Abstract
We prove weighted \({L^p}\)-Liouville theorems for a class of second-order hypoelliptic partial differential operators \({\mathcal{L}}\) on Lie groups \({\mathbb{G}}\) whose underlying manifold is \({n}\)-dimensional space. We show that a natural weight is the right-invariant measure \(\check{H}\) of \({\mathbb{G}}\). We also prove Liouville-type theorems for \({C^{2}}\) subsolutions in \({L^{p}(\mathbb{G},\check{H})}\). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator \({\mathcal{L}-\partial_{t}}\).
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Bonfiglioli, A., Kogoj, A.E. Weighted \({{L^p}}\)-Liouville theorems for hypoelliptic partial differential operators on Lie groups. J. Evol. Equ. 16, 569–585 (2016). https://doi.org/10.1007/s00028-015-0313-3
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DOI: https://doi.org/10.1007/s00028-015-0313-3