Wirtinger operators for functions of several quaternionic variables

We introduce Wirtinger operators for functions of several quaternionic variables. These operators are real linear partial differential operators which behave well on quaternionic polynomials, with properties analogous to the ones satisfied by the Wirtinger derivatives of several complex variables. Due to the non-commutativity of the variables, Wirtinger operators turn out to be of higher order, except the first ones that are of the first order. In spite of that, these operators commute with each other and satisfy a Leibniz rule for products. Moreover, they characterize the class of slice-regular polynomials, and more generally of slice-regular quaternionic functions. As a step towards the definition of the Wirtinger operators, we provide Almansi-type decompositions for slice functions and for slice-regular functions of several variables. We also introduce some aspects of local slice analysis, based on the definition of locally slice-regular function in any open subset of the $n$-dimensional quaternionic space.