Minimax private density estimation with mixed anisotropy

We determine the minimax density-estimation rate under sequentially interactive local differential privacy and an anisotropic negative Sobolev loss, when the density and discriminator smoothness vectors are proportional. The result covers coordinates whose discriminator smoothness lies on opposite sides of the ambient dimension. The harmonic mean of discriminator smoothness determines the rate, with a single logarithmic factor at the critical value. A weighted randomized response for individual Fourier coefficients gives the upper bound. The noncritical lower bounds follow from Albert, Chevallier, Laurent and Sacko; at the critical value, an orthogonal information bound for the full private history yields the matching logarithmic lower bound.