The study of singularities of distributions and PDE solutions at the combined location–frequency (phase space) level using tools such as the wavefront set, pseudodifferential and Fourier integral operators, and symplectic geometry to track how singularities propagate and interact.

Principle

Localize simultaneously in space and frequency: singularities are described by their position and cotangent directions (the wavefront set), and their propagation under linear PDEs follows bicharacteristic flow in phase space determined by the principal symbol; microlocal ellipticity gives local regularity in phase space.

Demonstration

For the wave equation on a smooth manifold, singularities of initial data propagate along null-bicharacteristics; applying a pseudodifferential parametrix one controls the wavefront set and proves finite-speed propagation and reflection/transmission laws at smooth interfaces.

Misapplication

Relying only on pointwise Fourier transforms or purely spatial localization and ignoring directional frequency information; or treating pseudodifferential calculus heuristically without checking symbol classes and microlocal ellipticity, which produces incorrect conclusions about propagation.

Consequence

Gives sharp regularity and propagation theorems, enables construction of parametrices, clarifies scattering and diffraction phenomena, and connects PDE behavior with symplectic/topological invariants used in index theory and quantum/classical correspondences.

Reversal

Purely local (physical-space) analysis that ignores frequency content cannot resolve directional singularities or propagation along bicharacteristics; global harmonic analysis captures frequency but loses spatial localization needed for boundary/interface phenomena.

Boundary

Applies primarily to linear PDEs, distributions and linear operators admitting symbol calculus (pseudodifferential/FIO framework); it is less directly applicable to fully nonlinear PDEs without linearization or to purely probabilistic models lacking a phase-space description.

Semantic Tension

Overlaps with harmonic analysis and geometrical optics: harmonic analysis emphasizes global frequency decompositions, geometrical optics uses high-frequency asymptotics; microlocal analysis distinguishes itself by combining local position and directional frequency to resolve singular structures.

Synthesis

Microlocal analysis is the phase-space calculus that describes where and in which directions solutions fail to be smooth, using symbol methods and symplectic dynamics to predict and control the propagation, interaction, and resolution of singularities.