Polyharmonicity, Almansi-type decompositions and Fueter-Sce theorem for several Clifford variables
Giulio Binosi
TL;DR
The paper extends slice-regular and Fueter–Sce theory from a single Clifford variable to several variables by deriving explicit polyharmonicity formulas for spherical derivatives and slice functions, and by developing Almansi-type decompositions in multiple variables. It establishes that, for odd $m$ and $\gamma_m=(m-1)/2$, the spherical derivative $f'_s$ is $\gamma_m$-polyharmonic and slice regularity induces corresponding polyharmonic decompositions, which are organized through ordered and simultaneous constructions. The work provides two proofs of a multi-variable Fueter–Sce theorem, showing that $\Delta_{m+1}^{\gamma_m} f$ is axially monogenic when $f$ is slice regular in several variables, thereby strengthening the bridge between slice and monogenic function theories in higher dimensions. Together, these results offer new decomposition tools and PDE connections in Clifford analysis, with potential applications in multi-variable hypercomplex function theory and related PDE problems.
Abstract
We study some harmonic properties of slice regular functions in one and several Clifford variables and give explicit formulas of the iterated Laplacian applied to slice regular functions and to their spherical derivative, which are new also in the one variable context. We propose several Almansi-type decompositions for slice functions in several Clifford variables. As a consequence, we establish a several variables version of Fueter-Sce theorem.