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The central aim of this paper is to study (regional) fractional Poincar\'etype inequalities on unbounded domains satisfying the finite ball condition.Both existence and non existence type results are established depending onvarious conditions on domains and on the range of $s \in (0,1)$. The bestconstant in both regional fractional and fractional Poincar\'e inequality ischaracterized for strip like domains $(\omega \times \mathbb{R}^{n-1})$, andthe results obtained in this direction are analogous to those of the localcase. This settles one of the natural questions raised by K. Yeressian in[\textit{Asymptotic behavior of elliptic nonlocal equations set in cylinders,Asymptot. Anal. 89, (2014), no 1-2}].

Study of fractional Poincar\'e inequalities on unbounded domains