  We study spatial decay properties for solutions of the Pelinovski-Stepanyants
equation posed on the cylinder. We establish the maximum polynomial decay
admissible for solutions of such a model. It is verified that the equation on
the cylinder propagates polynomial weights with different restrictions than the
model set in $\mathbb{R}^2$. For example, a local well-posedness theory is
deduced which contains the line solitary wave provided by solutions of the
Benjamin-Ono equation extended to the cylinder. Key results for our analysis
are obtained from a detailed study of the dispersive effects of the linear
equation in weighted Sobolev spaces. The results in this paper appear to be one
of the first studies of polynomial decay for nonlocal models in the cylinder.
