L1-isomorphism of free spaces over highly sublinear distortions

Determine whether the Lipschitz free space LF(X, w∘d) is isomorphic to l1 for every infinite compact metric space (X, d) when w: [0, ∞) → [0, ∞) is a distortion function satisfying w(t) = o(t^α) as t → 0 for every α ∈ (0, 1) (for example, w(t) = 1/ log(1/t) for sufficiently small t), even though the distorted space (X, w∘d) generally fails to be doubling.

Background

The paper recalls that for any infinite compact doubling space (X, d) and exponent α ∈ (0, 1), the snowflake (X, d^α) has free space isomorphic to l1. It then considers more singular distortions w that are smaller than any power t^α near 0 (e.g., w(t) = 1/log(1/t)), noting that (X, w∘d) typically fails to be doubling.

In this context, the authors explicitly note that the validity of LF(X, w∘d) ≅ l1 for such distortions was an open question (citing Weaver) and proceed to resolve it under finite Nagata-dimension hypotheses via Theorem 4.18.