Hypoellipticity of a system of complex vector fields

Abstract

We are concerned with the system generated by the complex vector fields $L_1$, …, $L_m$. We find the sufficient conditions for the hypoellipticity of the single partial differential operator $P = ∑_{j=1}^mL_jL_j$ which are also the sufficient conditions for the hypoellipticity of the system L. We establish the hypoellipticity of L by giving some conditions on the solution of the system of the Cauchy problems $L_jz=0$, $z\mid_{t=0}$=0,1 ≤j ≤m$ Finally we give some examples concerning this situtation.