Gaussian rigidity and width of measures

Abstract

Given $0\le m<n$ and a positive Radon measure $\mu$ on ${\mathbb{R}}^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal{D}(\mu ):=\underset{R\to \infty }{\mathrm{limsup}}[R^2\underset{x\in \mathrm{spt}\mu }{\sup }|G_R^{\mu }(x)-1|],$$ where $G_R^{\mu }(x):=(4\pi R^2{)}^{-m/2}{\int }_{{\mathbb{R}}^n}{e}^{-|z-x{|}^2/(4R^2)}d\mu (z)$. Provided that $\mu$ admits a flat blow-down ${\mathcal{H}}^m└V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\le \delta <\infty$, the so-called width of $\mu$, such that $$\mathrm{spt}\mu \subseteq {\bar{B}}_{\delta }(V+x_0),{\delta }^2\le C(m,n)\mathcal{D}(\mu ),$$ where ${\bar{B}}_{\delta }(V+x_0)$ denotes the closed $\delta$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal{D}(\mu )=0$, the support of $\mu$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $\mu$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $\mu$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

arXiv comment19 pages; comments are welcome