Riemann–Hilbert Problems for Hardy Space of Meta-Analytic Functions on the Unit Disc

Abstract

The aim of this paper is to study Riemann–Hilbert problems for Hardy space of a class of meta-analytic functions defined on the unit disc. Here, the meta-analytic functions we focus on are null-solutions to a class of polynomially Cauchy–Riemann equations. We first establish decomposition theorems for Hardy space of meta-analytic functions defined on the unit disc, and use them to characterize the boundary behavior of Hardy space of meta-analytic functions defined on the unit disc. Then, we make full use of these decomposition theorems and a transform constructed to solve the Riemann–Hilbert problem for Hardy space of a class of meta-analytic functions in two different cases of the parameter involved, separately. Finally, we give explicit integral expressions of solutions and conditions of solvability, respectively.