Some questions about complex harmonic functions

Abstract

In this paper, 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 its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by \(HH^{2}(\mathbb {D})\) of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form \(\chi +\overline{\pi }\) where \(\chi ,\pi \) are analytic functions from \(\mathbb {D}\) into \(\mathbb {D}\). We also analyze the space of bounded linear operators on \(HH^{2}(\mathbb {D})\).