On a criterion for hypoellipticity

In this paper a criterion for hypoellipticity is proved which is formulated in terms of certain estimates in the $H_{(s)}$ norms, and which is a generalization of a criterion of Trèves. With the use of this criterion it is possible to prove the hypoellipticity of certain operators that do not satisfy Hörmander's criterion. It is proved, for example, that the operator $P=\partial^2/\partial x^2+\varphi^2(x)\partial^2/\partial y^2$ is hypoelliptic, where $\varphi(x)$ is an infinitely differentiable function that is not equal to zero for $x\ne0$ and has a zero of infinite order at $x=0$.
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