Multipliers of Dirichlet-type subspaces of Bloch space
Bull. Korean Math. Soc.
Published online November 8, 2019
Songxiao Li, Zengjian Lou, and Conghui Shen
Shantou University
Abstract : Let $M(X,Y)$ denote the space of multipliers from $X$ to $Y,$ where $X$ and $Y$ are analytic function spaces.
As we known, for Dirichlet-type spaces $\mathcal{D}_{\alpha}^p,$
$M(\mathcal{D}^p_{p-1},\mathcal{D}^q_{q-1})=\{0\},$ if $p\neq q,$ $0<p,q<\infty.$ If $0<p,q<\infty,$ $p\neq q,$ $0<s<1$ such that $p+s,q+s>1,$ then $M(\mathcal{D}^p_{p-2+s},\mathcal{D}^q_{q-2+s})=\{0\}.$ However, $X\cap\mathcal{D}^p_{p-1} \subseteq X\cap\mathcal{D}^q_{q-1}$ and $X\cap \mathcal{D}^p_{p-2+s} \subseteq X\cap \mathcal{D}^q_{q-2+s}$ whenever $X$ is a subspace of the Bloch space $\mathcal{B}$ and $0<p\leq q<\infty.$
This says that the multipliers $M(X\cap \mathcal{D}^p_{p-2+s},X\cap\mathcal{D}^q_{q-2+s})$ is nontrivial. In this paper, we study the multipliers $M(X\cap\mathcal{D}^p_{p-2+s},X\cap\mathcal{D}^q_{q-2+s})$ for
distinct classical subspaces $X$ of the Bloch space $\mathcal{B},$ where $X=\mathcal{B},$ $BMOA$ or $H^\infty.$
Keywords : Multipliers; Carleson measure; Dirichlet-type space; Bloch space.
MSC numbers : 47B38, 32A37, 30H30
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