Hardy Spaces for the Three-Dimensional Vekua Equation

  • Briceyda B. DelgadoEmail author
  • R. Michael Porter
Part of the Trends in Mathematics book series (TM)


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.


Monogenic functions Hardy spaces Main Vekua equation Vekua-Hardy spaces Vekua-Hilbert transform Conductivity equation 

Mathematics Subject Classification (2010)

Primary 30G20; Secondary 30H10 35J25 



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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasCinvestav-QuerétaroSantiago de QueretaroMexico

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