Cognitive radio technology has been recently proposed for sharing and utilising the spectrum in order to satisfy the increasing demands for spectrum access. In this radio technology, secondary users may be granted access to the spectrum bands occupied by a primary user as long as the interference power, inflicted on the primary receiver as an effect of the transmission of the secondary user, is deemed unharmful. In this paper the authors assume that the successful operation of the primary user requires a minimum rate to be guaranteed by its channel for a certain percentage of time and obtain the interference-power constraint that is required to be fulfilled by the secondary user. Considering the input transmit-power constraint, on average or peak power, for the secondary user, the authors investigate the capacity gains offered by this spectrum-sharing approach when only partial channel information of the link between the secondary's transmitter and primary's receiver is available to the former. In particular, the lower bounds on the capacity of a Rayleigh flat-fading channel with two different transmission techniques, namely channel inversion and optimum rate allocation with constant power transmission, are derived. Closed-form expressions for these capacity metrics are provided, and numerical simulations are conducted to corroborate the theoretical results.