Hypoelliptic

Hypoelliptic refers to a property of partial differential equations and their solutions. A linear partial differential operator P is called hypoelliptic if for any open set $\Omega$, any distribution $u$ such that $Pu$ is $C^\infty$ on $\Omega$ implies that $u$ is also $C^\infty$ on $\Omega$. In simpler terms, if the application of the operator smooths out a distribution, then the original distribution must have already been smooth. This is a weaker condition than elliptic, which requires that $Pu$ being $C^\infty$ implies $u$ is $C^\infty$ everywhere. A key characteristic of hypoelliptic operators is that they do not have characteristic surfaces where solutions can lose regularity. The heat operator is a classic example of a hypoelliptic operator. The property of hypoellipticity is crucial in the study of the regularity of solutions to differential equations and has significant applications in various areas of mathematics and physics, including the analysis of wave propagation and quantum mechanics. It is a fundamental concept in the theory of partial differential equations.