$\mathcal {N}_p$-Spaces in the Unit Disc $\mathbb {D}$

Abstract

In this chapter, the $\mathcal {N}_p$-spaces of holomorphic functions on the open unit disc $\mathbb {D}$ are studied. This family of spaces is also known as Bergman-type or Beurling-type spaces. The Bergman-type spaces and Bergman-type spaces have attracted a great deal of attention and have been considered by many mathematicians. These spaces, on the one hand, play an important role in both function theoretic and operator theoretic development of function spaces, and, on the other hand, have a closed connection to Bloch spaces which appear as the images of the bounded functions under the Bergman projections. Bloch spaces also play the role of the dual spaces of the Bergman spaces. Also among the Bergman- and Bergman-type spaces, the so-called $Q_p$-spaces are of great interest. Adopting the intrinsic relation between the classical Dirichlet space and the Bergman space, the appearance of $\mathcal {N}_p$-spaces is also related to $\mathcal {Q}_p$ spaces. Hence, some basic Banach space properties of the $\mathcal {N}_p$-spaces are discussed below. In particular, it will be showed that, for $p\in (0,1)$, the $\mathcal {N}_p$-spaces are all different topological vector spaces with independent interest.