Liouville type theorem for solutions of linear partial differential equations with constant coefficients

We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.