The Teodorescu and the Π-operator in octonionic analysis and some applications

Abstract

In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the $L_2$-space decomposition demonstrated in this paper for octonion-valued functions. In the unit ball, we explicitly show that the intrinsic weight factor is crucial to obtain the reproduction property and that the latter precisely compensates an additional associator term that otherwise appears when leaving out the weight factor.

Furthermore, we study an octonionic Teodorescu transform and show how it is related to the unweighted version of the Bergman transform and establish some operator relations between these transformations. We apply two different versions of the Borel-Pompeiu formulae that naturally arise in the context of the non-associativity. Next, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling operator, also known as the Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all prominent hypercomplex function theories. Then we describe some basic mapping properties arising in context with the $L_2$-space decomposition discussed before.

Finally, we explore several applications of the octonionic Π-operator. Initially, we demonstrate its utility in solving the octonionic Beltrami equation, which characterizes generalized quasi-conformal maps from $R^8$ to $R^8$ in a specific analytical sense. Subsequently, analogous results are presented for the hyperbolic octonionic Dirac operator acting on the right half-space of $R^8$. Lastly, we discuss how the octonionic Teodorescu transform and the Bergman projection can be employed to solve an eight-dimensional Stokes problem in the non-associative octonionic setting.