Document Type

Article

Publication Date

3-6-2014

Publication Title

Advances in Mathematics

Abstract

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where p≥2, $\ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure μ, and X∗i denotes the adjoint of Xi with respect to μ. Our estimates are derived assuming that (i) the control distance dgenerated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the μ-measure of d−metric balls and (iii) the validity of a Poincar\'e inequality involving $\X$ and μ. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

Volume

257

Issue

1

First Page Number

25

Last Page Number

65

DOI

10.1016/j.aim.2014.02.018

Version

Preprint

Rights

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

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