Regularity of laws for distribution dependent SDEs with fractional noises

In this paper, we investigate the regularity of laws for a class of distribution dependent SDEs driven by two independent fractional noises BH and B˜H˜ with Hurst parameters H∈(0,1) and H˜∈(1∕2,1). We establish the log-Harnack inequalities and Bismut formulas for the Lions derivative to this type of equations with distribution dependent noise, in both non-degenerate and degenerate cases. Our proofs consist of utilizing coupling arguments which are indeed backward couplings introduced by F.-Y. Wang [36], together with a careful analysis of fractional derivative operator.