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 results regional inequality are established depending various conditions range \begin{document}$ s \in (0,1) $\end{document}. The best constant in both characterized for strip like id="M2">\begin{document}$ (\omega \times \mathbb{R}^{n-1}) $\end{document}, obtained direction analogous those local case. This settles one natural questions raised by K. Yeressian [Asymptotic behavior elliptic nonlocal equations set cylinders, Asymptot. Anal. 89, (2014), no 1-2].