In this paper, we consider a two-tier cellular network consisting of a primary macro cell base station (PMBS) which is overlaid by cognitive small cell base stations (CSBSs) to achieve efficient spectrum utilization. The deployment of two-tier cellular networks can provide higher capacity for the system but also causes cross-tier, intra-tier, and inter-tier interference within the cellular networks. Thus, we employ transmit and receive beamforming in the considered two-tier cellular network to mitigate interference. We first design the receive beamforming vector for a primary user (PU) such that it cancels all inter-tier interference from other PUs. Then, the transmit beamforming vectors at the secondary users (SUs) are designed to null out the cross-tier interference to the PUs. Further, the receive beamforming vectors at the SUs are designed to mitigate the crosstier interference from the PUs to the SUs. Finally, the transmit beamforming vector at the PMBS is designed to maximize the signal-to-interference-plus-noise ratio at the PUs. To quantify the performance of the system, we derive an expression for the channel capacity in the downlink from the CSBSs to the SUs. Numerical results are provided to reveal the effect of network parameters such as intra-tier interference distances, fading conditions, and number of antennas on the channel capacity of the SUs.