Quantitative Alberti representations in spaces of bounded geometry

Abstract

A metric measure space $(X,d,\mu )$ is said to be $A_{\infty }$ on curves if there exist constants $\tau <1$ and $\theta >0$ with the following property. For every $x\in X$, $0<r\le \mathrm{diam}(X)$, and a Borel set $S\subset B(x,r)$ with $\mu (S)>\tau \mu (B(x,r))$, there exists a continuum $\gamma \subset X$ of length $\le r$ satisfying ${\mathcal{H}}_{\infty }^1(\gamma \cap S)\ge \theta r$. I first observe that spaces of $Q$-bounded geometry, $Q>1$, are $A_{\infty }$ on curves. Then, I show that any complete, doubling, and quasiconvex space $(X,d,\mu )$ which is $A_{\infty }$ on curves has Alberti representations with $L^p$-densities for some $p>1$, depending only on the doubling and $A_{\infty }$-constants. More precisely, any normalised restriction of $\mu$ to a ball $B\subset X$ can be written as ${\mu }_B=f_{B}d{\nu }_B$, where ${\nu }_B$ is a convex combination of measures of linear growth supported on continua of length $\le \mathrm{diam}(B)$, and $\Vert f_B{\Vert }_{L^p({\nu }_B)}\le C$ for some constant $C\ge 1$ independent of $B$.

15 pages

References in corpus (1)

• Cheeger's differentiation theorem via the multilinear Kakeya inequality