Fourier transforms of spherical distributions on compact symmetric spaces
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In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. We characterize the singular support of a distribution in terms of its Fourier transform. Finally, we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator.
New version. Our previous results were for symmetric spaces of the compact type. The new version including an appendix sketching the extension of the Paley-Wiener theorem for K-invariant functions and distributions to compact symmetric spaces
New version. Our previous results were for symmetric spaces of the compact type. The new version including an appendix sketching the extension of the Paley-Wiener theorem for K-invariant functions and distributions to compact symmetric spaces