Clark measures on polydiscs associated to product functions and multiplicative embeddings

TL;DR

This paper explores Clark measures on the unit polydisc, focusing on their relationship with product functions and multiplicative embeddings, and introduces new examples of multivariate inner functions.

Contribution

It provides new insights into Clark measures for multivariate inner functions and examines their connection with multiplication and composition in the polydisc setting.

Findings

Clark measures linked to product functions are characterized.

New examples of multivariate inner functions are analyzed.

Relationships between Clark measures and multiplicative embeddings are established.

Abstract

We study Clark measures on the unit polydisc, giving an overview of recent research and investigating the Clark measures of some new examples of multivariate inner functions. In particular, we study the relationship between Clark measures and multiplication; first by introducing compositions of inner functions and multiplicative embeddings, and then by studying products of one-variable inner functions.