BMO with respect to Banach function spaces

Abstract

For every cube $Q\subset {\mathbb{R}}^n$ we let $X_Q$ be a quasi-Banach function space over $Q$ such that $\Vert {\chi }_Q{\Vert }_{X_Q}\simeq 1$, 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 $$\mathrm{BMO}={\mathrm{BMO}}_X={\mathrm{BMO}}_X^{\ast }.$$ 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 ${\mathrm{BMO}}_X^{\ast }\hookrightarrow \mathrm{BMO}$. Our main theorems recover and improve all previously known results in this area.

29 pages

References in corpus (1)

• Operator-free sparse domination