A note on Harnack inequalities and propagation set for a class of hypoelliptic operators

Cinti, Chiara ; Nystrom, Kaj ; Polidoro, Sergio (2009) A note on Harnack inequalities and propagation set for a class of hypoelliptic operators. [Preprint]
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In this paper we are concerned with Harnack inequalities for non-negative solutions to a class of second order hypoelliptic ultraparabolic partial differential equations in the form $$L u:= X_1^2 u + ... + X_m^2 u + X_0 u - \partial_t u = 0$$ where the vector fields $X_1, \dots, X_m$ and $X_0 - \partial_t$ are invariant with respect to a suitable homogeneous Lie group on $R^{N+1}$. Our main goal is the following result: consider any domain $Omega$ of $R^{N+1}$ and fix any $(x_0,t_0)$ in $Omega$. We give a geometric sufficient condition on the compact subsets $K$ of $Omega$ for which the Harnack inequality $$\sup_{K} u \le C_K u(x_0,t_0)$$ holds for all non-negative solutions $u$ to the equation $L u=0$ in $Omega$. We also compare our result with an abstract Harnack inequality from potential theory.

Document type
Cinti, Chiara
Nystrom, Kaj
Polidoro, Sergio
Harnack inequality, hypoelliptic operators, potential theory
Deposit date
14 Sep 2009 08:48
Last modified
16 May 2011 12:11

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