Parametrices

Parametrices are objects in the theory of partial differential equations that play the role of approximate inverses to differential operators. If P is a differential operator of order m on a smooth manifold, a parametrix Q is an operator of order −m such that P Q − I and Q P − I are smoothing operators (that is, belong to the class of operators with infinitely smoothing kernels). Equivalently, P Q = I − R and Q P = I − S with R and S smoothing.

For elliptic operators, parametrices exist in the pseudodifferential calculus. If P is elliptic, one can construct Q locally (and globally on compact manifolds) so that P Q and Q P differ from the identity by smoothing operators. The construction relies on inverting the principal symbol p_m(x,ξ) and producing an asymptotic expansion of the symbol q_{-m}(x,ξ) ≈ p_m(x,ξ)^{-1} modulo lower order terms.

Consequences of having a parametrix include mapping properties and regularity results. The parametrix Q maps Sobolev spaces H^s to H^{s+m}, while P maps H^s to H^{s−m} up to smoothing remainders. Therefore elliptic regularity follows: if P u is smooth, then u is smooth; more generally, the microlocal structure of u can be described in terms of that of P u. Parametrices are central in microlocal analysis, spectral theory, and inverse problems, and they underpin the construction of fundamental solutions in related settings.

Non-uniqueness: any two parametrices differ by a smoothing operator. Local existence is standard via partitions of unity and coordinate charts.

Example: for the Laplace operator P = −Δ on R^n, there exists a parametrix Q whose kernel behaves like the Newton potential in appropriate dimensions; in the pseudodifferential framework, Q has order −2 and P Q = I − R with R smoothing.