The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan (Q808375)

Let \(G\) be a real semisimple Lie group, connected, noncompact, with finite center, and let \(K\) be a maximal compact subgroup. The paper provides a ten pages proof of a deep theorem in spherical harmonic analysis on \(G\): for \(0<p\leq 2\), the spherical Fourier transform is a topological isomorphism between the \(L^ p\) Schwartz space of bi-\(K\)-invariant functions on \(G\), and a suitable Schwartz space on the dual of a Cartan subspace (in the Lie algebra). This result is due to Harish-Chandra for \(p=2\); it has been extended to \(0<p<2\) by P. C. Trombi and V. S. Varadarajan. Their proofs were long and difficult. The author was able to deduce this theorem from the corresponding Paley- Wiener theorem for the space of \(C^{\infty}\) compactly supported bi-\(K\)- invariant functions. Although the original proof of the latter result, by S. Helgason and R. Gangolli, relied on the former (with \(p=2)\), an independent and much shorter proof follows from the work of J. Rosenberg. The difficult Harish-Chandra, Trombi, Varadarajan theorem thus becomes, with the present paper, a very accessible result. First, the author slightly improves this Paley-Wiener theorem. Then, he proves a precise estimate between seminorms in the Schwartz spaces, by means of an increasing sequence of compact subsets in \(G\). The result follows by a density argument.