Multipliers of Dirichlet subspaces of the Bloch space

Abstract

For $0<p<\infty$ we let ${\mathcal{D}}_{p-1}^p$ denote the space of those functions $f$ which are analytic in the unit disc $\mathbb{D}$ and satisfy $\int_\mathbb D (1-| z|)\sp {p-1}| f'(z)| \sp p\,dA(z)<\infty $. It is known that, whenever $p\ne q$, the only multiplier from ${\mathcal{D}}_{p-1}^p$ to ${\mathcal{D}}_{q-1}^q$ is the trivial one. However, if $X$ is a subspace of the Bloch space and $0<p\le q<\infty$, then $X\cap {\mathcal{D}}_{p-1}^p\subset X\cap {\mathcal{D}}_{q-1}^q$, a fact which implies that the space of multipliers $\M(\mathcal D^p_{p-1}\cap X, \mathcal D^q_{q-1} \cap X)$ is non-trivial. In this paper we study the spaces of multipliers $\M(\mathcal D^p_{p-1}\cap X, v\cap X)$ ($0<p,q<\infty$) for distinct classical subspaces $X$ of the Bloch space. Specifically, we shall take $X$ to be $H^{\infty }$, $BMOA$ and the Bloch space $\mathcal{B}$.

arXiv comment26 pages