Geometric criteria for $C^{1,\alpha }$ rectifiability

Abstract

We prove criteria for ${\mathcal{H}}^k$-rectifiability of subsets of ${\mathbb{R}}^n$ with $C^{1,\alpha }$ maps, $0<\alpha \le 1$, in terms of suitable approximate tangent paraboloids. We also provide a version for the case when there is not an a priori tangent plane, measuring on dyadic scales how close the set is to lying in a $k$-plane. We then discuss the relation with similar criteria involving Peter Jones' $\beta$ numbers, in particular proving that a sufficient condition is the boundedness for small $r$ of $r^{-\alpha }{\beta }_p(x,r)$ for ${\mathcal{H}}^k$-a.e. $x$ and for any $1\le p\le \infty$.

References in corpus (2)

• Rectifiability and approximate differentiability of higher order for sets
• Menger curvatures and $C^{1,\alpha }$ rectifiability of measures

Cited by in corpus (2)

• Cones, rectifiability, and singular integral operators
• Cone and paraboloid points of arbitrary subsets of Euclidean space