Microlocal analysis

Summary

Microlocal analysis studies functions, distributions and linear operators on $\R^n$ or on a manifold by localizing both in position $x$ and in frequency direction $\xi$. Its natural home is phase space, the cotangent bundle, whose points are pairs $(x, \xi)$. Its main tools are the wave front set, pseudo-differential operators and Fourier integral operators. It explains where and in which directions solutions of linear partial differential equations are singular, and how these singularities move.