H-measures and their variants

When a sequence converges weakly to 0 in but not

strongly ( being a subset of ), 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 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 (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.