On the local solvability of a vector field and the hypoellipticity of the differential operator

Abstract

In this paper, we give a necessary and sufficient condition for the local solvability of a vector field $L = a(x,y)\frac{∂}{∂ x} + ib(x,y) \frac{∂}{∂ y}$. In particular, in case of $L = \frac{∂}{∂ x} + ib(x) \frac{∂}{∂ y}$, we give a simple proof of the necessary and sufficient condition for the local solvability of L. Also, we give a partial positive result to Schwartz`` question about the hypoellipticity of a differential operator.