Multipliers of Dirichlet subspaces of the Bloch space

For $0<p<\infty$ we let $\mathcal D^p_{p-1}$ 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 $p\neq q$, the only multiplier from $\mathcal D^p_{p-1}$ to $\mathcal D^q_{q-1}$ is the trivial one. However, if $X$ is a subspace of the Bloch space and $0<p\le q<\infty$, then $X \cap \mathcal D^p_{p-1}\subset X\cap \mathcal D^q_{q-1}$, 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 $\mathcal B$.

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