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On the second order derivatives of convex functions on the Heisenberg group

Cristian E. GutiérrezAnnamaria Montanari — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In the euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous –convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for...

Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators

Bruno FranchiCristian E. GutiérrezRichard L. Wheeden — 1994

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong A weight with respect to the metric associated with the operator. Roughly speaking, the strong A condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

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