Page 87 - Textos de Matemática Vol. 34

A SURVEY ON EMBEDDINGS OF BESSEL-POTENTIAL-TYPE SPACES 77
[11], the authors established sharpness of embedding theorems for Bessel-potential spaces modelled upon Lorentz-Karamata spaces and proved non-compactness of such embeddings, where the target spaces are either Lorentz-Karamata spaces or Lipschitz-H¨older spaces. Moreover, they also extended the embedding results of [18]-[20] since the definition of LK-spaces (see Section 2) is more general than that given in [18]-[20]. As in [12], it is not assumed any symmetry of slowly varying functions involved in the quasi-norms of LK-spaces. In [11], there is as well an improvement (cf. Theorem 5.5 below) of an embedding of Bessel-potential spaces modelled upon LK-spaces into H¨older-type spaces in sub-limiting case proved in [20].
In this survey we summarise the embedding theorems from [18]-[20] and [10]-[11]. Moreover, we extend some results from [18]-[20] and give the proofs of some of the embeddings from [10]-[11], which were not proved there but referred to the proofs of the results in [18]-[20].
In Section 2 we present some notation. Section 3 deals with slowly varying functions and results about Lorentz-Karamata spaces. In Section 4, we collect some properties of the Bessel-potential spaces modelled upon Lorentz-Karamata spaces. In the last section, we consider the embedding results.
2. Notation and Preliminaries
As usual, Rn is the Euclidean n-dimensional space. Let μn be the n-dimensional
Lebesgue measure in Rn and let Ω be a μn-measurable subset of Rn. We de-
note by χΩ the characteristic function of Ω and write |Ω|n = μn(Ω). The fam-
ily of all extended scalar-valued (real or complex) μn-measurable functions on
Ω will be denoted by M(Ω), and M+(Ω) will stand for the subset of M(Ω)
consisting of all those functions which are non-negative a.e. Let f ∈ M(Ω).
The non-increasing rearrangement of f is the function f∗ defined on [0,+∞) by
f∗(t)=inf{λ≥0:|{x∈Ω:|f(x)|>λ}|n ≤t}forallt≥0,andthemaximalfunc-
tion f∗∗ of f∗ is defined by f∗∗(t) = t−1 t f∗(τ)dτ, t > 0. Clearly, f∗(t) ≤ f∗∗(t), 0
t > 0, and we also have the inequality
(2.1) (f + g)∗∗(t) ≤ f∗∗(t) + g∗∗(t) for all t > 0,
cf. [2, Theorem II.3.4]. For general facts about (rearrangement-invariant) Banach function spaces with Banach function norm ρ over a measure space (R, μ) we refer to [2, Chapters. 1 & 2].
Let p ∈ (0,+∞]. We denote by Lp(Rn) the Lebesgue space endowed with the (quasi-) norm ∥.∥p.
The space of all scalar-valued (real or complex), bounded and continuous func- tions on Rn is denoted by CB(Rn) and it is equipped with the L∞(Rn) norm. For each h ∈ Rn, let ∆h be the difference operator defined on scalar functions f on Rn