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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## Abstract

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.

Abstract

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

Preprint

Creators

Keywords

Harnack inequality, hypoelliptic operators, potential theory

Subjects

DOI

Deposit date

14 Sep 2009 08:48

Last modified

16 May 2011 12:11

URI

## Other metadata

Document type

Preprint

Creators

Keywords

Harnack inequality, hypoelliptic operators, potential theory

Subjects

DOI

Deposit date

14 Sep 2009 08:48

Last modified

16 May 2011 12:11

URI

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