Multipliers of Dirichlet subspaces of the Bloch space

TL;DR

This paper investigates the multiplier spaces between intersections of Dirichlet-type spaces and classical subspaces of the Bloch space, revealing non-trivial multipliers under certain conditions and analyzing their structure.

Contribution

It characterizes the multiplier spaces between intersections of Dirichlet spaces and classical Bloch subspaces, extending known trivial cases to new non-trivial scenarios.

Findings

Multiplier spaces are non-trivial when intersecting Dirichlet spaces with subspaces of the Bloch space.

The paper provides explicit descriptions of these multiplier spaces for various classical subspaces.

Results highlight the relationships between different analytic function spaces and their multipliers.

Abstract

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 $.