We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.
"Improved fractional Poincaré type inequalities on John domains." Ark. Mat. 57 (2) 285 - 315, October 2019. https://doi.org/10.4310/ARKIV.2019.v57.n2.a3