Recent zbMATH articles in MSC 30H80https://www.zbmath.org/atom/cc/30H802022-05-16T20:40:13.078697ZWerkzeugCorona theorem for the Dirichlet-type spacehttps://www.zbmath.org/1483.300952022-05-16T20:40:13.078697Z"Luo, Shuaibing"https://www.zbmath.org/authors/?q=ai:luo.shuaibingSummary: 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
\[
\{ f_1,\ldots ,f_n\} \subseteq M\big( D(\mu )\big) \quad \text{and}\quad \inf_{z\in \mathbb{D}}\sum_{j=1}^n |f_j (z)|>0
\]
then
\[
\exists \,\{ g_1,\ldots,g_n\}\subseteq M\big( D(\mu)\big) \quad \text{such that}\quad \sum_{j=1}^n f_j g_j =1,
\]
thereby generalizing \textit{L. Carleson}'s corona theorem for \(M(H^2)=H^{\infty}\) in [Ann. Math. (2) 76, 547--559 (1962; Zbl 0112.29702)] and \textit{J.Xiao}'s corona theorem for \(M(\mathscr{D})\subset H^{\infty}\) in [Manuscr. Math. 97, No. 2, 217--232 (1998; Zbl 1049.30025)] thanks to
\[
D(\mu )=
\begin{cases}
\text{Hardy space } H^2 \quad & \text{as}\quad \text{d}\mu (z)=(1-|z|^2)\,\text{d}A(z)\quad \forall z\in \mathbb{D}; \\
\text{Dirichlet space}\; \mathscr{D} & \text{as}\quad \text{d}\mu (z)=|\text{d}z|\quad \forall z\in \mathbb{T} =\partial\mathbb{D}. \end{cases}
\]Corona theorem for strictly pseudoconvex domainshttps://www.zbmath.org/1483.320162022-05-16T20:40:13.078697Z"Gwizdek, Sebastian"https://www.zbmath.org/authors/?q=ai:gwizdek.sebastianSummary: Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that \textit{M. Kosiek} and \textit{K. Rudol} [``Corona theorem'', Preprint, \url{arXiv:2106.15683}] obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved ``abstract'' Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.