Hypoelliptic differential operators with generalized constant coefficients

The space G of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter epsilon. There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on epsilon and behave like powers of epsilon as epsilon --> 0. The solutions of such family of equations should also satisfy the power order estimate with, respect to epsilon.