Metric currents and the Poincar\'e inequality

TL;DR

This paper establishes an equivalence between the support of a weak 1-Poincaré inequality and the existence of a pencil of curves in complete doubling metric spaces, using currents and recent decomposition results.

Contribution

It introduces the notion of generalized pencils of curves and proves their equivalence to the weak 1-Poincaré inequality in metric spaces.

Findings

Weak 1-Poincaré inequality implies existence of generalized pencils of curves.

Existence of generalized pencils of curves implies existence of pencils of curves.

Decomposition of normal 1-currents leads to the construction of pencils of curves.

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.