Quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms

In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when $u$ is a non-trivial solution to $\triangle u + W \cdot \nabla u + V u = 0$ in some open, connected subset of $\mathbb R^n$, where $n \geq 3$, we characterize the vanishing order of solutions in terms of the norms of $V$ and $W$ in their respective Lebesgue spaces. Using these maximal order of vanishing estimates, we also establish quantitative unique continuation at infinity results for solutions to $\triangle u + W \cdot \nabla u + V u = 0$ in $\mathbb R^n$. The main tools in our work are new versions of $L^p\to L^q$ Carleman estimates for a range of $p$- and $q$-values.