On smooth solutions of a class of almost hypoelliptic equations of constant strength

Abstract

In this paper we state a new theorem about smoothness of solutions of almost hypoelliptic and hypoelliptic by Burenkov equation P(x',D)u = 0, where the coefficients of the linear differential operator P(x,D) = P(x₁, ..., x_n, D₁, ..., D_n) of uniformly constant strength depend only on the variables x' = (x₁, ..., x_k), k ≤ n: if the operator P(x', D) is hypoelliptic by Burenkov and almost hypoelliptic for any   x'∈ E_k, then all the solutions of the differential equation P(x', D)u = 0 belonging to a certain weighted Sobolev class are infinitely differentiable functions.