  This paper continues the research started in \cite{LW16}. In the framework of
the convolution structure density model on $\bR^d$, we address the problem of
adaptive minimax estimation with $\bL_p$--loss over the scale of anisotropic
Nikol'skii classes. We fully characterize the behavior of the minimax risk for
different relationships between regularity parameters and norm indexes in the
definitions of the functional class and of the risk. In particular, we show
that the boundedness of the function to be estimated leads to an essential
improvement of the asymptotic of the minimax risk. We prove that the selection
rule proposed in Part I leads to the construction of an optimally or nearly
optimally (up to logarithmic factor) adaptive estimator.
