Schwarz Problems for Poly-Hardy Space on the Unit Ball

Abstract

In this paper we study the Schwarz boundary value problems \({\big(}\) for short BVP\({\big)}\) for the poly-Hardy space defined on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\) . We first discuss the boundary behavior of functions belonging to the poly-Hardy class. Then we construct the Schwarz kernel function, and describe the boundary properties of the Schwarz-type integrable operator. Finally, we study the Schwarz BVPs for the Hardy class and the poly-Hardy class on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\), and obtain the expressions of solutions, explicitly. As an application, the monogenic signals considered for the Hardy spaces defined on the unit sphere are reconstructed when the scalar- and sub-algebra-valued data are given, which is the extension of the analytic signals for the Hardy spaces on the unit circle of the complex plane.