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Cognitive radio networks (CRNs) have the ability to utilise the radio spectrum efficiently by allowing secondary users (SUs) to communicate in the licensed frequency bands. In this study, downlink spatial multiplexing techniques are proposed to enable multiple SUs to share spectrum simultaneously without harmfully interfering the primary users (PUs). The multiuser transmitter beamformers are designed by setting constraints on the interference temperatures of the PUs and signal-to-interference and noise ratios (SINRs) of the SUs. The proposed beamformers minimise the total transmit power while achieving the required quality of services (QoSs) for each SU. Since spatial multiplexing techniques require channel state information (CSI) at the basestation, which could normally be in error, a robust spatial multiplexing technique using worst-case performance optimisation is proposed for underlay CRNs. The proposed robust beamformer has the ability to maintain SINRs of all SUs above a set of target values and interference leakage to PUs below a threshold for all possible CSI errors within a convex hull. Both the robust and the non-robust designs are formulated into a convex optimisation framework using semidefinite constraints.