Inequalities of John—Nirenberg type in doubling spaces

The concept of an H-chain set in a doubling spaceX, which generalizes that of a Hölder domain in Euclidean space, is defined and investigated. We show that every H-chain set is mean porous and that its outer layer has measure bounded by a power of its thickness. As a consequence, we show that a John-Nirenberg type inequality holds on an open subset Ω ofX if, and often only if, Ω is an H-chain set