In this paper, we consider a problem of estimation. The model is the same as in the first part of this paper. Our goal is to construct an adaptive estimator over a family of Hölder spaces if an additionnal information is known. Typically, we suppose that the "effective smoothness parameter" is known. A knowledge of this type is to be understood as follows: If you want to estimate with a given rate (precision), you have to fix this parameter. We construct an estimator which is minimax over the union of all Hölder spaces defined using this effective smoothness. This problem is linked to the maxiset theory.