Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

Abstract

Solving a problem of L. Schwartz, those constant coefficient partial differential operators P(D) are characterized that admit a continuous linear right inverse on ℰ(Ω) or 𝒟 ′ (Ω), Ω an open set in R n . For bounded Ω with C 1 -boundary these properties are equivalent to P(D) being very hyperbolic. For Ω=R n they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial P.