Uniqueness in an inverse boundary problem for a magnetic Schr\"odinger operator with a bounded magnetic potential

We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in $\R^n$, $n\ge 3$, for the magnetic Schr\"odinger operator with $L^\infty$ magnetic and electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schr\"odinger operator with a gain of two derivatives.