We consider Hölder continuous circulant ( 2 × 2 ) matrix functions G 2 1 defined on the Ahlfors-David regular boundary Γ of a domain Ω in ℝ 2 n . The main goal is to study under which conditions such a function G 2 1 can be decomposed as G 2 1 = G 2 1 + - G 2 1 - , where the components G 2 1 ± are extendable to two-sided H -monogenic functions in the interior and the exterior of Ω , respectively. H -monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. H -monogenic functions then are the null solutions of a ( 2 × 2 ) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.