Hardy type spaces on infinite dimensional group orbits (Q2896642)

The Hardy type space \(\mathcal H^2_\chi\) with a Haar measure \(\chi\) on a locally compact group \(G\) acting on a Hilbert space was introduced by \textit{O. Lopushansky} [Topology 48, No. 2-4, 169--177 (2009; Zbl 1195.46045)]. In the paper under review, the authors consider the case where a \(G\)-invariant measure \(\chi\) is defined on a unitary orbit of a locally compact group \(G\) acting on an infinite-dimensional Hilbert space \(E\). They establish a Cauchy type integral formula, which for every function \(f\in \mathcal H^2_\chi\) on an orbit gives its unique analytic extension onto the open unit ball in \(E\). The existence of radial boundary values is also proved. As an example, the case of the reduced Heisenberg group \(\mathbb H=\mathbb R^2\times \mathbb T\) is considered in detail.