Extension of the Fefferman-Stein decomposition of BMO, 2

Abstract

In this section we extend the argument in Sections 22 and 24 to certain weighted BMO functions. Definition 26.1. Let $w \in L_{{\text{loc}}}^1 \left( {{\text{R}}^n ,{\text{R}}} \right)$ and let ω(x) > 0 a.e. x. For a measurable set E ⊂ R ⁿ, for ε> 0 and for $\vec g \in L_{{\text{loc}}}^1 \left( {{\text{R}}^n ,{\text{R}}^m } \right)$ let

$$ w\left( E \right) = \mathop {{\text{ess}}{\text{.sup }}w\left( y \right),}\limits_{y \in E} $$

,

$$ \left\| {\vec g} \right\|_{{\text{BMO}}_{w,\varepsilon } } = \mathop {{\text{sup}}}\limits_{B:\ell \left( B \right) \leqslant \varepsilon } {\text{ }}\mathop {\inf }\limits_{\vec c \in {\text{R}}^m } \int {\left| {\vec g\left( x \right) - \vec c} \right|} dx/\left( {w\left( B \right)\left| B \right|} \right) $$

, where B is taken over all balls in R ⁿ with its radius ≤ ε, and let

$$ \left| {\left\| {\vec g} \right\|} \right|_{{\text{BMO}}w,\varepsilon } = \left\| {\vec g} \right\|_{{\text{BMO}}_{w,\varepsilon } } + \mathop {{\text{sup}}}\limits_{x: \in {\text{R}}^n } \frac{{\left| {\vec g} \right| * \left( \chi \right)_\varepsilon \left( x \right)}} {{w\left( {B\left( {x,\varepsilon } \right)} \right)}},{\text{where }}\chi {\text{ = }}\chi _{B\left( {0,1} \right)} $$

.