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

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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.

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Acknowledgements

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, postdoctoral grant from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010, National Natural Science Foundation of China (11326087, 11601525), Natural Science Foundation of Hunan Province (2017JJ3406), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Authors and Affiliations

  1. Department of Mathematics, CIDMA, University of Aveiro, 3810-193, Aveiro, Portugal

    Min Ku

  2. Department of Computing Science, University of Radboud, 6525 EC, Nijmegen, Netherlands

    Min Ku

  3. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Fuli He

  4. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430072, China

    Ying Wang

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  1. Min Ku
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  2. Fuli He
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  3. Ying Wang
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Correspondence to Ying Wang.

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Communicated by Heinrich Begehr.

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Ku, M., He, F. & Wang, Y. Riemann–Hilbert Problems for Hardy Space of Meta-Analytic Functions on the Unit Disc. Complex Anal. Oper. Theory 12, 457–474 (2018). https://doi.org/10.1007/s11785-017-0705-1

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