On Directional Hilbert Operators for Regular Quaternionic Functions on $${\mathbb R}^3$$

Abstract

In this paper we define directional quaternionic Hilbert operators on the three–dimensional space \({\mathbb H}_0 = \langle i, j, k \rangle \cong {\mathbb R}^{3}\). We consider functions in the kernel of the Cauchy-Riemann operator $$ {\mathcal D} = 2 \left(\frac{\partial}{\partial \bar{z}_{1}} + j \frac {\partial}{\partial \bar{z}_{2}}\right)= \frac{\partial}{\partial {x_0}}+i \frac{\partial}{\partial {x_1}} + j \frac{\partial}{\partial {x_2}}-k \frac{\partial}{\partial {x_3}}$$ a variant of the Cauchy. Fueter operator. This choice is motivated by the strict relation existing between this type of regularity and holomorphicity w.r.t. the whole class of complex structures on \({\mathbb H}\). For every imaginary unit \(p \in {\mathbb S}^2\), let Jp be the corresponding complex structure on \({\mathbb H}\). Given a domain \(\Omega \subseteq {\mathbb H}\), every holomorphic map from (ΩJp) to (\({\mathbb H}\), Lp), where Lp is defined by left multiplication by p, is a regular function. We combine the quaternionic Cayley transformation, that maps the unit ball to the right half–space \({\mathbb H}^{+} = \{q \in {\mathbb H} \, | \, Re(q) > 0 \}\) with the Hilbert operators introduced in [16] on the unit sphere S of \({\mathbb H}\) in order to define directional Hilbert operators for (boundary values of) regular functions on \({\mathbb H}_0 \cong {\mathbb R}^3\).