Malliavin differentiability of McKean-Vlasov SDEs with common noise

We establish the Malliavin differentiability of McKean-Vlasov stochastic differential equations (MV-SDEs) with common noise under the global Lipschitz assumption in the space variable and the measure variable. Our result gives also meaning to the Malliavin derivative of the conditional law with respect to the common noise. As an application, we derive an integration by parts formula on the Wiener space for the class of common noise MV-SDEs under consideration.