Microlocal analysis is a branch of mathematical analysis that studies functions, generalized functions and partial differential equations by localizing them both in position and in frequency. The main idea is to describe a singularity not only by the point at which it occurs, but also by the cotangent direction in which it occurs. The term microlocal refers to localization at a small scale in phase space. The information of position and covector in which a singularity occurs is encoded by the wave front set of a distribution, a conic subset of the cotangent bundle with the zero section removed.

Microlocal analysis was developed from the 1950s through the 1970s in connection with linear partial differential equations, Fourier transform methods, hyperfunctions and pseudo-differential operators. It is concerned with elliptic regularity, propagation of singularities, Fourier integral operators, geometric optics, scattering theory, spectral theory, semiclassical analysis and inverse problems.

Microlocal smoothness

, this is expressed using the Fourier transform.

Thus microlocal smoothness is a weaker notion than ordinary smoothness: a distribution can be microlocally smooth in some directions at a point and not others.

.

Wave front set

The wave front set.

. Thus the wave front set contains the ordinary location of singularities, but also records their directions.

The wave front set of a Dirac delta distribution:

, its wave front set is contained in the conormal bundle. The singularity is therefore concentrated in directions normal to the hypersurface.

Pseudodifferential operators

Pseudo-differential operators generalize differential operators by allowing more general functions of the frequency variable. Locally, such an operator has the form

is called the symbol of the operator.

of the cotangent bundle if its principal symbol is nonzero there. The set where the principal symbol vanishes is the characteristic set of the operator.

Pseudodifferential operators are microlocal in the sense that they do not create new singularities in arbitrary directions:

. Conversely, elliptic pseudodifferential operators can be inverted microlocally, which leads to microlocal forms of elliptic regularity.

Propagation of singularities

on the cotangent bundle determines curves called bicharacteristics.

This theorem gives a rigorous version of geometric optics. For hyperbolic equations, singularities travel along the rays determined by the Hamiltonian flow of the principal symbol. In the case of the wave equation on a Riemannian manifold, these rays are related to the geodesic flow. In Lorentzian geometry, the corresponding statement is that singularities of wave solutions propagate along null geodesics (null bicharacteristics).

Microlocalization functor

See also

• Algebraic analysis

• Microfunction

• Microdifferential operator

References

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External links

• lecture notes by Richard Melrose

• newer lecture notes by Richard Melrose