密度单调变换下的平方函数与可求长性

中文摘要 AI 辅助

设μ为ℝ^d中的n-AD正则测度，Chousionis、Garnett、Le与Tolsa[CGLT]证明，当且仅当由密度差Δ_μ(x,r)=μ(B(x,r))/r^n - μ(B(x,2r))/(2r)^n构造的平方函数满足Carleson条件时，μ是一致n-可求长的。本文证明，若密度先与由AD正则性常数c₀确定的区间[c₀⁻¹,c₀]上的双Lipschitz函数F复合，则上述刻画仍成立。主要例子为[Le]中引入的F=log，此时平方函数取尺度不变形式Δ_μ^log(x,r)=log(μ(B(x,r))/μ(B(x,2r)))+n log2。我们给出完整证明，将结论推广至[CGLT]中的光滑平方函数（其中密度替换为μ与高斯核或更一般径向核的卷积），讨论F非双Lipschitz时的情况，并处理μ(ℝ^d)<∞的情形（此时F在零点附近的行为仅影响两个等价命题中的一个）。我们还证明，Tolsa与Toro[TT]提出的、基于μ几乎处处相同平方函数的n-可求长测度定性刻画，在与任意局部双Lipschitz的F复合后仍然成立，该结论无需AD正则性或双倍性条件；对于F=log，条件lim_{r→0}Δ_μ(x,r)=0转化为lim_{r→0}μ(B(x,r))/μ(B(x,2r))=2⁻ⁿ。

英文摘要

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 $\lim_{r\to0}μ(B(x,r))/μ(B(x,2r))=2^{-n}$.

发表机构

• Spatiolyx LLC（Spatiolyx有限责任公司）

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