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

Abstract

A linear differential operator P(x, D) = P(x₁,... x_n, D₁,..., D_n) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in Eⁿ is called formally almost hypoelliptic in Eⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. 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.