A nonconstant coefficients differential operator associated to slice monogenic functions

Slice monogenic functions have had a rapid development in the past few years. One of the main properties of such functions is that they allow the definition of a functional calculus, called $S$-functional calculus, for (bounded or unbounded) noncommuting operators. In the literature there exist two different definitions of slice monogenic functions that turn out to be equivalent under suitable conditions on the domains on which they are defined. Both the existing definitions are based on the validity of the Cauchy-Riemann equations in a suitable sense. The aim of this paper is to prove that slice monogenic functions belong to the kernel of the global operator defined by $G(x):=\vert\underline {x}\vert^2\frac {\partial }{\partial x_0} \ + \ \underline {x} \ \sum _{j=1}^n x_j\frac {\partial }{\partial x_j},$ where $\underline {x}$ is the 1-vector part of the paravector $x=x_0+\underline {x}$ and $n\in \mathbb{N}$. Despite the fact that $G$ has nonconstant coefficients, we are able to prove that a subclass of functions in the kernel of $G$ have a Cauchy formula. Moreover, we will study some relations among the three classes of functions and we show that the kernel of the operator $G$ strictly contains the functions given by the other two definitions.