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Liouville-type theorems for real sub-Laplacians. (English) Zbl 1016.35014
Let $${\mathcal L}$$ be a real sub-Laplacian on $$\mathbb{R}^N$$, $$N\geq 3$$, and denote by $$G= (\mathbb{R}^N,0)$$ its related homogeneous group. Let $$Q$$ be the homogeneous dimension of $$G$$. The main result is the following generalization of the classical Harnack inequality. Let $$Q/2< p\leq\infty$$. There exist positive constants $$C$$ and $$\theta$$ such that $\sup_{|x|\leq r} u(x)\leq C\Biggl(\inf_{|x|\leq r} u(x)+ r^{2-{Q\over p}}\|{\mathcal L}u\|_{L^p(D(0,\Theta_r))}\Biggr)$ for every nonnegative function $$u\in C^2(\mathbb{R}^n,\mathbb{R})$$ and for every $$r> 0$$.
A representation formula for functions $$u$$ for which $${\mathcal L}u$$ is a polynomial is also shown. As a consequence, some conditions are given ensuring that $$u$$ is a polynomial whenever $${\mathcal L}u$$ is a polynomial.

##### MSC:
 35H20 Subelliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J70 Degenerate elliptic equations 43A80 Analysis on other specific Lie groups
##### Keywords:
generalized Harnack inequality; representation formula
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