Smoothness testing in the Sobolev sense

We want to develop a test to determine whether a function lying in a fixed L2-Sobolev-type ball of smoothness t, and generating a noisy signal, is in fact of a given smoothness s larger than t or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness t, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness t. The functions that we remove are functions of smoothness strictly smaller than s, but that are very close to s-smooth functions. This problem has been considered in the case of specific Besov bodies where it is easier, and we plan to extend it to more usual Sobolev ellipsoids.