New composition products for complex harmonic functions, the dynamic with respect to these and composition operators induced

TL;DR

This paper introduces new composition products for complex harmonic functions, enabling their composition to remain within the same class, and explores associated composition operators and their potential for future research in semigroups and evolution families.

Contribution

It defines novel composition products for complex harmonic functions and studies composition operators with symbols in Hardy spaces, laying groundwork for future semigroup analysis.

Findings

Composition of two complex harmonic functions remains harmonic.

Defined and analyzed composition operators with Hardy space symbols.

Established a foundation for studying semigroups of harmonic functions.

Abstract

In this work we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here we begin the study of the iterations of the functions of this class showing briefly their potential to be a topic of future research. In parallel, we define and study composition operators whose symbols belong to a Hardy space of complex harmonic functions also introduced in the work. All this constitutes a previous work for the research of semigroups and evolutionary families composed of complex harmonic functions.