Compactness of Sobolev-type embeddings with measures

Abstract

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in [Formula: see text] with respect to upper Ahlfors regular measures [Formula: see text], that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure [Formula: see text] and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

Compactness of Sobolev embeddings and decay of norms

“…This follows from the fact that lim sup t→0 + ψ(t) φ(t) = 1 and from (2.10). Therefore, W m X(Ω) → M ψ (Ω, ν) is not compact thanks to [8,Theorem 4.1]. Hence Y (Ω, ν) = M ψ (Ω, ν) has the desired properties.…”

“…Due to [8,Theorem 4.1], W m L p,q;α (Ω) → Y (Ω, ν) is not compact if and only if Y L p,q;α (Ω, ν) → Y (Ω, ν) is not almost-compact.…”

“…Lastly, as Z(Ω, ν) ̸ ⊆ Λ ψ (Ω, ν), the embedding Λ ψ (Ω, ν) ∩ Z(Ω, ν) → Λ ψ (Ω, ν) is not almost-compact [19,Lemma 3.7], and so W m X(Ω) → Λ ψ (Ω, ν) noncompactly owing to [8,Theorem 4.1].…”

“…We shall consider Sobolev-type embeddings having the form W m X(Ω) → Y (Ω, ν), where W m X(Ω) is the mth order, m ∈ N, Sobolev-type space built upon a rearrangement-invariant space X(Ω), and Y (Ω, ν) is a rearrangement-invariant space on Ω endowed with a d-Ahlfors measure ν (for more details, see Section 2). Such Sobolev-type embeddings and their compactness were recently studied in [12,13,8].…”

Discretization and antidiscretization of Lorentz norms with no restrictions on weights

“…Combining our results with [4, Theorem 5.1] is a way to obtain some sharper Sobolev-Lorentz trace embedddings into Lorentz Λ-spaces on low-dimensional sets. Furthermore, our characterizations of (1.2) may also be used to get better compactness results for such embeddings (see [3,Remark 5.4]).…”