Quantitative unique continuation for non-regular perturbations of the Laplacian

In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$, where $d \geq 3$. Here, $V \in L^{q_0}$, $W_1 \in L^{q_1}$, and $W_2 \in L^{q_2}$ with $q_0 > d/2$, $q_1 > d$, and $q_2 > d$. Our aim is to provide an explicit quantification of the unique continuation property with respect to the norms of the potentials. To achieve this, we revisit the Carleman estimates established in [Dehman-Ervedoza-Thabouti-2023] and prove a refined version of them, and we combine them with an argument due to T. Wolff introduced in [Wolff-1992] for the proof of unique continuation for solutions of equations of the form $\Delta u = V u + W_1 \cdot \nabla u$.