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arXiv · 1807.02969

Metric currents and the Poincaré inequality

Katrin Fässler·Tuomas Orponen
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Abstract

We show that a complete doubling metric space $(X,d,μ)$ supports a weak $1$-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points $s,t \in X$. This notion was introduced by S. Semmes in the 90's, and has been previously known to be a sufficient condition for the weak $1$-Poincaré inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining $s$ and $t$ is a normal $1$-current $T$, in the sense of Ambrosio and Kirchheim, with boundary $\partial T = δ_{t} - δ_{s}$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\|T\| \ll μ$, with $$\frac{d\|T\|}{dμ}(y) \lesssim \frac{d(s,y)}{μ(B(s,d(s,y)))} + \frac{d(t,y)}{μ(B(y,d(t,y)))}.$$ We show that the $1$-Poincaré inequality implies the existence of GPCs joining any pair of points in $X$. Then, we deduce the existence of PCs from a recent decomposition result for normal $1$-currents due to Paolini and Stepanov.

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Katrin Fässler, Tuomas Orponen. 2018-10-08. Metric currents and the Poincaré inequality. https://arxiv.org/abs/1807.02969

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