Regularity Estimates and Intrinsic-Lions Derivative Formula for Singular McKean-Vlasov SDEs

Regularity estimates and Bismut formula of $L^k$ ($kge 1$) intrinsic-Lions derivative are presented for singular McKean-Vlasov SDEs, where the noise coefficient belongs to a local Sobolev space, and the drift contains a locally integrable time-space term as well as a time-space-distribution term Lipschitz continuous in the space and distribution variables. The results are new also for classical SDEs.