On the Schwartz correspondence for Gelfand pairs of polynomial growth

Let $(G,K)$ be a Gelfand pair, with $G$ Lie group of polynomial growth, and let $\\Sigma\\subset\\mathbb R^\\ell$ homeomorphic image the spectrum, obtained by choosing generating system $D1,\\dots,D\\ell$ $G$-invariant differential operators on $G/K$ associating to bounded spherical function $\\phi$ $\\ell$-tuple its eigenvalues under action $D_j$'s. We say that property (S) holds for if transform maps bi-$K$-invariant Schwartz space $\\mathcal S(K\\backslash G/K)$ isomorphically onto S(\\Sigma)$, restrictions $\\Sigma$ functions $\\mathbb R^\\ell$. This is known hold many nilpotent pairs, i.e., pairs where $G=K\\ltimes N$, $N$ nilpotent. In this paper we enlarge scope analysis outside range stating basic setting general growth then focussing strong pairs.