Carleman Estimates for Critical Electromagnetic Schr\"odinger Operators on Conic Manifolds

Abstract

We establish an $L^p$--$L^q$-type Carleman estimate for the Schr\"odinger operator $H_{A,V}$ with scaling-critical electromagnetic singular potentials on a conical singular space $(X,g)$ of dimension $d\geq3$, where the metric is $g=dr^2+r^2h$ and $X=C(Y)=(0,\infty)_r\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$. More precisely, we prove $$ \|e^{\tau\phi}u\|_{L^{2d/(d-2)}(X)}\le C\,\|e^{\tau\phi}H_{A,V}u\|_{L^{2d/(d+2)}(X)} $$ with the quadratic weight $\phi(r)=\tfrac14(\log r)^2$, uniformly in $\tau$. As an application of this Carleman estimate, we prove the unique continuation property for this electromagnetic Schr\"odinger operator when the cross-section is $Y=\mathbb{S}^{d-1}$.