Generalized Harmonic Function Structures
Markus Klintborg
TL;DR
This work characterizes rotation-commuting operators in the second Weyl-algebra $A_2(K)$ by showing they correspond to a one-dimensional reduction $T_{m,D}$ in $A_1$ acting on $|z|^2$-polynomials, and defines generalized harmonic polynomials as the joint kernel of a natural subalgebra $H_2$. It develops a detailed structural theory for polynomials in $K[z,\bar{z}]$, including a graded module decomposition over $K[|z|^2]$, Almansi-type cells, and a $(\gamma_1,\gamma_2)$-harmonic framework expressed via hypergeometric bases. The paper proves a key invariant-action result: any rotation-commuting operator in the rotation-commuting algebra $R_2$ preserves $K^*$-submodules generated by the monomials $\xi_m(z)$, reducing analysis to one-dimensional ODEs. It extends these algebraic constructions to polyharmonic polynomials (the cellular decomposition) and further to smooth functions on the unit disc, establishing a comprehensive cellular representation for polyharmonics in $C^{\infty}(\mathbb{D})$ with a unique decomposition. The results provide a unified algebraic-analytic framework connecting Weyl-algebra structure, harmonic decompositions, and symmetry, enabling explicit constructions and reductions of generalized harmonic problems under rotational symmetry.
Abstract
We model generalized harmonic functions on rings of differential operators and complex function spaces. The differential operators in the second Weyl-algebra that commute with rotations are described and leads to a natural notion for such functions. We also investigate how such functions are related, and retrieve the cellular decomposition for polyharmonic functions.