Almansi‐type theorems in Clifford analysis

Abstract

Abstract In this paper, we consider functions defined in a star‐ike omain Ω⊂ℝ n with values in the Clifford lgebra C 𝓁 0, n which are polymonogenic with respect to the (left) Dirac operator D = ∑ j =1 n e j ∂ / ∂x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a ecomposition of the form f = f 1 + xf 2 +···+ x k −1 f k , where x = x 1 e 1 +···+ x n e n and fj , j =1,…, k , are monogenic functions. This generalizes classical Almansi theorem for polyharmonic functions as well e Fischer decomposition of polynomials. Similar results tained for the powers of weighted Dirac operators of the form D̃ =∣ x ∣ −α xD , α ∈ℝ\{0}. Copyright © John Wiley & Sons, Ltd.