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Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. (English) Zbl 1049.43004
Let \(G\) be a noncompact semisimple Lie group and \(K\) a maximal compact subgroup of G. S. Helgason [Geometric analysis on symmetric spaces (Mathematical Surveys and Monographs 39, Am. Math. Soc., Providence, Rhode Island) (1994; Zbl 0809.53057)] gave a Paley-Wiener theorem for the Fourier transform, which characterizes the image of the compactly supported, infinitely differentiable functions on \(X = G/K\) in terms of holomorphic extensions and their growth at infinity, analogous to the classical case. The question addressed by the present paper is what type of Paley-Wiener type theorem is available for the inverse Fourier transform?
When restricted to \(K\)-invariant functions, the Fourier transform on \(X\) reduces to the spherical transform on \(G\). For the complex, rank one case, A. Pasquale [Pac. J. Math. 193, No. 1, 143–176 (2000; Zbl 1014.22010)] has proved a Paley-Wiener theorem for the inverse spherical transform. The present author proves a real Paley-Wiener theorem for the inverse Fourier transform for general Riemannian symmetric spaces. Briefly, it is that for smooth \(f\) in \(L^2(X)\) to have compactly supported Fourier transform it must satisfy \(\lim\| \Delta^n f\|_2^{1/2n} < \infty\) where \(\Delta\) is the Laplace-Beltrami operator.

43A85 Harmonic analysis on homogeneous spaces
22E46 Semisimple Lie groups and their representations
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