Lévy measures for Dirichlet-type spaces on the unit bidisc

Abstract

For the Dirichlet-type space $\mathcal D(ρ^{(1)},ρ^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $ρ^{(1)}$ and $ρ^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the Lévy measure $ν_{(\mathscr M_z,1)}$ associated with the completely alternating multisequence $\left\{\|z^α\|_{\mathcal D(ρ^{(1)},ρ^{(2)})}^2 \right\}_{α\in\mathbb Z_+^2}$ admits the explicit representation \begin{equation*} dν_{(\mathscr M_z,1)}(x)=\frac{1}{1-x_1}d((S_{*}ρ^{(1)})\timesδ_1)(x)+\frac{1}{1-x_2}d(δ_1\times (S_{*}ρ^{(2)}))(x) \end{equation*} on $[0,1]^2\backslash\{(1,1)\},$ where $S_{*}ρ^{(i)},$ $i=1,2$, denotes the pushforward measure of $ρ^{(i)}$ by the map $S:\overline{\mathbb D}\rightarrow [0,1]$ defined by $S(z)=|z|^2,$ $z\in \overline{\mathbb D},$ and $δ_1$ denotes the Dirac measure at 1.