Big-Pieces-of-Lipschitz-Images Implies a Sufficient Carleson Estimate in a Metric Space

TL;DR

This paper demonstrates that for 1-Ahlfors-regular sets, having large bi-Lipschitz pieces is equivalent to satisfying a Carleson measure condition, providing a key link in geometric measure theory.

Contribution

It establishes the equivalence between Big Pieces of bi-Lipschitz Images (BPBI) and a Carleson condition in the context of 1-Ahlfors-regular sets, supplementing previous bi-Lipschitz decomposition results.

Findings

BPBI condition is equivalent to a Carleson estimate

Provides a new characterization of Lipschitz images in metric spaces

Enhances understanding of geometric measure theory in regular sets

Abstract

This note is intended to be a supplement to the bi-Lipschitz decomposition of Lipschitz maps shown in [Sch]. We show that in the case of 1-Ahlfors-regular sets, the condition of having `Big Pieces of bi-Lipschitz Images' (BPBI) is equivalent to a Carleson condition.