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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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