BMO with respect to Banach function spaces

Andrei K. Lerner; Emiel Lorist; Sheldy Ombrosi

doi:10.1007/s00208-023-02628-4

BMO with respect to Banach function spaces

Andrei K. Lerner, Emiel Lorist, Sheldy Ombrosi

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For every cube Q⊂ Rⁿ we let X_Q be a quasi-Banach function space over Q such that ‖χQ‖XQ≃1 , and for X= { X_Q} define ‖f‖BMOX:=supQ‖f-1|Q|∫Qf‖XQ,‖f‖BMOX∗:=supQinfc‖f-c‖XQ. We study necessary and sufficient conditions on X such that BMO=BMOX=BMOX∗. In particular, we give a full characterization of the embedding BMO↪BMOX in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMOX∗↪BMO . Our main theorems recover and improve all previously known results in this area.

Original language	English
Journal	Mathematische Annalen
DOIs	https://doi.org/10.1007/s00208-023-02628-4
State	Accepted/In press - 2023

Bibliographical note

Funding Information:
Andrei K. Lerner was supported by ISF Grant no. 1035/21. Emiel Lorist was supported by the Academy of Finland through Grant no. 336323. Sheldy Ombrosi was partially supported by ANPCyT PICT 2018-2501.

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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T1 - BMO with respect to Banach function spaces

AU - Lerner, Andrei K.

AU - Lorist, Emiel

AU - Ombrosi, Sheldy

PY - 2023

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AB - For every cube Q⊂ Rn we let XQ be a quasi-Banach function space over Q such that ‖χQ‖XQ≃1 , and for X= { XQ} define ‖f‖BMOX:=supQ‖f-1|Q|∫Qf‖XQ,‖f‖BMOX∗:=supQinfc‖f-c‖XQ. We study necessary and sufficient conditions on X such that BMO=BMOX=BMOX∗. In particular, we give a full characterization of the embedding BMO↪BMOX in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMOX∗↪BMO . Our main theorems recover and improve all previously known results in this area.

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BMO with respect to Banach function spaces

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