C 1,α-rectifiability in low codimension in Heisenberg groups

Abstract

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