Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , 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.