Microlocal defect functionals (H-measures, H-distributions, semiclassical measures etc.) are objects which determine, in some sense, the lack of strong compactness for weakly convergent L^p sequences. Recently, Luc Tartar introduced one-scale H-measures, a generalisation of H-measures with a characteristic length, which also comprehend the notion of semiclassical measures.
We present a self-contained introduction to one-scale H-measures, carrying out some alternative proofs, and strengthening some results, comparing these objects to known microlocal defect functionals. Furthermore, we develop the localisation principle for these objects in a rather general form, from which we are able to derive the known localisation principles for both H-measures and semiclassical measures. Moreover, it enables us to obtain a variant of compactness by compensation suitable for equations with a characteristic length.
