Almansi‐type theorems in Clifford analysis

Abstract

SUMMARYIn this paper, we consider functions deÿned in a star-like domain ⊂ R n with values in the Cli ord algebra C' 0; n which are polymonogenic with respect to the (left) Dirac operator D = n j=1 e j @=@x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a decomposition of the formwhere x = x 1 e 1 + · · · + xnen and f j ; j = 1; : : : ; k; are monogenic functions. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. Similar results are obtained for the powers of weighted Dirac operators of the formD = |x| − xD; ∈ R\{0}.

Fischer decomposition by inframonogenic functions

“…Indeed, suppose that h(x) is a harmonic function in a star-like domain Ω ⊂ R m . By the Almansi decomposition (see [12,15]), we have that h(x) admits a decomposition of the form…”

“…In this section we will derive a similar decomposition but in terms of inframonogenic homogeneous polynomials. For other generalizations of the Fischer decomposition we refer the reader to [5,7,8,9,10,12,14,17,18].…”

On the solutions of a class of iterated generalized Bers–Vekua equations in Clifford analysis

“…(1) to the iterated Bers-Vekua equation for which in [8] a corresponding representation theorem was proved. For the iterated Dirac operator∂ k with∂ = n j=1 e j ∂/∂x j a similar decomposition was proved in [9], whereas in [10] a unified approach to decomposing kernels of iterated operators was investigated.…”

Fock spaces, Landau operators and the time-harmonic Maxwell equations

“…These sorts of spaces are proper subspaces of the so-called poly-monogenic functions with respect to the C ∞ -topology (cf [26]). On the other hand, from lemma 3.1 and from relation (20), it follows that D − annihilates φ(x)f (x) for any f ∈ kerD:…”