From $ H^\infty$ to N. Pointwise properties and algebraic structure in the Nevanlinna class

TL;DR

This survey explores how classical properties and objects of the algebra of bounded analytic functions can be extended to the Nevanlinna class, highlighting the role of harmonic functions in these analogies.

Contribution

It systematically characterizes the analogues of key $H^ $ properties within the Nevanlinna class, establishing a general rule for transposing results via harmonic control.

Findings

Analogues of interpolating sequences and Corona theorem are characterized in N.

The transposition rule from $H^ $ to N often involves replacing bounds with harmonic functions.

Some classical results do not extend directly, indicating exceptions.

Abstract

This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for $H^\infty$ can be transposed to N by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.