Local solvability

Summary

A linear differential operator $P$ is locally solvable at a point if $P u = f$ has a solution near that point for every smooth $f$. Operators with constant coefficients are always locally solvable, but Hans Lewy found a first-order operator with polynomial coefficients that is locally solvable at no point. This discovery started the study of necessary and sufficient conditions for local solvability. See the definition.