Parametrix

Summary

A parametrix of a differential operator is an approximate inverse: an operator whose composites with the given operator differ from the identity by smoothing operators. For operators on $\R^n$ it is the same as an approximate fundamental solution, one whose error is a smooth function instead of zero. Every elliptic operator has a parametrix, which can be built as a pseudo-differential operator from the inverse of the principal symbol. Parametrices are the main tool for proving elliptic regularity and the Fredholm property of elliptic operators on compact manifolds.