Morrey Sequence Spaces: Pitt’s Theorem and Compact Embeddings

Abstract

Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces mu,p = mu,p(Z d ), 0 < p ≤ u < ∞, which have yet been considered almost nowhere. They are defined as natural generalisations of the classical p spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterise the compactness of embeddings of related finite-dimensional spaces.

Proper inclusions of Morrey spaces

“…Remark 2.2. As mentioned before, a similar result for discrete Morrey spaces on Z d can be found in [7]. Here, we give a different proof of the necessary condition for this inclusion property.…”

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We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in [3,7]. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of p-summable sequences.

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“…In addition, it is shown that ℓ p 2 q 2 ⊆ ℓ p 1 q 1 =⇒ q 2 ≤ q 1 whenever 1 ≤ p 1 ≤ q 1 < ∞, 1 ≤ p 2 ≤ q 2 < ∞, and p 1 ≤ p 2 . Recently, these inclusion results are extended to discrete Morrey spaces on Z d in [7]. Moreover, the authors in [7] give a necessary condition for inclusion relations of discrete Morrey spaces on Z d .…”

“…Recently, these inclusion results are extended to discrete Morrey spaces on Z d in [7]. Moreover, the authors in [7] give a necessary condition for inclusion relations of discrete Morrey spaces on Z d . They also prove that the embedding (1.2) is never compact.…”

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

“…See [, Chapter 7] for further details, examples and references. We just mention here that , and , for , , are pairwise essentially incomparable by the Pitt–Rosenthal Theorem, and, by the recent Pitt–Rosenthal like theorem of , certain discrete Morrey spaces introduced in are essentially incomparable to as well. See for a comparison with other notions of incomparability of Banach spaces.…”