Connectivity conditions and boundary Poincaré inequalities

Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets $\mathrm{\Omega }\subset {ℝ}^{n+1}$, with codimension-$1$ Ahlfors–David regular boundaries. First, we prove that if $\mathrm{\Omega }$ satisfies both the local John condition and the exterior corkscrew condition, then $\mathrm{\Omega }$ also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if $\mathrm{\Omega }$ is a $2$-sided chord-arc domain, then the boundary $\partial \mathrm{\Omega }$ supports a Heinonen–Koskela-type weak $1$-Poincaré inequality. We also construct an example of a set $\mathrm{\Omega }\subset {ℝ}^{n+1}$ such that the boundary $\partial \mathrm{\Omega }$ is Ahlfors–David regular and supports a weak boundary $1$-Poincaré inequality but $\mathrm{\Omega }$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.