Biregular Mappings on H × H : Domains of Hyperholomorphy, Integral Representations, and Runge Approximation

We develop a PDE and boundary integral framework for quaternion-valued fields on product domains Ω ⊂ H × H governed by the mixed left/right Cauchy–Fueter system We identify the natural compatibility condition and prove local solvability with quantitative H 1 estimates, as well as global weak solvability on admissible products U x × U y . Motivated by these estimates, we introduce domains of hyperholomorphy and hyper-conjugates for data that are harmonic in each factor ( Δ x u = Δ y u = 0 ), and we establish Carleman-type quantitative unique continuation tools (boundary blow-up, three-balls, and doubling), including a propagation-of-smallness principle across the two factors. On the potential-theoretic side, we construct a double boundary integral representation for biregular fields with kernel K ( ξ , η ; x , y ) = E ( ξ − x ) E ( y − η ) , establish mapping and jump relations for the associated layer potentials on Lipschitz boundaries, and obtain a Fredholm boundary integral equation for the boundary density in the smooth admissible regime. Finally, we prove a constructive Runge approximation theorem on admissible products and outline a practical discretization workflow consistent with the analysis.