Inner Functions with Derivatives in the Weak Hardy Space

TL;DR

This paper characterizes exponential Blaschke products as the inner functions with derivatives in the weak Hardy space, linking their properties to logarithmic means and derivative behavior in model spaces.

Contribution

It provides a new characterization of exponential Blaschke products based on the weak Hardy space membership of their derivatives.

Findings

Exponential Blaschke products have derivatives in the weak Hardy space.

Characterization involves logarithmic means and derivative behavior in model spaces.

Links between inner functions and weak Hardy space derivatives are established.

Abstract

It is proved that exponential Blaschke products are the inner functions whose derivative is in the weak Hardy space. Exponential Blaschke products are described in terms of their logarithmic means and also in terms of the behavior of the derivatives of functions in the corresponding model space.