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

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.

💡 Research Summary

The paper establishes a precise equivalence between two central notions of uniform rectifiability in the setting of metric spaces: “Big Pieces of bi‑Lipschitz Images” (BPBI) and a Carleson‑type square‑function estimate involving Jones’ β‑numbers. The authors work under the hypothesis that the underlying set E is 1‑Ahlfors‑regular, i.e., the Hausdorff measure μ satisfies μ(B(x,r)∩E)≈r for all x∈E and 0<r≤diam E.

Main theorem.
For a complete metric space X and a 1‑Ahlfors‑regular subset E⊂X, the following are equivalent:

1. E has BPBI – there exist constants θ>0 and L≥1 such that for every ball B(x,r) with x∈E, a subset F⊂E∩B(x,r) with μ(F)≥θ r can be mapped onto a subset of the real line by an L‑bi‑Lipschitz map.
2. E satisfies the Carleson condition
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