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 $$ \mathcal{ L} = \frac{ 1}{ w(x)}\sum_{ i = 1}^N\frac{ \partial}{ \partial x_i} \left(\sum_{ j = 1}^Nw(x)a_{ ij}(x)\frac{ \partial}{ \partial x_j}\right), \quad x \in \mathbb{ R}^N, $$ where $w \in C^{ \infty}(\mathbb{ R}^N,\mathbb{ R})$ is a (smooth and) strictly positive function on the whole of $\mathbb{ R}^N$ and $A(x) := \begin{ pmatrix}a_{ ij}(x) \end{ pmatrix}$ is a symmetric and positive semi-definite $N\times N$ 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 $$ -\mathcal{ L} u + Vu = 0, $$ 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 $\mathbb{ G}\setminus\{ 0\}$.
Categories: math.AP
