On a class of differential operators with constant coefficients

A linear differential operator P(D) = P(D1, …, Dn) with constant coefficients is called almost hypoelliptic if all the derivatives DαP of the characteristic polynomial P(ξ1, …, ξn) can be estimated by P. The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P(D)u = f is again an infinitely differentiable function and P(ξ) → ∞ as ξ → ∞.