Oscar Pezzi: Local integrability of fundamental solutions

Abstract: In the theory of partial differential equations, the concept of fundamental solution — that is, a distribution E satisfying P(D)E = δ — plays a central role. The Malgrange–Ehrenpreis Theorem guarantees the existence of such solutions for any linear differential operator with constant coefficients.

However, a subtler and more delicate question remains: is the fundamental solution locally integrable? The question arises from the fact that when the operator P(D) is hypoelliptic — namely, when a solution u of P(D)u = f is smooth wherever f is smooth — then the fundamental solution of P(D) is smooth outside the origin. In other words, can we view these distributions as actual functions in some neighbourhood of the origin?

Based on Hörmander's article On the Local Integrability of Fundamental Solutions.