Rectifiability of measures and the $\beta_p$ coefficients

TL;DR

This paper investigates the limitations of characterizing n-rectifiability using $eta_p$ coefficients, providing counterexamples for p ≠ 2, and offers an alternative proof for rectifiability of measures with bounded lower density.

Contribution

It constructs counterexamples showing $eta_p$ coefficients do not characterize rectifiability for p ≠ 2 and presents an alternative proof for rectifiability of measures with bounded lower density.

Findings

Counterexamples for $eta_p$ coefficients when p ≠ 2

Characterization of n-rectifiability fails for p ≠ 2

Alternative proof for rectifiability of measures with bounded lower density

Abstract

In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\beta_2$ coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the $\beta_p$ coefficients with $p\neq2$. This is in strong contrast with what happens in the case of uniform $n$-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber and Valtorta about the $n$-rectifiability of measures with bounded lower $n$-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.