e., when p = n, this inclusion does not hold for q = +∞, unless k
p=1sothatk=n. However,wedohavethat
(1.1) Wk (Ω)→Lq(Ω) forall q, p≤q<+∞,
see for example [1, Corollary 5.13 and Lemma 5.14]. Therefore, the optimal inte- grability conditions satisfied by functions in Wk cannot be specified as simple Lq
n/k
conditions.
In 1967 Trudinger [29] (see also Pohoˇzaev [24] and Yudovich [30]) found refine-
ments of (1.1) expressed in terms of Orlicz spaces of exponential type. He was able to prove that a continuous embedding of the form
(1.2) Wpk(Ω) → LΦ(Ω),
where k p = n and Ω is a bounded domain in Rn, n > 1, with a smooth boundary,
holds for the Orlicz space LΦ (Ω) generated by the function Φ(t) = exp tλ for large
t, where λ = n for all k ∈ N. Such an Orlicz space is clearly contained in Lq(Ω) n−1
2000 Mathematics Subject Classification. 26A12, 46E30, 46E35, 26A15, 26A16, 47B38, 26D10, 26D15.
Key words and phrases. Slowly varying functions, Lorentz-Karamata spaces, (fractional) Sobolev-type spaces, Bessel potentials, embedding theorems.
This work was supported by CMUC (Centre of Mathematics of the University of Coimbra).
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n/k