An elementary proof of the Trombi theorem for the Fourier transform of \({\mathcal C}^ p(G:F)\)
Let G be a real, connected, semisimple Lie group with finite center. Let \(G=KAN\) be an Iwasawa decomposition of G, \(g=k\oplus a\oplus n\), the corresponding Lie algebra decomposition. For any \(F\subset \hat K\), \(| F| <\infty\), and \(0<p\leq 2\) let \({\mathcal C}^ p(G:F)\) be the \(L^ p\)-Schwartz space of type F. The image of \({\mathcal C}^ p(G:F)\) by the Fourier transform has been characterized by \textit{P. Trombi} [J. Funct. Anal. 40, 84-125 (1981; Zbl 0465.22006)]. To prove the continuity of the inverse Fourier transform (which is the delicate part in the proof) Trombi uses the asymptotic behavior of the Eisenstein integrals at infinity [cf. \textit{P. Trombi}, ibid. 30, 83-105 (1978; Zbl 0392.43011)]. The purpose of the present paper is to give an alternative and simpler proof of the continuity of the inverse Fourier transform, by using the Gangolli estimates for the coefficients of the Harish-Chandra expansion of the Eisenstein integrals obtained by \textit{M. Eguchi}, \textit{M. Hashizume} and \textit{S. Koizumi} [Hiroshima Math. J. 17, 457-469 (1987; Zbl 0655.22007)].
scientific article; zbMATH DE number 4090121