Iterated Borel–Pompeiu Representation on Quaternionic Product Domains and a Distinguished Boundary Transform

Let U,V⊂H be bounded C1 domains, and let f be quaternion-valued on U×V. We study the mixed Cauchy–Fueter system DxLf=0 and fDyR=0 on product domains by iterating the classical one-variable Borel–Pompeiu formulas in an order consistent with quaternionic multiplication. Under closure regularity on U¯×V¯, we prove an iterated representation formula and show that, in the biregular case, the boundary contribution reduces to the distinguished boundary ∂U×∂V. This leads to a distinguished boundary transform, TU,V, on continuous boundary data. We prove that TU,V maps C(∂U×∂V;H) into C∞(U×V;H), establish compact subset estimates for mixed real derivatives, and derive a local approximation theorem within the transform range by finite sums of separated one-variable Cauchy transforms. The analysis is restricted to this representation framework. In particular, the paper does not address a general solvability theory for the mixed inhomogeneous system and does not characterize the full range of TU,V.