Corona Theorem for the Dirichlet-Type Space

This paper utilizes Cauchy’s transform and duality for the Dirichlet-type space D(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(\mu )$$\end{document} with positive superharmonic weight Uμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_\mu $$\end{document} on the unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document} to establish the corona theorem for the Dirichlet-type multiplier algebra M(D(μ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\big (D(\mu )\big )$$\end{document} that: if {f1,...,fn}⊆M(D(μ))andinfz∈D∑j=1n|fj(z)|>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{f_1,...,f_n\}\subseteq M\big (D(\mu )\big )\quad \text {and}\quad \inf _{z\in \mathbb {D}}\sum _{j=1}^n|f_j(z)|>0 \end{aligned}$$\end{document}then ∃{g1,...,gn}⊆M(D(μ))such that∑j=1nfjgj=1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \exists \,\{g_1,...,g_n\}\subseteq M\big (D(\mu )\big )\quad \text {such that}\quad \sum _{j=1}^nf_jg_j=1, \end{aligned}$$\end{document}thereby generalizing Carleson’s corona theorem for M(H2)=H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(H^2)=H^\infty $$\end{document} in Carleson (Ann Math (2) 76, 547–559, 1962) and Xiao’s corona theorem for M(D)⊂H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\mathscr {D})\subset H^\infty $$\end{document} in Xiao (Manuscr Math 97, 217–232, 1998) thanks to D(μ)=Hardy spaceH2asdμ(z)=(1-|z|2)dA(z)∀z∈D;Dirichlet spaceDasdμ(z)=|dz|∀z∈T=∂D.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D(\mu )={\left\{ \begin{array}{ll} \text {Hardy space}\ H^2\quad &{}\text {as}\quad \text {d}\mu (z)=(1-|z|^2)\,\text {d}A(z)\quad \forall \ z\in \mathbb {D};\\ \text {Dirichlet space}\; \mathscr {D}\ &{}\text {as}\quad \text {d}\mu (z)=|\text {d}z|\quad \forall \ z\in \mathbb {T}=\partial {\mathbb {D}}. \end{array}\right. } \end{aligned}$$\end{document}