Corona-type theorems and division in some function algebras on planar domains

TL;DR

This paper investigates corona-type theorems and division problems in specific function algebras on planar domains, introducing new spaces and classes of sets to solve these problems.

Contribution

It introduces the algebra of ar-smooth functions and solves division problems for certain function algebras, extending classical results to new contexts.

Findings

Complete solution for division in $A^m(K)$ when $n=1$

Introduction of the space $C_{ar{ ext{d}}, 1}(K)$

Elementary proof of the Nullstellensatz for $A(K)$

Abstract

Let $A$ be an algebra of bounded smooth functions on the interior of a compact set in the plane. We study the following problem: if $f,f_1,\dots,f_n\in A$ satisfy $|f|\leq \sum_{j=1}^n |f_j|$, does there exist $g_j\in A$ and a constant $N\in\N$ such that $f^N=\sum_{j=1}^n g_j f_j$? A prominent role in our proofs is played by a new space, $C_{\dbar, 1}(K)$, which we call the algebra of $\dbar$-smooth functions. In the case $n=1$, a complete solution is given for the algebras $A^m(K)$ of functions holomorphic in $K^\circ$ and whose first $m$-derivatives extend continuously to $\ov{K^\circ}$. This necessitates the introduction of a special class of compacta, the so-called locally L-connected sets. We also present another constructive proof of the Nullstellensatz for $A(K)$, that is only based on elementary $\dbar$-calculus and Wolff's method.