Uniform algebras generated by holomorphic and close-to-harmonic functions

TL;DR

This paper extends a classical approximation theorem for uniform algebras generated by holomorphic and harmonic functions to include certain non-harmonic perturbations, using plurisubharmonicity and polynomial convexity.

Contribution

It introduces a new approach that relaxes harmonicity conditions, allowing non-harmonic perturbations with small Laplacian to generate the same uniform algebra.

Findings

The uniform algebra generated by z and h+R equals the continuous functions on the disk closure.

The approach uses plurisubharmonicity and polynomial convexity to handle non-harmonic perturbations.

The result generalizes the Axler-Shields theorem to a broader class of functions.

Abstract

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a nowhere-holomorphic harmonic function on D that is continuous up to the boundary --- equals the algebra of continuous functions on closure(D). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is "small" in a certain sense.