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

Abstract

This survey shows how, for the Nevanlinna class $\mathcal{N}$ 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 $\mathcal{H}^{\infty}$ : 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 $\mathcal{H}^{\infty}$ can be transposed to $\mathcal{N}$ 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.