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We consider proper holomorphic mappings of equidimensional pseudoconvexdomains in complex Euclidean space, where both source and target can berepresented as Cartesian products of smoothly bounded domains. It is shown thatsuch mappings extend smoothly up to the closures of the domains, provided eachfactor of the source satisfies Condition R. It also shown that the number ofsmoothly bounded factors in the source and target must be the same, and theproper holomorphic map splits as product of proper mappings between the factordomains.

Condition R and proper holomorphic maps between equidimensional product domains