Some Hilbert spaces related with the Dirichlet space (Q301870)

Given a positive integer \(d\), let \({\mathcal D}_d\) be the reproducing kernel Hilbert space over the unit disc of the complex plane with the reproducing kernel \(k^d\) where \(k(w, z)= {1\over \overline z w}\log {1 \over {1-\overline z w}}\). When \(d=1\), the space \({\mathcal D}_1\) is the well-known Dirichlet space \(\mathcal D\).NEWLINENEWLINEIn this paper the authors first obtain some equivalent norms for the space \({\mathcal D}_d\) and identify the dual space, denoted by \(HS_d\), of \({\mathcal D}_d\) under the dual pairing of \(\mathcal D\). As applications they obtain the following results: (1) Exact Hilbert-Schmidt norm formulae for some Hankel-type operators; (2) Carleson measure characterizations for the spaces \({\mathcal D}_d\) and \(HS_d\); (3) The existence of norms on \({\mathcal D}_d\) under which \({\mathcal D}_d\) has the complete Nevanlinna-Pick property. The details are too technical to be described here.