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The deployment of cognitive radio networks enables efficient spectrum sharing and opportunistic spectrum access. It also presents new challenges to the classical problem of interference management in wireless networks. This paper develops an axiomatic framework for power allocation in cognitive radio networks based on four goals: QoS protection to primary users, opportunism to secondary users, admissibility to secondary users, and autonomous operation by individual users. Two additional goals, licensing and versatility, which are desirable rather than essential, are also presented. A general class of Duo Priority Class Power Control (DPCPC) policies that satisfy such goals is introduced. Through theoretical analysis and simulation, it is shown that a specific interference-aware power-control algorithm reaches such goals.