Poincar\'e-type Inequalities and Finding Good Parameterizations

TL;DR

This paper links geometric regularity of sets with analytic parameterizations, showing that certain oscillation conditions on tangent planes imply the set can be mapped bi-Lipschitzly to a plane, extending Poincaré inequalities.

Contribution

It establishes a new connection between Carleson-type conditions, Poincaré inequalities, and bi-Lipschitz parameterizations for Ahlfors regular sets, generalizing previous geometric measure theory results.

Findings

Carleson conditions ensure bi-Lipschitz parameterization.

Poincaré inequalities are equivalent on metric measure spaces.

Set oscillations control geometric regularity.

Abstract

A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. In 1960, E. R. Reifenberg proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-H\"older 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 Poincar\'{e}-type inequality involving the tangential derivative. 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 Poincar\'e-type inequality considered here and show that it is in fact equivalent to other Poincar\'e-type inequalities considered on general metric measure spaces.