Certain Invariant Spaces of Bounded Measurable Functions on a Sphere

TL;DR

This paper characterizes weak*-closed unitarily and M"obius invariant spaces and algebras of essentially bounded functions on a sphere, extending previous work on continuous and L^p functions to the L^∞ setting.

Contribution

It provides an analogous characterization for L^∞ functions, expanding the understanding of invariant spaces and algebras on the sphere.

Findings

Characterization of weak*-closed invariant spaces of L^∞ functions.

Analysis of invariant algebras of L^∞ functions.

Extension of Nagel and Rudin's results to the L^∞ case.

Abstract

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and M\"obius invariant spaces of continuous and $L^p$-functions on a sphere, for $1\leq p<\infty$. In this paper we provide an analogous characterization for the weak*-closed unitarily and M\"obius invariant spaces of $L^\infty$-functions on a sphere. We also investigate the weak*-closed unitarily and M\"obius invariant algebras of $L^\infty$-functions on a sphere.