Vector-valued operators, optimal weighted estimates and the $C_p$ condition

Sharp weighted estimates are obtained for vector-valued extensions of the Hardy-Littlewood maximal operator, Calder'on-Zygmund operators and Coifman-Rochberg-Weiss commutator. Those estimates will rely upon suitable pointwise estimates in terms of sparse operators. We also prove some new results for the $C_p$ classes introduced by Muckenhoupt and later extended by Sawyer, in particular we extend the result to the full expected range $p>0$, to the weak norm, to other operators and to the their vector-valued extensions.