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We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping “behaves like a projection mapping” along with a “garbage set” that isarbitrarily smallin an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper “Hard Sard”, and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings.
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Lower bounds on mapping content and quantitative factorization through trees
We give a simple quantitative condition, involving the “mapping content” of Azzam–Schul, which implies that a Lipschitz map from a Euclidean space to a metric space must be close to factoring through a tree. Using results of Azzam–Schul and the present authors, this gives simple checkable conditions for a Lipschitz map to have a large piece of its domain on which it behaves like an orthogonal projection. The proof involves new lower bounds and continuity statements for mapping content, and relies on a “qualitative” version of the main theorem recently proven by Esmayli–Hajłasz.
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- PAR ID:
- 10320470
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/jlms.12595
