Study of fractional Poincaré inequalities on unbounded domains

The aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results on regional fractional inequality are established depending on various conditions on domains and on the range of <tex-math id="M1">{document}s ∈ (0,1){document}</tex-math>. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains <tex-math id="M2">{document}(ω × Rⁿ⁻¹){document}</tex-math>, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2].