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

Abstract

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.

57 pages, 5 figures, title changed, minor editing

References in corpus (1)

• Interpolating sequences in spaces with the complete Pick property

Cited by in corpus (6)

• Bi-parameter Carleson embeddings with product weights
• Improved surrogate bi-parameter maximum principle
• Dyadic bi-parameter repeated commutator and dyadic product BMO
• Dyadic bi-parameter simple commutator and dyadic little BMO
• Counterexamples for bi-parameter Carleson embedding
• Differences between the potential theories on a tree and on a bi-tree