Corona theorem for the Dirichlet-type space

Abstract

This paper utilizes Cauchy's transform and duality for the Dirichlet-type space $D(\mu)$ with positive superharmonic weight $U_\mu$ on the unit disk $\mathbb{D}$ to establish the corona theorem for the Dirichlet-type multiplier algebra $M\big(D(\mu)\big)$ that: if $$\{f_1,...,f_n\}\subseteq M\big(D(\mu)\big)\quad\text{and}\quad \inf_{z\in\mathbb{D}}\sum_{j=1}^n|f_j(z)|>0$$ then $$ \exists\,\{g_1,...,g_n\}\subseteq M\big(D(\mu)\big)\quad\text{such that}\quad \sum_{j=1}^nf_jg_j=1, $$ thereby generalizing Carleson's corona theorem for $M(H^2)=H^\infty$ and Xiao's corona theorem for $M(\mathscr{D})\subset H^\infty$ thanks to $$ D(\mu)=\begin{cases} \text{Hardy space}\ H^2\quad &\text{as}\quad d\mu(z)=(1-|z|^2)\,dA(z)\ \ \forall\ z\in\mathbb{D};\\ \text{Dirichlet space}\ \mathscr{D}\ &\text{as}\quad d\mu(z)=|dz|\ \ \forall\ z\in\mathbb{T}=\partial{\mathbb{D}}. \end{cases} $$