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We consider the utility-maximization problem in the downlink of wireless mesh networks with orthogonal frequency-division multiple access (OFDMA). We formulate this problem as a cross-layer design of joint rate control and OFDMA scheduling to efficiently utilize the scarce radio resources. The problem is decoupled into a rate-control problem at the transport layer and a channel-aware and queue-aware scheduling problem at the Media Access Control/physical layer (MAC/PHY). The rate-control problem adjusts arrival rates to the base station (BS) queues, and the scheduling problem determines link rates, i.e., departure rates from the BS and other network nodes, through subcarrier and modulation rate assignment. Although the rate-control problem is locally solved at the BS, we propose a greedy algorithm that solves the scheduling problem in a distributed manner, at network nodes. Furthermore, we propose a heuristic algorithm for fast execution of the scheduling scheme at individual nodes. Numerical results show that the heuristic algorithm performs comparably with the greedy algorithm, whereas it has lower computational complexity. In addition, our proposed scheduling scheme, when it cooperates with the rate-control mechanism, improves the network performance in terms of end-to-end delay, aggregate utility, and fairness.