Potential Analysis

, Volume 25, Issue 2, pp 147–164 | Cite as

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: \(L_\epsilon=\sum_{i=1}^m X_i^{2} + \epsilon\Delta\), 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.

Mathematics Subject Classifications (2000)

35H10 35A08 43A80 35B45 

Key words

hypoelliptic operators Carnot groups fundamental solution a priori estimates 


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© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Dip. di MatematicaUniversità di BolognaBolognaItaly

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