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Inequalities of John-Nirenberg type in doubling spaces. (English) Zbl 0990.46019
The author introduces the concept of an \(H\)-chain set \(\Omega\) in a doubling space \(X\); roughly speaking this means that there exists a “fairly short” chain of balls from any \(x\in\Omega\) to a fixed \(x_0\in\Omega\). \(H\)-chain sets generalize the notion of Hölder domains in Euclidean space but are not necessarily connected. It is shown that every \(H\)-chain set \(\Omega\) is mean porous and that its outer layer has measure bounded by a power of its thickness. As a consequence the author shows that a John-Nirenberg type inequality holds on an open subset \(\Omega\) of a doubling space \(X\) if, and often only if, \(\Omega\) is an \(H\)-chain set.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
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