Hardy spaces of generalized analytic functions and composition operators

We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Omega. We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain partial derivative Omega. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C-phi on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol phi.