Some questions about complex harmonic functions

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 \(HH^{2}(\mathbb {D})\).