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

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 $$\mathcal{L}=\frac{1}{w(x)}\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(\sum _{j=1}^{N}w(x)a_{ij}(x)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}),x\in {\mathbb{R}}^N,$$ where $w\in C^{\mathrm{\infty }}({\mathbb{R}}^N,\mathbb{R})$ is a (smooth and) strictly positive function on the whole of ${\mathbb{R}}^N$ and $A(x):=\left(\begin{array}{c}{a}_{ij}(x)\end{array}\right)$ is a symmetric and positive semi-definite $N\times N$ matrix with real $C^{\mathrm{\infty }}$ entries. From such a inequality, it has been derived a result of unique continuation for the solutions of the equation $$-\mathcal{L}u+Vu=0,$$ where $\mathcal{L}$ is a left-invariant homogeneous PDO on a homogeneous Lie group $\mathbb{G}$ and $V$ is real-valued function defined on $\mathbb{G}$ and continuous on $\mathbb{G}\setminus \{0\}$.