Vector-valued operators, optimal weighted estimates and the Cp condition

In this paper some new results concerning the Cp classes introduced by Muckenhoupt (1981) and later extended by Sawyer (1983), are provided. In particular, we extend the result to the full expected range p < 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination, which in the vector-valued case are provided as well. We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.