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Harnack’s inequality for sum of squares of vector fields plus a potential. (English) Zbl 0795.35018
We study quantitative properties of solutions of operators of the type \({\mathcal L}= \sum_{j=1}^ p X_ j^ 2\), where \(X_ j\) are smooth vector fields in \(\mathbb{R}^ n\) satisfying Hörmander’s condition of hypoellipticity: rank Lie \([X_ 1,\dots, X_ p]=n\) at every \(x\in \mathbb{R}^ n\). Our main objective is to establish a uniform Harnack’s inequality for nonnegative solutions and the continuity of solutions of \((-{\mathcal L}+ V)u=0\), where \(V\) is a measurable function belonging to a suitable class.

MSC:
35H10 Hypoelliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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