Inclusions of Waterman–Shiba spaces into generalized Wiener classes

Abstract

The characterization of the inclusion of Waterman-Shiba spaces ΛBV (p) into generalized Wiener classes of functions BV (q; δ) is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.Let Λ = (λ i ) be a Λ-sequence, that is, a nondecreasing sequence of positive numbers such that

Relations between Schramm spaces and generalized Wiener classes

“…It was shown by Kita and Yoneda in [9] that the embedding BV p ⊆ BV (pn↑∞) is both automatic and strict for all 1 ≤ p < ∞. Furthermore, Goginava characterized the embedding ΛBV ⊆ BV (qn↑∞) in [6], and a characterization of the embedding ΛBV (p) ⊆ BV (qn↑q) (1 ≤ q ≤ ∞) was given by Hormozi, Prus-Wiśniowski and Rosengren in [8]. In this paper, we investigate the embeddings ΛBV (p) ⊆ ΓBV (qn↑q) and ΦBV ⊆ BV (qn↑q) (1 ≤ q ≤ ∞).…”

“…A natural and important problem is to determine relations between the above-mentioned classes; see [21], [12], [4], [9], [6], [13], [8] and [5] for some results in this direction. In particular, Perlman and Waterman found the fundamental characterization of embeddings between ΛBV classes in [12].…”

Relationships between ΦBV and ΛBV

“…There are many papers devoted to studying relationships of classes of functions of generalized bounded variation (see [13,9,4,2]). Pierce and Velleman gave sufficient and necessary conditions for inclusions…”