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Schwarz Problems for Poly-Hardy Space on the Unit Ball

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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.

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Authors and Affiliations

  1. CIDMA, Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Uwe Kähler & Min Ku

  2. Faculty of Science and Technology, University of Macau, Macao, China

    Tao Qian

Authors
  1. Uwe Kähler
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  2. Min Ku
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  3. Tao Qian
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Corresponding author

Correspondence to Min Ku.

Additional information

This work was supported by Portuguese funds through the CIDMA-Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project UID/MAT/ 0416/2013, by University of Macau MYRG115(Y1-L4)-FST13-QT, and by the Postdoctoral Foundation from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010.

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Kähler, U., Ku, M. & Qian, T. Schwarz Problems for Poly-Hardy Space on the Unit Ball. Results Math 71, 801–823 (2017). https://doi.org/10.1007/s00025-016-0575-2

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