Rx3 has to decode $m_3$, Rx1 can decode $m_3$ via $\tilde{Y}_3$. Therefore, $(R_1,R_2,R_3)$ is contained in the capacity region of a MAC with common information from Tx1 and Tx3 to Rx1 with $X_2$ as a receiver side information, where $R_1$ is the common rate, $R_3$ is the private rate for Tx3, and the private rates for Tx1 is zero. Therefore, the sum-rate $R_1+R_3$ is bounded as (\[eqn:Cap2\_II\]). From the maximum-entropy theorem [@CovTho06] (or [@ElgKim11 P. 21]), this bound is largest for the Gaussian inputs and is evaluated to (\[eqn:Gaus\_Cap1\_II\]). In a similar manner, we can obtain (\[eqn:Gaus\_Cap1\_III\]) at Rx2. The bound in (\[eqn:Gaus\_Cap1\_I\]) follows by applying the standard methods as in (\[eqn:cap\_set1\_fanoI\]).

[22]{}

Goldsmith A., Jafar S. A., Maric I., and Srinivasa S., “Breaking Spectrum Gridlock with Cognitive Radios: An Information Theoretic Perspective,” *Proceedings of the IEEE, invited*, May 2009, **97**, (5), pp. 894–914

Mitola J., *Cognitive radio architecture*, John Wiley Sons, Inc.: New York, 1991

Devroye N., Mitran P., Tarokh V., “Achievable rates in cognitive radio channels,” *IEEE Trans. Inf. Theory*, May 2006, **52**, pp. 1813–1827

Gel’fand S. and Pinsker M., “Coding for channels with random parameters,” *Prob. Contr. Info. Theory*, 1980, **9**, (1), pp. 19–31

Han T. S. 