Quantitative Alberti representations in spaces of bounded geometry

Tuomas Orponen

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arXiv · 1907.06903

Quantitative Alberti representations in spaces of bounded geometry

Tuomas Orponen

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Abstract

A metric measure space $(X,d,μ)$ is said to be $A_{\infty}$ on curves if there exist constants $τ< 1$ and $θ> 0$ with the following property. For every $x \in X$, $0 < r \leq \mathrm{diam}(X)$, and a Borel set $S \subset B(x,r)$ with $μ(S) > τμ(B(x,r))$, there exists a continuum $γ\subset X$ of length $\leq r$ satisfying $\mathcal{H}^{1}_{\infty}(γ\cap S) \geq θr$. 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 $(X,d,μ)$ 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 $μ$ to a ball $B \subset X$ can be written as $μ_{B} = f_{B} \, dν_{B}$, where $ν_{B}$ is a convex combination of measures of linear growth supported on continua of length $\le \mathrm{diam}(B)$, and $\|f_{B}\|_{L^{p}(ν_{B})} \leq C$ for some constant $C \geq 1$ independent of $B$.

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BibTeX RIS

Tuomas Orponen. 2019-07-16. Quantitative Alberti representations in spaces of bounded geometry. https://arxiv.org/abs/1907.06903

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