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The emerging proxy-based wireless content delivery networks (CDNs) should to be designed to download huge-size files over fading-affected channels. However, from a radio resource management point of view, several basic problems still need to be solved for such wireless delivery systems to operate efficiently. Specifically, due to the fading nature of the downlink channel, a still open basic problem is how to design optimal energy-allocation (for example, scheduling) policies that minimize the requested download time when constraints on the total available energy and peak energy are simultaneously active. In this contribution, this problem is solved for application scenarios where the downlink channel is slotted and continuous-state, the carried out traffic is elastic, and the resulting conveyed throughput is measured by any desired increasing concave rate function. Specifically, the optimal energy-allocation policy minimizing the download time is computed in closed form, and its performance is compared against that of a basic on-off heuristic energy scheduler on some Rayleigh-faded multi-antenna delivery systems of practical interest. The carried out performance comparisons point out that the presented optimal policy typically may outperform the heuristic one up to two orders of magnitude, especially when the delivery system is strongly energy limited.