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Let be a Gelfand pair, with a Lie group of polynomial growth, and let be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system of -invariant differential operators on and associating to a bounded spherical function the -tuple of its eigenvalues under the action of the 's.
We say that property (S) holds for if the spherical transform maps the bi--invariant Schwartz space isomorphically onto , the space of restrictions to of the Schwartz functions on . This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where , with nilpotent.
In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs.
Cite this article
Francesca Astengo, Bianca Di Blasio, Fulvio Ricci, On the Schwartz correspondence for Gelfand pairs of polynomial growth. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 79–96DOI 10.4171/RLM/927