BMO with respect to Banach function spaces

LERNER, ANDREI K.; LORIST, EMIEL; OMBROSI, SHELDY

MATHEMATISCHE ANNALEN

SPRINGER

Año: 2024

0025-5831

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.