Abstract:
In this paper we study scalar pseudodifferential operators for which the gradient $\operatorname{grad}_{x,\xi}p^0(x,\xi)$ of the principal part of the symbol does not vanish and is not proportional to a real vector at any characteristic point $(x,\xi)\in\Omega\times\{\mathbf R^n\setminus0\}$. Such operators are called nondegenerate. It is assumed in addition that for each point of $\Omega\times\{\mathbf R^n\setminus0\}$ there exists an operator in the Lie algebra generated by the operators $P$ and $P^*$ the principal part of the symbol of which does not vanish at this point. For these operators we present here hypoellipticity conditions, conditions for the local solvability of the equation $Pu=f$, a theorem on the smoothness of the solutions of this equation, and so on. All of the conditions obtained have a simple algebraic character and are exact, necessary and sufficient.
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