For your first question, have a look at chapter 1 of the book of Jean-Marc Delort, "F.B.I. Transformation - Second Microlocalization and Semilinear Caustics", Springer Lecture Notes in Mathematics **1522** (1992), particularly Corollary 1.4, page 11. Delort also provides a characterization of the Sobolev wave front sets using the FBI transform (Theorem 1.2, pp. 8-11) attributed to Patrick Gérard ("Moyennisation et régularité deux-microlocale", Ann. Scient. Éc. Norm. Sup. 4ème série **23** (1990) 89-121).

The FBI transform is essentially the Fourier transform followed by a convolution with a Gaussian. The latter operation mollifies the singularities of distributions (and even hyperfunctions) in the *analytic* category, as already noticed by H. Whitney in his work concerning the extension of differentiable functions off closed subsets in the 30's (the extension happens to be analytic away from the original domain, thanks to the Gaussian convolution).

Jacques Bros and Daniel Iagolnitzer's original motivation to introduce the FBI transform in the 70's was to analyse the *analytic* wave front set of certain distributions / hyperfunctions that appear in quantum field scattering theory (in their work, they followed Sato's school and called the analytic wave front set of a distribution its *essential support*, due to the Holmgren-type theorem proven by Kashiwara that links the support and the analytic wave front set of a hyperfunction). When working at the hyperfunction level, where everything is boundary values of holomorphic functions, it's pretty natural to convolute using a Gaussian weight, which is entire analytic and intrinsically linked to the geometry of the complex domain. The holomorphic extension of the Gaussian weight off real arguments $x$ then automatically includes the imaginary exponential appearing in the Fourier transform, as follows:

$$\exp(-\lambda\langle z,z\rangle)=\exp(-\lambda\langle x+i\xi,x+i\xi\rangle)=e^{\lambda\langle \xi,\xi\rangle}e^{-\lambda\langle x,x\rangle}e^{-2i\lambda\langle x,\xi\rangle}\ .$$

At the rhs of the second equality, we notice the exponential factor that appears in the FBI transform characterizations of the wave front set. In fact, the FBI transform has the advantage of providing a relatively simple and unified characterization of all known wave front sets, ranging from the Sobolev to the analytic and Gevrey wave front sets.