A Paley-Wiener theorem for the analytic wave front set (Q1840693)

It is shown that the convex hull of the analytic wave front \(WF_A(u)\) of a hyperfunction \(u\) with compact support in \(\mathbb{R}^n\) can be described by the asymptotic behaviour of the Fourier-Laplace transform \(\widehat u\) of \(u\). Namely, for any \(\eta\in \mathbb{R}^n\) and \(\xi\in\mathbb{R}^n\setminus \{0\}\), \[ \begin{multlined}\sup\{\langle x,\eta\rangle: (x,\xi)\in WF_A(u)\}= \lim_{\delta\to 0} \varliminf_{t\to 0} \sup_{|\widetilde\xi- \xi|<\delta} i_{\widehat u}(\widetilde\xi+ it\eta)/t=\\ \lim_{t\to 0,\widetilde\xi\to \xi} i_{\widehat u}(\widetilde\xi+ it\eta)/t= \lim_{t\to 0,\widetilde\xi\to \xi,\widetilde\eta\to \eta} i_{\widehat u}(\widetilde\xi+ it\widetilde\eta)/t,\end{multlined} \] where \(i_{\widehat u}\) is the indicator function of \(u\). In addition, it is proved that the convex hull of the support of \(u\) coincides with the union of convex hulls of all sets \(\{x:(x, \xi)\in WF_A(u)\}\), \(\xi\in \mathbb{R}^n\setminus\{0\}\).