Uniform Estimates of the fundamental solution for a family of hypoelliptic operators

In this paper we are concerned with a family of elliptic operators L-epsilon represented as sum of square vector fields, whose limit is an Hormander type operator L. It is well known that each operator L_epsilon admits a fundamental solution. Here we establish a priori estimates uniform in epsilon of the fundamental solution, using a generalization of the freezing and lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in epsilon, for solutions of the approximated equation. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.