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

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 $P(x, D) = P(x_1,\dots, x_n, D_1,\dots, D_n)$ of uniformly constant strength depend only on the variables $x' = (x_1,\dots, x_k)$, $k \leqslant n$: if the operator $P(x', D)$ is hypoelliptic by Burenkov and almost hypoelliptic for any $x'\in\mathbb{E}^k$, then all the solutions of the differential equation $P(x', D)u = 0$ belonging to a certain weighted Sobolev class are infinitely differentiable functions.