  We consider the problem of nonparametric density estimation under privacy
constraints in an adversarial framework. To this end, we study minimax rates
under local differential privacy over Sobolev spaces. We first obtain a lower
bound which allows us to quantify the impact of privacy compared with the
classical framework. Next, we introduce a new Coordinate block privacy
mechanism that guarantees local differential privacy, which, coupled with a
projection estimator, achieves the minimax optimal rates.
