Morrey smoothness spaces: A new approach

Nuclear Embeddings of Morrey Sequence Spaces and Smoothness Morrey Spaces

“…The most recent approach to Morrey smoothness spaces can be found in [20]: for −d ≤ < 0, 0 < p < ∞, 0 < q ≤ ∞, spaces of type A s p,q (R d ) and A s p,q (R d ) were introduced there, which satisfy that…”

“…There are also extensions to different parameters 1 , 2 , cf. [20,Theorem 6.19]. Recall that = −d corresponds to the classical situation, i.e., -A s p,q = A s p,q if = −d.…”

“…Then condition (2.29) reads exactly as the well-known necessary and sufficient condition for id : A s 1 p 1 ,q 1 ( ) → A s 2 p 2 ,q 2 ( ) to be compact. We coined for such phenomena in [20] the notion Slope-d-rule. In other words, the compact embeddings of Morrey smoothness spaces of the above type is just an example of such a Slope-d-rule.…”

“…Remark 5. 5 We briefly return to the new spaces -A s p,q introduced in [20] and check whether the so-called Slope-d-rule observed in Remark 2.19 for the compactness assertion remains true also for the nuclearity. Then Theorems 5.1 and 5.3 amount to the following observation.…”

“…Although the above scales are defined in different ways, they are closely connected and share a number of properties. Both approaches can be seen as examples of more general scales of Morrey smoothness spaces, we refer to the recent paper [20] in this context. Further details, definitions, properties and references can be found in Sect.…”