On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group

TL;DR

This paper extends a fundamental theorem linking boundedness of certain operators to uniform rectifiability from Euclidean spaces to parabolic spaces and the Heisenberg group, broadening the scope of geometric measure theory.

Contribution

It generalizes the David-Semmes theorem to non-Euclidean settings, specifically parabolic space and the Heisenberg group, establishing new connections in geometric measure theory.

Findings

Extended the theorem to parabolic space

Extended the theorem to the Heisenberg group

Established boundedness implies uniform rectifiability in new settings

Abstract

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg group.