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

TL;DR

This paper solves the Carleson measure problem for the Dirichlet space on the bidisc by establishing a sharp capacitary criterion in a bi-parameter setting. It builds a bi-parameter potential theory on the bitree, proving a Strong Capacitary Inequality despite the failure of the Maximum Principle, and then transfer these discrete results back to the continuous bidisc to characterize Carleson measures and multipliers. The approach hinges on a careful discretization via the bitree, duality arguments, and a Maz'ya–Stegenga capacitary framework adapted to a product geometry. The results yield a complete description of the Dirichlet space Carleson measures on the bidisc and a corresponding multiplier characterization, with potential extensions to polydiscs and other bi-parameter function spaces. The work lays a rigorous foundation for bi-parameter potential theory and its connections to operator theory in several complex variables.

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.

Figures (4)

• Figure 1: Discretized disc
• Figure 2: Graph $\mathfrak{G}$
• Figure 3:
• Figure 4:

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.1
• Theorem 1.6
• Lemma 2.1
• Proposition 2.1
• Lemma 2.2