Metric currents and the Poincar\'e inequality

Abstract

We show that a complete doubling metric space $(X,d,\mu)$ supports a weak $1$-Poincar\'e 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 $1$-Poincar\'e 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 $\partial T = \delta_{t} - \delta_{s}$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\|T\| \ll \mu$, with $$\frac{d\|T\|}{d\mu}(y) \lesssim \frac{d(s,y)}{\mu(B(s,d(s,y)))} + \frac{d(t,y)}{\mu(B(y,d(t,y)))}.$$ We show that the $1$-Poincar\'e 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.