The capacity region of a product Gaussian broadcast channel with degraded message sets

A product Gaussian broadcast channel with three receivers and degraded message sets is considered. A single-letter characterization of the capacity region was obtained in a recent work of Nair and El Gamal based on superposition coding and indirect decoding. Through Lagrangian theory and an extremal entropy inequality, the single-letter expression of Nair and El Gamal is evaluated for the specific product Gaussian model, which leads to an explicit characterization of the capacity region.