Holomorphic functions and regular quaternionic functions on the hyperkähler space H

Let H be the space of quaternions, with its standard hypercomplex structure. Let $\mathcal R(\Omega)$ be the module of \emph{$\psi$-regular} functions on $\Omega$. For every $p\in H$, $p^2=-1$, $\mathcal R(\Omega)$ contains the space of holomorphic functions w.r.t. the complex structure $J_p$ induced by $p$. We prove the existence, on any bounded domain $\Omega$, of $\psi$-regular functions that are not $J_p$-holomorphic for any $p$. Our starting point is a result of Chen and Li concerning maps between hyperk\"ahler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes $J_p$-holomorphic functions among $\psi$-regular functions

An application of biregularity to quaternionic Lagrange interpolation

We revisit the concept of totally analytic variable of one quaternionic variable introduced by Delanghe \cite Delanghe} and its application to Lagrange interpolation by G\"uerlebeck and Spr\"ossig \cite{GS}. We consider left-regular functions in the kernel of the Cauchy-Riemann operator $$\mathcal D=2\left(\frac{\partial}{\partial{\bar z_1}}+j\frac{\partial}{\partial{\bar z_2}}\right)=\frac{\partial}{\partial{x_0}}+i\frac{\partial}{\partial{x_1}}+j\frac{\partial}{\partial{x_2}}-k\frac{\partial}{\partial{x_3}}.$$ For every imaginary unit p\in {\Sp}^2, let {\CC}_p=\langle 1,p\rangle\simeq {\CC} and let $J_p=p_1J_1+p_2J_2+p_3J_3$ be the corresponding complex structure on {\HH}. We identify totally regular variables with real--affine holomorphic functions from ({\HH},J_p) to ({\CC}_p,L_p), where $L_p$ is the complex structure defined by left multiplication by $p$. We then show that every $J_p$--biholomorphic map, which is always a biregular function, gives rise to a Lagrange interpolation formula at any set of distinct points in {\HH}. Publisher version at: http://link.aip.org/link/?APCPCS/1048/691/

Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on $L^2$ traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions. Published in Journal of Mathematical Analysis and Applications (http://www.elsevier.com/wps/find/journaldescription.cws_home/622886/description#description

A four dimensional Bernstein Theorem

We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein's inequality and give a formulation of this last result in terms of four-dimensional zonal harmonics and Gegenbauer polynomials