Vector-valued operators, optimal weighted estimates and the $C_p$ condition
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In this paper some new results concerning the $C_p$ classes introduced by Muckenhoupt and later extended by Sawyer, are provided. In particular we extend the result to the full range expected $p>0$, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination results that in some cases we provide as well. We will also provide sharp weighted estimates for vector valued extensions relying on those sparse domination results.