H-measures and their variants

When a sequence converges weakly to 0 in L²(Ω) but not strongly (Ω being a subset of R^N), one says that it contains oscillations or concentration effects depending upon if the limit of its square has a N-dimensional density or not. One quantitative way to analyze these oscillations or concentration effects is to use H-measures (which I have introduced for a few applications and which Patrick GÉRARD has introduced independently for other ones); they are measures on Ω ⨯ S^N-1 which have a quadratic microlocal character (and make more precise the quadratic theorem of Compensated Compactness Theory).

For dealing with problems with one characteristic length, Patrick GÉRARD then introduced a variant, called semi-classical measures, living on Ω ⨯ R^N (quite similar to the idea which I had proposed independently to use H-measures after adding one dimension). It was then shown by Pierre-Louis LIONS and Thierry PAUL that the semi-classical measures could be introduced using WIGNER transform. I then found with Patrick GÉRARD how to use correlations instead, and why these objects are not good enough when at least two scales are present.
Some other variants of H-measures can be defined, adapted to different questions, and I will describe the advantages and defects of some of these variants.