Square Functions and Rectifiability under Monotone Transformations of the Density
Abstract
Let μ be an n-AD-regular measure in ℝd. Chousionis, Garnett, Le and Tolsa [CGLT] proved that μ is uniformly n-rectifiable if and only if the square function built from the density differences Δμ(x,r)=μ(B(x,r))/rn−μ(B(x,2r))/(2r)n satisfies a Carleson condition. In this paper we show that the same characterization holds if the density is first composed with a function F which is bi-Lipschitz on the interval [c0−1,c0] determined by the AD-regularity constant c0. The main example is F=log, introduced in [Le], for which the square function takes the scale-invariant form Δμlog(x,r) = log(μ(B(x,r))/μ(B(x,2r)))+nlog 2. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of μ with a Gaussian or a more general radial kernel, discuss what happens when F is not bi-Lipschitz, and treat the case μ(ℝd)<∞, where the behavior of F near zero enters in only one of the two implications. We also show that the qualitative characterization of n-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at μ-almost every point, holds after composition with any locally bi-Lipschitz F. This requires neither AD-regularity nor doubling, and for F=log the condition limr→0Δμ(x,r)=0 becomes limr→0μ(B(x,r))/μ(B(x,2r))=2−n.
Related
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BibTeX
@misc{le2026square,
title = {{Square Functions and Rectifiability under Monotone Transformations of the Density}},
author = {Triet M. Le},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30166},
eprint = {2608.30166},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30166},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Square Functions and Rectifiability under Monotone Transformations of the Density
AU - Triet M. Le
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30166
UR - https://arxiv.org/abs/2608.30166
AB - Let μ be an n-AD-regular measure in ℝ^d. Chousionis, Garnett, Le and Tolsa [CGLT] proved that μ is uniformly n-rectifiable if and only if the square function built from the density differences Δ_μ(x,r)=μ(B(x,r))/r^n−μ(B(x,2r))/(2r)^n satisfies a Carleson condition. In this paper we show that the same characterization holds if the density is first composed with a function F which is bi-Lipschitz on the interval [c_0^−1,c_0] determined by the AD-regularity constant c_0. The main example is F=log, introduced in [Le], for which the square function takes the scale-invariant form Δ_μ^log(x,r) = log(μ(B(x,r))/μ(B(x,2r)))+nlog 2. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of μ with a Gaussian or a more general radial kernel, discuss what happens when F is not bi-Lipschitz, and treat the case μ(ℝ^d)<∞, where the behavior of F near zero enters in only one of the two implications. We also show that the qualitative characterization of n-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at μ-almost every point, holds after composition with any locally bi-Lipschitz F. This requires neither AD-regularity nor doubling, and for F=log the condition lim_r→0Δ_μ(x,r)=0 becomes lim_r→0μ(B(x,r))/μ(B(x,2r))=2^−n.
ER -