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

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