Quantitative rectifiability characterization for doubling measures via logarithmic square functions

Determine whether a Carleson-type condition on the logarithmic density-difference square function \(\Delta^{\log}_\mu\), for doubling measures that are not AD-regular, characterizes a quantitative form of rectifiability.

Background

For AD-regular measures, the paper proves that composing density differences with F=log⁡F=\log preserves the uniform rectifiability characterization because the logarithm is bi-Lipschitz on the uniformly controlled density interval. For general doubling measures, the logarithmic quantity depends only on the scale-invariant ratio μ(B(x,r))/μ(B(x,2r))\mu(B(x,r))/\mu(B(x,2r)), whereas the ordinary density difference need not be uniformly comparable to it. The paper’s qualitative theorem applies under pointwise density assumptions but does not provide a quantitative Carleson characterization in the broader doubling setting.

References

It is not known to us whether a Carleson-type condition on $\Delta{\log}_\mu$ characterizes some quantitative form of rectifiability for doubling measures which are not AD-regular, in the spirit of and , and the arguments in this paper do not apply to this case; Section~\ref{sec:qualitative} gives only the qualitative statement.

— Square Functions and Rectifiability under Monotone Transformations of the Density  (2608.30166 - Le, 31 Aug 2026) in Section 7, Concluding remarks