BMO with respect to Banach function spaces

TL;DR

This paper characterizes when BMO spaces associated with Banach function spaces coincide with classical BMO, providing conditions for their equivalence and embedding properties, thus advancing the understanding of function space relationships.

Contribution

It offers a complete characterization of the conditions under which BMO spaces related to Banach function spaces are equivalent to classical BMO and establishes criteria for their embeddings.

Findings

Characterization of BMO and BMO_X equivalence via sparse cube collections

Weak sufficient conditions for BMO_X^* embedding into BMO

Recovery and improvement of previous results in BMO space theory

Abstract

For every cube $Q \subset \mathbb{R}^n$ we let $X_Q$ be a quasi-Banach function space over $Q$ such that $\|\chi_Q\|_{X_Q} \simeq 1$, and for $X= \{X_Q\}$ define \begin{align*} \|f\|_{\mathrm{BMO}_X} &:=\sup_Q \,\|f-{\textstyle\frac{1}{|Q|}\int_Qf} \|_{X_Q},\\ \|f\|_{\mathrm{BMO}_X^*} &:=\sup_Q \,\inf_c\, \|f-c\|_{X_Q}. \end{align*} We study necessary and sufficient conditions on $X$ such that $$ \mathrm{BMO} = \mathrm{BMO}_X = \mathrm{BMO}_{X}^*. $$ In particular, we give a full characterization of the embedding $\mathrm{BMO} \hookrightarrow \mathrm{BMO}_X$ in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding $\mathrm{BMO}_X^* \hookrightarrow \mathrm{BMO}$. Our main theorems recover and improve all previously known results in this area.