Metric currents and the Poincaré inequality

Abstract

We show that a complete doubling metric space $(X,d,\mu )$ 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 $\partial T={\delta }_t-{\delta }_s$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\Vert T\Vert \ll \mu$, with $$\frac{d\Vert T\Vert }{d\mu }(y)\lesssim \frac{d(s,y)}{\mu (B(s,d(s,y)))}+\frac{d(t,y)}{\mu (B(y,d(t,y)))}.$$ We show that the $1$-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.

19 pages. v2: Upgraded main result

References in corpus (1)

• Decomposition of acyclic normal currents in a metric space