Gaussian rigidity and width of measures

Source abstract

Given $0\le m<n$ and a positive Radon measure $μ$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional \emph{Gaussian discrepancy}, defined as $$\mathcal D(μ):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}μ}|G_R^μ(x)-1|\Bigr],$$ where $G_R^μ(x):=(4πR^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,dμ(z)$. Provided that $μ$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\leδ<\infty$, called the \emph{width} of $μ$, such that $$\mathrm{spt}μ\subseteq \bar B_δ(V+x_0),\quad δ^2\le C(m,n)\mathcal{D}(μ),$$ where $\bar B_δ(V+x_0)$ denotes the closed $δ$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(μ)=0$, the support of $μ$ 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 $μ$, 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 $μ$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.