We consider the jointly optimal design of a transmission scheduling and admission control policy for adaptive video streaming over small cell networks. We formulate the problem as a dynamic network utility maximization and observe that it naturally decomposes into two subproblems: admission control and transmission scheduling. The resulting algorithms are simple and suitable for distributed implementation. The admission control decisions involve each user choosing the quality of the video chunk asked for download, based on the network congestion in its neighborhood. This form of admission control is compatible with the current video streaming technology based on the DASH protocol over TCP connections. Through simulations, we evaluate the performance of the proposed algorithm under realistic assumptions for a small-cell network.