Differentiability, porosity and doubling in metric measure spaces. (English) Zbl 1272.30083
The authors prove that a metric measure space has a Lipschitz differentiable structure in the sense of Cheeger if and only if
(a) it has a certain approximate differentiable structure and
(b) porous sets have measure zero,
where (b) in turn implies a pointwise doubling condition almost everywhere. However, (a) alone does not guarantee pointwise doubling, as the authors show by an elaborate counterexample based on Laakso spaces.
(a) it has a certain approximate differentiable structure and
(b) porous sets have measure zero,
where (b) in turn implies a pointwise doubling condition almost everywhere. However, (a) alone does not guarantee pointwise doubling, as the authors show by an elaborate counterexample based on Laakso spaces.
Reviewer: Tuomas Hytönen (Helsinki)
MSC:
| 30L99 | Analysis on metric spaces |
| 49J52 | Nonsmooth analysis |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
Keywords:
metric measure space; differentiable structure; pointwise doubling condition; porous set; Laakso spaceReferences:
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