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

$$\begin{aligned} \{f_1,...,f_n\}\subseteq M\big (D(\mu )\big )\quad \text {and}\quad \inf _{z\in \mathbb {D}}\sum _{j=1}^n|f_j(z)|>0 \end{aligned}$$

then

$$\begin{aligned} \exists \,\{g_1,...,g_n\}\subseteq M\big (D(\mu )\big )\quad \text {such that}\quad \sum _{j=1}^nf_jg_j=1, \end{aligned}$$

thereby generalizing Carleson’s corona theorem for $M(H^2)=H^\infty$ in Carleson (Ann Math (2) 76, 547–559, 1962) and Xiao’s corona theorem for $M(\mathscr {D})\subset H^\infty$ in Xiao (Manuscr Math 97, 217–232, 1998) thanks to

$$\begin{aligned} D(\mu )={\left\{ \begin{array}{ll} \text {Hardy space}\ H^2\quad &{}\text {as}\quad \text {d}\mu (z)=(1-|z|^2)\,\text {d}A(z)\quad \forall \ z\in \mathbb {D};\\ \text {Dirichlet space}\; \mathscr {D}\ &{}\text {as}\quad \text {d}\mu (z)=|\text {d}z|\quad \forall \ z\in \mathbb {T}=\partial {\mathbb {D}}. \end{array}\right. } \end{aligned}$$