Carleman Estimates for Critical Electromagnetic Schrödinger Operators on Conic Manifolds

We establish an $L^p$–$L^q$-type Carleman estimate for the Schrödinger operator $H_{A,V}$ with scaling-critical electromagnetic singular potentials on a conical singular space $(X,g)$ of dimension $d\ge 3$, where the metric is $g=d{r}^2+r^{2}h$ and $X=C(Y)=(0,\mathrm{\infty }{)}_r\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$. More precisely, we prove $$\mathrm{\parallel }{e}^{\tau \phi }u{\mathrm{\parallel }}_{L^{2d\mathrm{/}(d-2)}(X)}\le C\mathrm{\parallel }{e}^{\tau \phi }{H}_{A,V}u{\mathrm{\parallel }}_{L^{2d\mathrm{/}(d+2)}(X)}$$ with the quadratic weight $\phi (r)=\frac{1}{4}(\log r{)}^2$, uniformly in $\tau$. As an application of this Carleman estimate, we prove the unique continuation property for this electromagnetic Schrödinger operator when the cross-section is $Y={\mathbb{S}}^{d-1}$.