Quantitative Alberti representations in spaces of bounded geometry

A metric measure space $(X,d,\mu )$ is said to be $A_{\mathrm{\infty }}$ on curves if there exist constants $\tau <1$ and $\theta >0$ with the following property. For every $x\in X$, $0<r\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(X)$, and a Borel set $S \subset B(x,r)$with$\mu(S) > \tau \mu(B(x,r))$,thereexistsacontinuum$\gamma \subset X$oflength$\leq r$satisfying$$\mathcal{H}$^{1}{\infty}(\gamma \cap S) \geq \theta r$. I first observe that spaces of $Q$-bounded geometry, $Q>1$, are $A_{\mathrm{\infty }}$ on curves. Then, I show that any complete, doubling, and quasiconvex space $(X,d,\mu )$ which is $A_{\mathrm{\infty }}$ on curves has Alberti representations with $L^p$-densities for some $p>1$, depending only on the doubling and $A_{\mathrm{\infty }}$-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 $\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(B)$, and $\mathrm{\parallel }{f}_B{\mathrm{\parallel }}_{L^p({\nu }_B)}\le C$ for some constant $C\ge 1$ independent of $B$.