The theory of the operator \(G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum _{j=1}^n x_j \frac{\partial }{\partial x_j}\) is deeply associated with the slice monogenic function theory and has grown in recent years. In particular, for \(n=3\) the quaternionic version of G has been recently used to study the quaternionic slice regular function theory. This work extends the study of the G operator in two senses: a) Clifford’s analysis structure. The function theory induced by the operator \(\begin{aligned} {\mathcal {H}}_a (x) = {{\underline{a}}} ( {x}) \frac{\partial }{\partial x_0} - \sum _{i=1}^n \left( \sum _{j=1}^n a_j ( {x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a ( {x}) \right) \frac{\partial }{\partial x_i}, \end{aligned}\) where a is a function with certain properties with domain in \({\mathbb {R}}^{n+1}\) is presented extending the already known results of the G (if \(a=I\) is the identity function we can see that \(G=-\underline{x}{\mathcal {H}}_I\) ) such as Splitting Lemma, Representations Theorem, Cauchy formula, a characterization of the zero sets of \({\mathcal {H}}_a\) and a power series development. Also, some properties of the material derivative: \(\begin{aligned} {D}_u = \frac{\partial }{\partial x_0} + \sum _{j=1}^n u_j \frac{\partial }{\partial x_j}, \end{aligned}\) where u is a certain \({\mathbb {R}}^{n}\) -valued function with domains in \({\mathbb {R}}^{n}\) , or \({\mathbb {R}}^{n+1}\) , are presented as consequences of function theory induced by \({\mathcal {H}}_a\) . b) Structure of quaternionic analysis. In particular, the case \(n=3\) is approached from the point of view of quaternionic analysis presenting results such as Splitting Lemma, Representations Theorem, Borel-Pompeiu, Stokes and Cauchy formulas, a conformal covariant property and a characterization of the zero sets of the quaternonic version of \({\mathcal {H}}_a\) and its consequences for \(D_u\) .
