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Providing strong QoS guarantees for wireless multi-hop networks is very challenging, due to many factors such as use of a shared communication medium, variability in wireless link quality, and so on. However, wireless mesh technology gives the opportunity to alleviate some of these problems, due to lack of mobility in the wireless infrastructure, and presence of natural centralization points in the network. The main contribution of this paper is the definition of a simple framework that exploits these features to provide provable, strong QoS guarantees to network clients. In particular, admitted clients are guaranteed a certain minimum bandwidth and maximum delay on their connections. The framework is based on STDMA scheduling at the MAC layer, which is periodically executed at the network manager to adapt to changes in traffic demand. While scheduling computation is centralized, admission control is performed locally at the wireless backbone nodes, thus reducing signaling. We propose two bandwidth distribution and related admission control policies, which are at opposite ends of the network utilization/spatial fairness trade-off. Through extensive simulations, we show that the proposed framework achieves its design goals of providing strong QoS guarantees to VoIP clients while not sacrificing throughput in a realistic mesh network scenario, also in presence of highly unbalanced load at the backbone nodes. To the best of our knowledge, this is the first proposal with similar features for wireless mesh networks.