On the Strong unique continuation property of a degenerate elliptic operator with Hardy type potential

In this paper we prove strong unique continuation for the following degenerate elliptic equation \begin{equation}\label{e0} \Delta_zu +|z|^2\partial_t^2u = Vu,\quad (z,t) \in $\mathbb{R}$^N \times $\mathbb{R}$ \end{equation} where the potential $V$ satisfies either of the following growth assumptions \begin{align} & |V(z,t)| \leq \frac{f(\rho(z,t))}{\rho(z,t)^2},
$\text{where }f\text{ satisfies the Dini integrability condition as in (1.3)}$ \ & $\text{or when }$ \notag \ & |V(z,t)| \leq C\frac{\psi(z,t)^{\epsilon}}{\rho(z,t)^2},\ \text{for some $ϵ>0$ with $\psi$ as in (2.6) and $N$ even.} \notag \end{align} This extends some of the previous results obtained in [G] for this subfamily of Baouendi-Grushin operators. As corollaries, we obtain new unique continuation properties for solutions $u$ to $${\mathrm{\Delta }}_{\mathbb{H}}u=Vu$$ with certain symmetries as expressed in (1.6) where ${\mathrm{\Delta }}_{\mathbb{H}}$ corresponds to the sub-Laplacian on the Heisenberg group ${\mathbb{H}}^n$.