Lévy measures for Dirichlet-type spaces on the unit bidisc

Abstract

For the Dirichlet-type space $\mathcal{D}({\rho }^{(1)},{\rho }^{(2)})$ on the unit bidisc ${\mathbb{D}}^2,$ where ${\rho }^{(1)}$ and ${\rho }^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb{D}},$ we show that the Lévy measure ${\nu }_{({\mathcal{M}}_z,1)}$ associated with the completely alternating multisequence ${\left\{\Vert z^{\alpha }{\Vert }_{\mathcal{D}({\rho }^{(1)},{\rho }^{(2)})}^2\right\}}_{\alpha \in {\mathbb{Z}}_{+}^2}$ 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 $S_{\ast }{\rho }^{(i)},$ $i=1,2$, denotes the pushforward measure of ${\rho }^{(i)}$ by the map $S:\overline{\mathbb{D}}\to [0,1]$ defined by $S(z)=|z{|}^2,$ $z\in \overline{\mathbb{D}},$ and ${\delta }_1$ denotes the Dirac measure at 1.

10 pages; comments are welcome