Higher Microlocalization and Propagation of Singularities

Abstract

A good point to start is to discuss the range of applicability of the Fourier transform, or rather of its inverse. We define the inverse Fourier transform formally by

$$h\left( x \right) = {{\left( {2\pi } \right)}^{{ - n}}}{{\smallint }_{{{{R}^{n}}}}}{{e}^{i}}\left\langle {x,\xi } \right\rangle f\left( \xi \right)d\xi $$

(1)

and the problem is to see for which classes of functions (or distributions) we can give a reasonable meaning to (1). In the last century f would have been assumed to be integrable; nowadays we would write the corresponding condition in terms of Lebesgue integrals and ask for f ∈ L ¹ (R ⁿ). There is a natural extension of this to f ∈ L ²(R ⁿ) using Plancherel’s theorem. A far reaching extension was achieved by L.Schwartz, who took f ∈ S’(R ⁿ).