Suppose that \(G\) is a connected nilpotent Lie group. For any irreducible unitary representation \(\pi\) of \(G\), denote by \(N_\pi\) its kernel. The main theorem of the paper under review is that there exists \(p\) in \([2,\infty)\), depending only on \(G\), such that the matrix coefficients \(g\mapsto\langle\pi(g)\xi,\eta\rangle\) lie in \(L^p(G/N_\pi)\), for all \(\xi\) and \(\eta\) in a dense subspace of the Hilbert space \({\mathcal H}_\pi\) of \(\pi\). This should be compared to the situation for semisimple groups, for which an analogous theorem holds, but on the one hand, the index \(p\) may depend on \(\pi\), while on the other hand, the matrix coefficients lie in \(L^p(G)\) for all \(\xi\) and \(\eta\) in \({\mathcal H}_\pi\). -- Results characterizing the image of the Schwartz space on \(G\) in the space of operators on \({\mathcal H}_\pi\) are given for the case where \(p=2\).