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 $M(D(\mu ))$ that: if $$\{f_1,...,f_n\}\subseteq M(D(\mu ))\text{and}\underset{z\in \mathbb{D}}{\inf }\sum _{j=1}^n|f_j(z)|>0$$ then $$\exists \{g_1,...,g_n\}\subseteq M(D(\mu ))\text{such that}\sum _{j=1}^n{f}_j{g}_j=1,$$ thereby generalizing Carleson's corona theorem for $M(H^2)=H^{\infty }$ and Xiao's corona theorem for $M(\mathcal{D})\subset H^{\infty }$ thanks to $$D(\mu )=\{\begin{array}{ll}\text{Hardy space}{H}^2& \text{as}d\mu (z)=(1-|z{|}^2)dA(z)\forall z\in \mathbb{D};\\ \text{Dirichlet space}\mathcal{D}& \text{as}d\mu (z)=|dz|\forall z\in \mathbb{T}=\partial \mathbb{D}.\end{array}$$