Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X. Eguchi et al. 1] characterized the image of $\text{E}$′(X)-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K-finite members of E′(X). We apply this Paley-Wiener type theorem to invariant differential equations on X.