On infinite differentiability of solutions of nonhomogeneous almost hypoelliptic equations

Abstract

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.