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Poincaré-type inequalities and finding good parameterizations

  • Jessica MerhejEmail author
Article

Abstract

A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. Reifenberg (Bull Am Math Soc 66:312–313, 1960) proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an n-Ahlfors regular rectifiable set \(M \subset \mathbb {R}^{n+d}\) that satisfies a Poincaré-type inequality involving Lipschitz functions and their tangential derivatives. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of M guarantees that M is contained in a bi-Lipschitz image of an n-plane. We also explore the Poincaré-type inequality considered here and show that it is in fact equivalent to other Poincaré-type inequalities considered on general metric measure spaces.

Keywords

Rectifiable set Carleson-type condition Poincaré-type condition p-Poincaré inequality Ahlfors regular Bi-Lipschitz image 

Mathematics Subject Classification

49Q15 51F99 

Notes

Acknowledgements

The author would like to thank T. Toro for her supervision, direction, and numerous insights into the subject of this project. The author was partially supported by the National Science Foundation DMS-0856687 and DMS-1361823 Grants, and by Notre Dame University.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsNotre Dame UniversityZouk MosbehLebanon

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