Corona theorem for the Dirichlet-type space

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