Fundamental solutions of evolutionary PDOs and rapidly decreasing distributions

TL;DR

This paper establishes a characterization of fundamental solutions for certain partial differential operators with constant coefficients, linking the boundedness of roots to the existence and uniqueness of rapidly decreasing solutions supported in a half-space.

Contribution

It proves an equivalence between the boundedness of polynomial roots' real parts and the existence of a unique fundamental solution with specific support and decay properties.

Findings

Boundedness of roots' real parts is equivalent to existence of a special fundamental solution.

Such fundamental solutions are unique when supported in the half-space.

The solutions have properties expressed in Schwartz space of rapidly decreasing distributions.

Abstract

Let $P(\partial_0,\partial_1,...,\partial_n)$ be a PDO on $\symR^{1+n}$ with constant coefficients. It is proved that (i) the real parts of the $\lambda$-roots of the polynomial $P(\lambda,i\xi_1,...,i\xi_n)$ are bounded from above when $(\xi_1,...,\xi_n)$ ranges over $\symR^n$ if and only if (ii) $P$ has a fundamental solution with support in $H_+=\{(x_0,x_1,\allowbreak..., x_n)\in \symR^{1+n}:x_0\ge0\}$ having some special properties expressed in terms of the L. Schwartz space $\calO^{\prime}_C$ of rapidly decreasing distributions. Moreover, it is proved that the fundamental solution with support in $H_+$ having these special properties is unique.