A Hardy Inequality for subelliptic operators with global fundamental solution, and an application to Unique Continuation

arXiv1512.07559MaRDI QIDQ6268688

Stefano Biagi, Andrea Bonfiglioli

Publication date: 23 December 2015

Abstract: This is a chapter from PhD Thesis by Stefano Biagi (advisor: prof. A. Bonfiglioli). We overview existing results showing that it is possible to generalize the classical Hardy's Inequality to more general linear partial differential operators (PDOs, in the sequel), possibly degenerate-elliptic, of the following quasi-divergence form ${\displaystyle }$ mathcal{L} = frac{1}{w(x)}sum_{i = 1}^Nfrac{partial}{partial x_i} left(sum_{j = 1}^Nw(x)a_{ij}(x)frac{partial}{partial x_j} ight), quad x in mathbb{R}^N, ${\displaystyle }$ where $win{C}^{infty}(mathbb{R}^N,mathbbR)$ is a (smooth and) strictly positive function on the whole of $mathbb{R}^N$ and $$ is a symmetric and positive semi-definite $NimesN$ matrix with real $C^{infty}$ entries. From such a inequality, it has been derived a result of unique continuation for the solutions of the equation ${\displaystyle }$ -mathcal{L} u + Vu = 0, ${\displaystyle }$ where $mathcalL$ is a left-invariant homogeneous PDO on a homogeneous Lie group $mathbbG$ and $V$ is real-valued function defined on $mathbbG$ and continuous on $mathbbGsetminus0$.

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