On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

摘要

A linear differential operator P(x, D) = P(x (1),... x (n) , D (1),..., D (n) ) = a(alpha) gamma (alpha)(x)D (alpha) with coefficients gamma (alpha)(x) defined in E (n) is called formally almost hypoelliptic in E (n) if all the derivatives D (nu) (xi) P(x, xi) can be estimated by P(x, xi), and the operator P(x, D) has uniformly constant power in E-n. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.