## 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.

### MSC:

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

### Citations:

Zbl 0809.53057; Zbl 1014.22010
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