Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz’ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory.