Basic functional properties of certain scale of rearrangement‐invariant spaces

Hana Turčinová

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3652-3675

Abstract: We define a new scale of function spaces governed by a norm‐like functional based on a combination of a rearrangement‐invariant norm, the elementary maximal function, and powers. A particular instance of such spaces surfaced recently in connection with optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures; however, a thorough analysis of these structures has not been carried out. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings, and, in a particular situation, a characterization of their associate structures. We discover a new one‐parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.

Date: 2023
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https://doi.org/10.1002/mana.202000463

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