Microlocal orders of singularities for distributions and an application to Fourier integral operators (Q2640797)

Two different ways of quantifying the singular behaviour of distributions \(f\in {\mathcal E}'\) are introduced: \[ OS_ 1(f)=\inf \{r| \lim_{| \xi | \to \infty}(1+| \xi |)^{-r}\hat f(\xi)=0\},\quad OS_ 2(f)=\inf \{r| f\in H^{-r}\}, \] (where \(H^{-r}\) denotes the Sobolev space). For general distributions corresponding local (OS(f;x)) and microlocal (OS(f;x,\(\xi\))) orders are introduced as \(\lim_{n\to \infty}OS(\alpha_ nf)\), where \(\alpha_ n\) is a localizing sequence of \(C_ 0^{\infty}\)-functions (to x) or pseudodifferential operators in \(S_ 0\) (to (x,\(\xi\))). The order \(OS_ 2(f;x,\xi)\) is shown to equal \(-s_ f^*(x,\xi)\) of \textit{J. J. Duistermaat} and \textit{L. Hörmander} [Acta Math. 128, 183-269 (1972; Zbl 0232.47055)]. Hypoellipticity and propagation of singularities are discussed in light of the orders introduced.