• Corpus ID: 211258815

Quantitative decompositions of Lipschitz mappings into metric spaces

@article{David2020QuantitativeDO,
  title={Quantitative decompositions of Lipschitz mappings into metric spaces},
  author={Guy C. David and Raanan Schul},
  journal={arXiv: Metric Geometry},
  year={2020}
}
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 is arbitrarily small in 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… 
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References

SHOWING 1-10 OF 25 REFERENCES
Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of…
Lipschitz and Bi-Lipschitz maps from PI spaces to Carnot groups
This paper deals with the problem of finding bi-Lipschitz behavior in non-degenerate Lipschitz maps between metric measure spaces. Specifically, we study maps from (subsets of) Ahlfors regular PI…
REGULAR MAPPINGS BETWEEN DIMENSIONS
The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by "regular mappings"…
Bi-Lipschitz Pieces between Manifolds
A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large…
A quantitative metric differentiation theorem
The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a…
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the…
Measure-preserving quality within mappings
In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on…
BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups
Abstract Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such…
Lectures on Analysis on Metric Spaces
1. Covering Theorems.- 2. Maximal Functions.- 3. Sobolev Spaces.- 4. Poincare Inequality.- 5. Sobolev Spaces on Metric Spaces.- 6. Lipschitz Functions.- 7. Modulus of a Curve Family, Capacity, and…
Some Results on Maps That Factor through a Tree
Abstract We give a necessary and sufficient condition for a map deffned on a simply-connected quasi-convex metric space to factor through a tree. In case the target is the Euclidean plane and the map…
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