Bi-parameter Potential theory and some applications to holomorphic spaces

The bidisc carries its own natural Potential Theories, which are not developed in depth. The main obstructions are the lack of the usual basic tools: covering theorems, maximum principles, inequalities for weighted maximal functions... In the case of the bi-disc, we prove a Capacitary Strong Inequality and deduce from it a characterization of the trace inequalities for a Bessel Potential Space. As a consequence, we characterize the multipliers for the Holomorphic Dirichlet Space on the bidisc.

Work in collaboration with Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti.