Sobolev Capacities of Configurations with Multiple Points in Poisson Space

In this work, we study the difference between the space of all configurations and the space of configurations without multiple points, in the sense of topological properties, Poisson measures, and capacities generated by Sobolev functions. We prove that, under certain conditions, the set of configurations having multiple points has zero Sobolev $C_{r,p}$ capacity in the space of configurations on $\mathbb R^d$ with Poisson measure.