Abstract— In this paper, we study the coexistence and optimization of a multicell cognitive radio network (CRN) which is overlaid with a multicell primary radio network (PRN). We propose a PRN-willingness-based design framework for coexistence and subchannel sharing, and a Lagrange duality based technique to optimize the weighted sum rate (WSR) of secondary users (SUs) over multiple cells. First, to avoid unacceptable SU interference to primary users (PUs), the PRN determines its interference margin based on its target performance metric and channel conditions, and broadcasts this information to the CRN. Second, each CRN cell optimizes its WSR and implements intercell iterative waterfilling (IC-IWF) to control the intercell interference. To account for the interference and transmit power limits at SUs, multilevel waterfilling (M-WF) and direct-power truncation (DPT) duality schemes are developed. Third, we develop a serial dual update technique which enables lowcomplexity and fast-...