This site is maintained by François Monard and Plamen Stefanov. Our goal is to present basic ideas of Microlocal Analyis with a number of applications and illustrations.
Microlocal and semiclassical analysis are variants of analysis in the phase space which allow us to make refined statements about the behavior and regularity of functions or distributions. In inverse problems, they are key toolboxes to address injectivity (analytic microlocal analysis), stability statements (recoverable singularities), sampling issues (semiclassical sampling), and even to understand and describe reconstruction artifacts in various geometric situations. This website, through a number of examples, aims to illustrate some applications of Microlocal Analysis with an emphasis on Inverse Problems.
What is a "phase space"? It is the set of points, generically denoted by \(x\) and (co)directions generically denoted by \(\xi\not=0\). Roughly speaking, we want to understand the behavior of a distribution at \(x\) in the (co)direction \(\xi\). This is NOT taking a directional derivative! The inspiration comes from the properties of the Fourier Transform (FT). If \(f\) is compacrly supported, we can tell the regularity of \(f(x)\) from its FT \(\hat{f}(\xi)\). A rapid decay \(|\hat{f}(\xi)|\le C_N\langle\xi\rangle^{-N}\) for every \(N>0\) means that \(f\) is smooth and vice versa.
This is not localized in space however. To do this, we consider the "windowed FT" of \(f\); i.e., we choose a point \(x_0\), and a "window" function \(\phi\in C_0^\infty\); and then consider the behavior of \(\widehat{\phi f}(\xi)\), as \(|\xi|\to\infty\). Then we would like to declare \(f\) microlocally smooth at \((x_0,\xi_0)\) (note that we are in phase space now) iff \(\widehat{\phi f}(\xi)\) decays rapidly when \(|\xi|\to\infty\) while going along the ray in its direction; more precisely, when it tends to \(0\) rapidly when \(\xi\) is replaced by \(r\xi\), as \(r\to\infty\).