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

Abstract

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: $L_\epsilon=\sum_{i=1}^m X_i^{2} + \epsilon\Delta$, in ${\mathbb R}^n$ where $\Delta$ is the Laplace operator, $m<n$, and the limit operator $L = \sum_{i=1}^m X_i^{2}$ is hypoelliptic. It is well known that $L_\epsilon$ admits a fundamental solution $\Gamma_\epsilon$. Here we establish some a priori estimates uniform in $\epsilon$ of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in $\epsilon$, for solutions of the approximated equation $L_\epsilon u = f$. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.