×

The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan. (English) Zbl 0732.43006

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.

MSC:

43A90 Harmonic analysis and spherical functions
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
22E46 Semisimple Lie groups and their representations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anker, J.-Ph, La forme exacte de l’estimation fondamentale de harish-chandra, C. R. acad. sci. Paris Sér. I, 305, 371-374, (1987) · Zbl 0636.22005
[2] Anker, J.-Ph, Lp Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of math., 132, 597-628, (1990) · Zbl 0741.43009
[3] Clerc, J.-L, Transformation de Fourier sphérique des espaces de Schwartz, J. funct. anal., 37, 182-202, (1980) · Zbl 0507.43002
[4] Ehrenpreis, L; Mautner, F.I, Some properties of the Fourier transform on semisimple Lie groups I, Ann. of math., 61, 406-439, (1955) · Zbl 0066.35701
[5] Ehrenpreis, L; Mautner, F.I, Some properties of the Fourier transform on semisimple Lie groups III, Trans. amer. math. soc., 90, 431-484, (1959) · Zbl 0086.09904
[6] Flensted-Jensen, M, Spherical functions on a real semisimple Lie group—A method of reduction to the complex case, J. funct. anal., 30, 106-146, (1978) · Zbl 0419.22019
[7] Flensted-Jensen, M, Analysis on non-Riemannian symmetric spaces, () · Zbl 0287.43015
[8] Gangolli, R, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of math., 93, 150-165, (1971) · Zbl 0232.43007
[9] Gangolli, R; Varadarajan, V.S, Harmonic analysis of spherical functions on real reductive groups, () · Zbl 0675.43004
[10] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. math., 80, 241-310, (1958) · Zbl 0093.12801
[11] Harish-Chandra, Spherical functions on a semisimple Lie group II, Amer. J. math., 80, 553-613, (1958) · Zbl 0093.12801
[12] Harish-Chandra, Discrete series for semisimple Lie groups II, Acta math., 116, 1-111, (1966) · Zbl 0199.20102
[13] Helgason, S, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. ann., 165, 297-308, (1966) · Zbl 0178.17101
[14] Helgason, S, A duality for symmetric spaces with applications to group representations, Adv. in math., 5, 1-154, (1971) · Zbl 0209.25403
[15] Helgason, S, Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press New York · Zbl 0451.53038
[16] Helgason, S, Groups and geometric analysis—integral geometry, invariant differential operators, and spherical functions, (1984), Academic Press New York · Zbl 0543.58001
[17] Rosenberg, J, A quick proof of harish-Chandra’s Plancherel theorem for spherical functions on a semisimple Lie group, (), 143-149 · Zbl 0322.43008
[18] Rouvière, F, Sur la transformation d’Abel des groupes de Lie semisimples de rang 1, Ann. scuola norm. sup. Pisa ser. IV, 10, 263-290, (1983) · Zbl 0527.43006
[19] Trombi, P.C, Spherical transform on symmetric spaces of rank one, () · Zbl 0572.22005
[20] Trombi, P.C; Varadarajan, V.S, Spherical transforms on semisimple Lie groups, Ann. of math., 94, 246-303, (1971) · Zbl 0203.12802
[21] Vretare, L, Elementary spherical functions on symmetric spaces, Math. scand., 39, 343-358, (1976) · Zbl 0387.43009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.