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Poincaré-type inequalities and finding good parameterizations. (English) Zbl 1431.49053

Summary: A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. E. R. Reifenberg [Bull. Am. Math. Soc. 66, 312–313 (1960; Zbl 0099.08601)] 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.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
51F99 Metric geometry

Citations:

Zbl 0099.08601
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References:

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