A cognitive radio network (CRN) is formed by either allowing the secondary users (SUs) in a secondary communication network (SCN) to opportunistically operate in the frequency bands originally allocated to a primary communication network (PCN) or by allowing SCN to coexist with the primary users (PUs) in PCN as long as the interference caused by SCN to each PU is properly regulated. In this paper, we consider the latter case, known as spectrum sharing, and study the optimal power allocation strategies to achieve the ergodic capacity and the outage capacity of the SU fading channel under different types of power constraints and fading channel models. In particular, besides the interference power constraint at PU, the transmit power constraint of SU is also considered. Since the transmit power and the interference power can be limited either by a peak or an average constraint, various combinations of power constraints are studied. It is shown that there is a capacity gain for SU under the average over the peak transmit/interference power constraint. It is also shown that fading for the channel between SU transmitter and PU receiver is usually a beneficial factor for enhancing the SU channel capacities.
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