Multipliers of Dirichlet-type subspaces of Bloch space
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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 set of 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 $\H^{\infty}.$
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 set of 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 $\H^{\infty}.$
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