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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.

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
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