Relatively compact sets in variable-exponent Lebesgue spaces

The Kolmogorov–Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces

“…w . Finally, we will obtain that our main theorem provides a generalization of the corresponding results, such as Bandaliyev [5], Bandaliyev and Górka [6], Górka and Macios [12], [13], Górka and Rafeiro [14] and Rafeiro [30].…”

Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

“…We say that the family F is p(•)-equi-integrable if for all ε > 0, there exists δ > 0, such that sup f ∈F f χ A L p(•) (X) < ε for any measurable set A of X with μ(A) < δ. Corollary 3.6. [3] Let (X, μ) be a finite measure space, and let p(•) ∈ P(X, μ) satisfy 0 < p − ≤ p + < ∞. Then, a subset F of L p(•) (X) is totally bounded in L p(•) (X) if and only if F is totally bounded in L 0 (X) and the family F is p(•)-equi-integrable.…”

“…In [35], some sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Note that in variable exponent Lebesgue spaces (that is for λ(•) := 0), Theorem 4.2 was proved in [3].…”

“…Rafeiro characterized precompact sets in variable exponent Lebesgue spaces on Euclidean spaces [29]. For example, we refer to [3,10,11] for the characterization of precompact sets in variable exponent Lebesgue spaces on an arbitrary doubling metric measure space. Recently, the precompactness in quasi-Banach function spaces was studied in [6,12,13] (see [4,14] as well).…”

Matrix weighted Kolmogorov-Riesz's compactness theorem

“…Since then, compactness criteria of subsets in Lebesgue spaces have been studied and applied in various settings, e.g. see [35,42] for some improvements and applications on Kolmogorov-Riesz's theorem, and see [31,19,20,2,3,1] for a series of works on compactness criteria in variable exponent function spaces.…”