microdifferential

A microdifferential is a mathematical operator that acts on functions. It is a generalization of the differential operator and plays a significant role in the theory of partial differential equations, particularly in the study of their local properties. The concept of microdifferential operators was introduced by Mikio Sato and his collaborators.

These operators are defined on the cotangent bundle of a manifold, which is a geometric space intimately related to the derivatives of functions. Specifically, microdifferential operators are constructed using a notion of "microlocalization," which allows for the analysis of functions and their derivatives at specific points in both position and direction. This contrasts with traditional differential operators, which operate on functions in the usual sense.

The theory of microdifferential operators provides a powerful framework for understanding the behavior of solutions to partial differential equations near singularities. It allows mathematicians to study phenomena such as wave propagation and the existence of generalized solutions in situations where classical methods might fail. This field has connections to various areas of mathematics and physics, including harmonic analysis, spectral theory, and quantum mechanics.