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In this paper, we study the optimal power control policies for fading channels in cognitive radio networks considering both the transmit and the interference power constraints. For each of the constraints, the peak power and the average power are investigated. We derive the optimal power allocation strategies in terms of maximizing the ergodic capacity of the secondary user when the channel state information is available to the transmitter and the receiver. It is shown that the optimal power adaption for each case has its own features, and is quite different from one to another. Finally, the numerical results are presented to validate the proposed power allocation methods for various power constraint region.