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 H1 estimates, as well as global weak solvability on admissible products Ux×Uy. Motivated by these estimates, we introduce domains of hyperholomorphy and hyper-conjugates for data that are harmonic in each factor (Δxu=Δyu=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.