Abstract
Abstract We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.
Highlights
Functions of bounded mean oscillation were introduced by John and Nirenberg in [19], and shown to satisfy the by- celebrated John-Nirenberg inequality
We study the space BMOG (X) in the general setting of a measure space X with a xed collection G of measurable sets of positive and nite measure, consisting of functions of bounded mean oscillation on sets in G
We develop the notion of a Denjoy family G, which guarantees that functions in BMOG (X) satisfy the John-Nirenberg inequality on the elements of G
Summary
Functions of bounded mean oscillation were introduced by John and Nirenberg in [19], and shown to satisfy the by- celebrated John-Nirenberg inequality. As the same is true with K in place of fG , by assumption, and μ(G ) > , it must be that fG = K Applying this to Gk− and Gk, ≤ k ≤ N, it follows that f = fGk− almost everywhere on Gk− and f = fGk almost everywhere on Gk. On the intersection Gk− ∩ Gk, which has strictly positive measure by the de nition of a nite chain, both f = fGk− and f = fGk must hold almost everywhere. If · BMOGp (X) is a norm modulo constants on BMOGp (X), every decomposition of X into sets of strictly positive measure is pairwise nitely connected. Assume that there exists a decomposition {Xω}ω∈Ω of X into sets of strictly positive measure for which there is a pair (Xa , Xb) that is not nitely connected
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