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Hardy Spaces for the Three-Dimensional Vekua Equation

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Book cover Topics in Clifford Analysis

Part of the book series: Trends in Mathematics ((TM))

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Abstract

Let Ω be a bounded Lipschitz domain in \(\mathbb {R}^3\). We introduce the Vekua-Hardy spaces \(H_f^p(\Omega )\) of solutions of the main Vekua equation \(DW=(Df/f)\overline {W}\) where 1 < p < ∞. Here W is quaternion-valued, D is the Moisil-Teodorescu operator, and the conductivity f is a bounded scalar function with bounded gradient. Using the Vekua-Hilbert transform \(\mathcal {H}_f\) defined in previous work of the authors, we give some characterizations of \(H_f^p(\Omega )\) analogous to those of the “classical” Hardy spaces of monogenic functions in \(\mathbb {R}^3\). The main obstacle is the lack of several fundamental analogues of properties of solutions to the special case DW = 0 (monogenic, or hyperholomorphic functions), such as power series and the Cauchy integral formula.

To Wolfgang Sprößig, with affection and respect

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Acknowledgements

The first author is pleased to express her gratitude to the APICS Team, INRIA, Sophia Antipolis, for the financial support provided for the stay realized in 2017. In particular, thanks to Dr. Juliette Leblond for her invaluable suggestions as well as for the introduction to this interesting topic of Hardy spaces of pseudo-analytic functions.

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  1. Departamento de Matemáticas, Cinvestav-Querétaro, Santiago de Queretaro, Mexico

    Briceyda B. Delgado & R. Michael Porter

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  1. Briceyda B. Delgado
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  2. R. Michael Porter
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Correspondence to Briceyda B. Delgado .

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  1. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Swanhild Bernstein

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Delgado, B.B., Porter, R.M. (2019). Hardy Spaces for the Three-Dimensional Vekua Equation. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_6

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