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In cognitive radio networks (CRN), underlay spectrum sharing allows secondary users (SUs) to utilize the spectrum on which the primary users (PUs) in primary radio networks (PRN) are working at the same time, without introducing intolerant interferences. In this paper, we study the joint transmission power and time allocation in CRN with underlay spectrum sharing technology. To maximize the total capacity of CRN and maintain the fairness for SUs, the system is modeled as an optimization problem with the constraints of interference and transmission power and time. Furthermore, we prove that the problem of joint optimization resource allocation under the constraints condition can be implemented independently from the time-frequency domain. Based on this study, we propose the optimal joint transmission power and time allocation method. Extensive simulation results show that the proposed optimal method can significantly improve the system capacity and maintain the fairness compared to the existing methods.