Unique determination of a magnetic Schr\"odinger operator with unbounded magnetic potential from boundary data

We consider the Gel'fand-Calder\'on problem for a Schr\"odinger operator of the form $-(\nabla + iA)^2 + q$, defined on a ball $B$ in $\mathbb R^3$. We assume that the magnetic potential $A$ is small in $W^{s,3}$ for some $s>0$, and that the electric potential $q$ is in $W^{-1,3}$. We show that, under these assumptions, the magnetic field $\operatorname{curl} A$ and the potential $q$ are both determined by the Dirichlet-Neumann relation at the boundary $\partial B$. The assumption on $q$ is critical with respect to homogeneity, and the assumption on $A$ is nearly critical. Previous uniqueness theorems of this type have assumed either that both $A$ and $q$ are bounded or that $A$ is zero.