Square Functions and Rectifiability under Monotone Transformations of the Density

Abstract

Let $μ$ be an $n$-AD-regular measure in $\mathbb{R}^d$. 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 $Δ_μ^{\log}(x,r) = \log\bigl(μ(B(x,r))/μ(B(x,2r))\bigr)+n\log 2$. 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 $μ(\mathbb{R}^d)<\infty$, 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 $\lim_{r\to0}Δ_μ(x,r)=0$ becomes $\lim_{r\to0}μ(B(x,r))/μ(B(x,2r))=2^{-n}$.