Surjectivity of Euler type differential operators on spaces of smooth functions

Abstract:

We develop a (global) solvability theory for Euler type linear partial differential equations $P(\theta )$ on $C^\infty (\Omega )$, with $\Omega$ an open subset of $\mathbb {R}^d$, i.e., for a special type of linear partial differential equation with polynomial coefficients. There is a natural closed upper bound $C^\infty _{I(P)}(\Omega )$ for the range of $P(\theta )$ on $C^\infty (\Omega )$. We characterize by $P(\theta )$-convexity type conditions those $\Omega$ such that $P(\theta )$ is surjective on $C^\infty _{I(P)}(\Omega )$. We also clarify when all shifted operators $P(c+\theta )$ are surjective on $C^\infty _{I(P(c+\ \cdot \ ))}(\Omega )$. We classify in geometric terms those $\Omega$ with $0\in \Omega$ such that every Euler operator $P(\theta )$ is surjective on $C^\infty _{I(P)}(\Omega )$. Moreover, we determine the operators $P(\theta )$ which are surjective onto $C^\infty _{I(P)}(\Omega )$ for every open set $\Omega \subseteq \mathbb {R}^d$. Under some mild assumptions on $\Omega$, we characterize those Euler operators which are invertible on $C^\infty (\Omega )$. Under the same assumptions we also calculate the spectrum of $P(\theta )$ on $C^\infty (\Omega )$. The results follow from the solvability theory for Hadamard type operators on the space of smooth functions and from a new general Mellin transform, both developed in this paper.