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.
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Citti, G., Manfredini, M. Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators. Potential Anal 25, 147–164 (2006). https://doi.org/10.1007/s11118-006-9014-4
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DOI: https://doi.org/10.1007/s11118-006-9014-4