C1,α-rectifiability in low codimension in Heisenberg groups

Abstract

A natural higher-order notion of C1,α{C}^{1,\alpha }-rectifiability, 0<α≤10\lt \alpha \le 1, is introduced for subsets of the Heisenberg groups Hn{{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of (CH1,α,H)\left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}})-regular surfaces. Using this, we prove a geometric characterization of C1,α{C}^{1,\alpha }-rectifiable sets of low codimension in Heisenberg groups Hn{{\mathbb{H}}}^{n} in terms of an almost everywhere existence of suitable approximate tangent paraboloids.

Keywords

• higher order
• approximate tangent paraboloids
• metric spaces
• 28a75 (primary)
• 43a80
• 53c17 (secondary)