On infinite differentiability of solutions of nonhomogeneous almost hypoelliptic equations

A linear differential operator $P(D)$ with constant coefficients is called almost hypoelliptic if all derivatives $P^{(\nu)}(\xi)$ of the characteristic polynomial $P(\xi)$ can be estimated above via $P(\xi)$. In this paper it is proved that all solutions of the equation $P(D)u=f$ where $f$ and all its derivatives are square integrable with a certain exponential weight, which are square integrable with the same weight, are also such that all their derivatives are square integrable with this weight, if and only if the operator $P(D)$ is almost hypoelliptic.